diff --git "a/EtFRT4oBgHgl3EQfyjjm/content/tmp_files/load_file.txt" "b/EtFRT4oBgHgl3EQfyjjm/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/EtFRT4oBgHgl3EQfyjjm/content/tmp_files/load_file.txt" @@ -0,0 +1,1091 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf,len=1090 +page_content='Nonlinear Optimization Filters for Stochastic Time-Varying Convex Optimization Andrea Simonetto a Paolo Massioni b February 1, 2023 Abstract We look at a stochastic time-varying optimization problem and we formulate online algorithms to find and track its optimizers in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The algorithms are derived from the intuition that standard prediction and correction steps can be seen as a nonlinear dynamical system and a measurement equation, respectively, yielding the notion of nonlinear filter design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The optimization algorithms are then based on an extended Kalman filter in the unconstrained case, and on a bilinear matrix inequality condition in the constrained case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Some special cases and variations are discussed, notably the case of parametric filters, yielding certificates based on LPV analysis and, if one wishes, matrix sum-of-squares relaxations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Supporting numerical results are presented from real data sets in ride-hailing scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The results are encouraging, especially when predictions are accurate, a case which is often encountered in practice when historical data is abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 1 Introduction We look at time-varying optimization problems of the form min xPRn fpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq ` gpxq, t ě 0, (1) where f : RnˆRd Ñ R is a smooth strongly convex function in x (for all y), parametrized over a time-varying data stream yptq P Rd (t represents the continuous time), and g is a closed convex and proper function (such as the indicator function, or an ℓ1 regularization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Problem (1) appears naturally in a number of application scenarios, where optimal decisions have to be taken online and they change as new data arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Examples stem from video processing [1] to robot control [2], and to large-scale optimal management of smart infrastructures [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Solving Problem (1) means to find and track the optimizer trajectory x‹ptq, as yptq changes and it is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In order to accomplish this, in the standard time-varying literature, one can sample Problem (1) at discrete time instant tk, k “ 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', with h :“ tk ´ tk´1 being the sampling time, and solve the sequence of time-invariant problems x‹ptkq “ arg min xPRn fpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptkqq ` gpxq, k P N, (2) as they are revealed in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Then one can set up online algorithms that find approximate x‹ptkq’s, say xk’s, that eventually converge to1 the solution trajectory, within an error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The reader is referred to the surveys [4, 5] for an ample treatment of the methods in both discrete and continuous time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' One of the important aspects to keep in mind is that online algorithms are sought that are computationally frugal, so that one can approximate the solution of x‹ptkq within the sampling time h, and the key performance metric is how good the algorithms are with respect to an algorithm that has infinite computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' A tacit assumption in the above methods is that one wants to converge to the solution trajectory generated by the evolution of the data stream yptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' However, this may be not the best course of action, since the data aUMA, ENSTA Paris, Institut Polytechnique de Paris, 91120 Palaiseau, France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='simonetto@ensta-paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' bUniv Lyon, INSA Lyon, Universit´e Claude Bernard Lyon 1, Ecole Centrale de Lyon, CNRS, Amp`ere, UMR5505, 69621 Villeurbanne, France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' paolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='massioni@insa-lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='fr 1Somebody could rather say: “track”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='13646v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='OC] 31 Jan 2023 are often noisy and convergence to a noisy optimizer may be not advisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' A better angle is to ask whether we can set up algorithms to converge to a filtered version of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In this context, we re-interpret Problem (1) as a stochastic problem, where we would like to find and track the filtered solution trajectory as ˆx‹ptq “ arg min xPRn EyPYptqrfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs ` gpxq, (3) where the expectation EyPYptqr¨s is with respect to the random variable y, which is drawn from a time-varying probability distribution Yptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Were the probability distribution time-invariant, Formulation (3) would be common in stochastic opti- mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' With our setting, we are considering instead a gradual distribution shift of an unknown distribution, which renders the formulation less common and more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Recent papers have started to look into this formulation [6–8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' especially the third one is the closest to our goal, however the authors “just” adapt the standard prediction-correction algorithms to the stochastic setting by properly tuning the prediction and the correction step sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Also they do not consider a non-smooth component as our function gpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In this paper, we look from a different angle and ask whether we can use a perturbed version of the optimality condition as a suitable dynamical model to do filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' This will have the advantage to combine prediction and correction in novel and more performant ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' To fix the ideas on this novel angle, consider the unconstrained problem: x‹ptq “ arg min xPRn fpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq, (4) with a strongly convex, doubly differentiable, and smooth function f in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' By perturbing the optimality condition ∇xfpx‹ptq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq “ 0n, we can derive the ordinary differential equation that describes how the optimizer evolves as [9]: d dtx‹ptq “ ´r∇xxfpx‹ptq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqqs´1∇yxfpx‹ptq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq d dtyptq “: Fpx‹ptq, yptq, 9yptqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (5) This is a nonlinear dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' To derive a filter, we need to couple this model with a measurement equation, which tells us how far from the optimizer trajectory we are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We can thus use, zptq “ ∇xfpx‹ptq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq, (6) as a measurement equation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', it will be different from zero at any point but on the optimizer trajectory, where zptq “ 0n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Armed with (5) and (6), we could design a dynamical filter to reconstruct x‹ptq based on a noisy data stream yptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 Contributions Starting from the continuous-time intuition from (5) and (6), we develop discrete-time filters for unconstrained and constrained time-varying problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In particular, ‚ We derive an extended Kalman filter for unconstrained and differentiable convex problems in the discrete- time setting starting from an algorithmic viewpoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ‚ We generalize the filtering procedure for constrained and non-differentiable problems leveraging the non- linear dynamical systems and nonlinear measurement equations coming from forward-backward algorithms and fixed-point residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We are then able to derive the optimal “Kalman-style” gain via both a worst-case approach and via dissipativity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We also present a possibly less conservative linear parameter-varying (LPV) methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ‚ We showcase the benefit of the approach with respect to state-of-the-art prediction-correction methods (which our filters generalize), in numerical simulations stemming from a ride-hailing example with multiple companies in New York City.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 Related work Time-varying and stochastic optimization are vibrant research fields, and we do not plan to give an exhaustive account here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The reader is referred to [4,5,10] for the first and [11–14] for a sub-sample of the second, and the many references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Jointly Stochastic and time-varying optimization is a less studied area, and as we have mentioned [6–8] scratch the surface in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The celebrated [15] paper is also somewhat in the general area of interest, though their approach does not include prediction, it uses restart and is more directed at finding optimal regret rates for non-stationary objectives rather than noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We will build on techniques from dissipativity theory for analyzing and designing optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' This is a recent and fertile area brought to fame by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Lessard and collaborators’ seminal paper [16], and now gaining momentum [17–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Our novel insight in this direction is to use the techniques to determine Kalman-style gains for optimization algorithms with errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Finally, we use LPV techniques from [23–25], especially in the context of matrix sum-of-squares relaxations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The remaining of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Section 2 discusses formulation and main assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We focus on the unconstrained case in Section 3, and on the general case in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Section 5 describes our gain design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We then conclude with some numerical simulations in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' All proofs are given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Notation is wherever possible standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For a differentiable function f, we define a step of the gradient method starting from a point xk as xk`1 “ rI ´ α∇xfp‚qsxk ” xk ´ α∇xfpxkq, where α ą 0 is the step size and I is the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Then, s steps are indicated as, xk`1 “ xs k “ rI ´ α∇xfp‚qs˝sxk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (7) We let also x0 k “ xk when needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We further indicate with proxαgpxq the proximal operator, proxαgpxq “ arg min vPRn !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' gpvq ` 1 2α}v ´ x}2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (8) Finally, spaces are indicated as R, N, probability distributions are calligraphic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', Y, matrices and vectors are boldfaced, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', x P Rn, A P Rnˆm, operators are in sans-serif, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', I, J, constants are in standard roman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2 Problem formulation and assumptions Let us now consider the sequence of problems, ˆx‹ptkq “ arg min xPRn EyPYptkqrfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs ` gpxq, k P N, (9) with a strongly convex, doubly differentiable and smooth cost function f uniformly in x and a generic convex function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Let us also introduce the shorthand notation ˆx‹ k “ ˆx‹ptkq, yk “ yptkq, and Yk “ Yptkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Notice that, as done in stochastic optimization, the data point yk “ yptkq is supposed to be a random vector drawn from the distribution Yk “ Yptkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' First, let us derive a discrete-time dynamical system on how the optimizers evolve in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Quite naturally, one could be attempted at discretizing (5), but the presence of the inverse of the Hessian and the derivative of the data stream makes it quite cumbersome, especially if one has then to linearize it for an extended Kalman filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Instead we use a Bayesian approach, assuming that we have a (noisy) prior on how the data stream evolves, and we start from what we can be computed algorithmically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Let us denote with Jk`1pxq : Rn Ñ Rn an approximation at time tk of EyPYk`1r∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Noisy prior on how the data evolves and how the gradient evolves can come from linear filters, or more sophisticated neural network models, or kernel models (see also [26] for what they call predictable sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For now think about one of the simplest model: Jk`1pxq “ 2∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ykq ´ ∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yk´1q (obtained via an extrapolation technique [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Then, we use an algorithmic view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' At time k, we would like to solve ˆx‹ k`1 “ arg min xPRn EyPYk`1rfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs ` gpxq, (10) 3 yet it is not possible with data up to time tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' So, we introduce the noisy dynamical system ˆx‹ k`1 “ rproxαgpI ´ αJk`1p‚qqs˝P ˆx‹ k ` qk “: Φk,gpˆx‹ kq ` qk, (11) where rproxαgpI ´ αJk`1p‚qqs˝P means the application of the proximal gradient method of step size α ą 0 for P times, and qk is the process error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The error comes both from a modelling error (truncating after P iterations), and from the noisy predicted gradient Jk`1pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We will see how to characterize the error later, but for now it is useful to keep in mind that qk is not-zero mean in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' If the gradient were exact and P Ñ 8, then Equation (11) would solve (10) with no noise (qk “ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The “pseudo”-dynamical system (11) will be our computationally affordable nonlinear dynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In par with (11), we introduce a measurement equation, zk`1 “ ´ˆx‹ k ` rproxβgpI ´ β∇xfp‚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yk`1qqs˝C ˆx‹ k ` rk “: Ψgpˆx‹ k, yk`1q ` rk, (12) where C is the number proximal gradient steps, β ą 0 is the correction step size, and rk is a noise term coming from the noisy character of yk`1 and it is in general not zero-mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Again, zk`1 “ 0n on the optimizer trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The right-hand side of (12) represents the fixed-point residual of our C-steps proximal gradient method, that we use to compute the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 Properties, requirements of the gradient approximators Before going further, it is useful to understand a bit better the properties of the gradient approximations Jk`1pxq and ∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yk`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Both are attempting at approximating EyPYk`1r∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs, but there are a few differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The first is based on data available up to time tk and it is in general a biased estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' This is not a problem per se, since even in deterministic prediction-correction methods, the predicted gradient is in general a biased prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Example 1 (Recurring stochastic example) Consider the case in which ∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yq is linear in y (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', for linear models and Least-Squares estimators, for example when fpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yq :“ 1 2}x ´ Ay}2 for a given matrix A), and more generally the case in which: fpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yq “ f 1pxq ` yTAx, with f 1pxq strongly convex and smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In this case, let }∇yxfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yq} “ }A} ď C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Further, suppose yptq is generated by a nominal (doubly differentiable in t) trajectory to which we add a Gaussian zero-mean noise at each sampling time tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' And in particular, set yptkq “ ¯yptkq ` ek, with ek „ Np0, Σkq for a given time-varying covariance matrix Σk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We also know that Er}ek}s ď a trpΣkq, and we set a trpΣkq ď Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Choose the extrapolation prediction: Jk`1pxq “ 2∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ykq ´ ∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yk´1q “ ∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2yk ´ yk´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Consider a nominal trajectory ¯yptq for which we assume the bounds: maxt}∇t ¯yptq} , }∇tt ¯yptq}u ď C, @x, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (13) Then, in the Appendix we show that: Er}Jk`1pˆx‹ k`1q ´ EyPYk`1r∇xfpˆx‹ k`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs}s “ EwPYk,zPYk´1r}∇xfpˆx‹ k`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2w ´ zq ` ´EyPYk`1r∇xfpˆx‹ k`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs}s ď C0Ch2 ` 3C0Σ, EyPYk`1r}∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yq ´ EyPYk`1r∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs}s ď C0Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ The second estimator, meaning using ∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yk`1q instead of EyPYk`1r∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs, is evaluated with data coming at time tk`1 and it is unbiased [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For the two approximations we require the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 4 Assumption 1 Let the cost function fpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yq be µ-strongly convex and L-smooth in x uniformly in y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', for all y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The chosen gradient predictor Jk`1pxq is then µ-strongly monotone and L-Lipschitz in x for all k’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Assumption 2 The noise processes and gradient prediction errors are bounded as follows: paq Er}Jk`1pˆx‹ k`1q ´ EyPYk`1r∇xfpˆx‹ k`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs}s ď τ, pbq EyPYk`1r}∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yq ´ EyPYk`1r∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs}s ď σ, for finite scalars τ and σ, for all k P N, and (b) for all x P Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Assumption 1 is often required for time-varying optimization [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The assumption on the predictor is also reasonable, for instance is verified in Example 1, and with Taylor-based and extrapolation-based predictions [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For Assumption 2: Property paq is in par with some deterministic and stochastic assumptions appeared in past years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For example, it can be seen in parallel with the quality of the hint or predictable sequences in [26,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In Example 1, Property paq is verified with τ “ C0Ch2 ` 3C0Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Stochastic versions of Property paq have appeared, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', in [8], with example-based constructions for determining a suitable Jk`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Property pbq is also commonly asked in stochastic frameworks [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Usually, one asks that EyPYk`1r}∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yq ´ EyPYk`1r∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs}2s ď σ2, but due to the convexity of p¨q2 and Jensen’s inequality, Property pbq is implied by the squared one (in fact pEr} ¨ }sq2 ď Er} ¨ }2s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For us Property pbq can be time-varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In Example 1, Property pbq is verified with σ “ C0Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Finally, Property pbq can be tightened to be valid only on the algorithm iterates pxkqkPN, and interpret it as gradient noise with a small theoretical effort [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 3 An extended Kalman filter We are now ready to derive a filter to track the filtered optimizer trajectory ˆx‹ptkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Since the linear system and the measurement equations are nonlinear, we will use an extended Kalman filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For it, we require that both ∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yq and Jk`1pxq are differentiable with respect to x, and we will require knowledge of the covariance of the noise processes qk, rk at all time instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We will also assume that g ” 0, to be able to differentiate, which puts us in an unconstrained differentiable problem setting, where we use gradient (the proximal operator is the identity in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Recall the notation xs k “ rI ´ α∇xfp‚, yk`1qs˝sxk defined in (7), indicating the effect of s steps of the gradient method, or an approximate gradient if fp‚, yk`1q is substituted with Jk`1p‚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Define the derivative quantities (we drop the mention to g, since it does not exist in this case), Fk “ ∇xΦkpxkq “ P ź p“1 ´ In ´ α∇xJk`1pxP ´p k q ¯ (14) Hk`1 “ ∇xΨpxk`1|k, yk`1q “ In ´ C ź c“1 ´ In ´ β∇xxfpxC´c k`1|k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yk`1q ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (15) We also let Rk P Rnˆn and Qk P Rnˆn be the covariance matrices of the noise processes rk and qk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' With this in place, the extended Kalman filter (TV-EKF) represented in Algorithm 1 is able to filter and track the optimizer trajectory ˆx‹ptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We notice that we have presented the filter with correction first, to highlight the standard workflow within a sampling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We notice also that the filter can be extended to include a filtering process for the data stream yptq, if a dynamical model for it is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Algorithmically, the presented TV-EKF requires several computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In the correction step, it comprises the computation of the Hessian of f at various points for determining Hk and taking a matrix inverse for determining Kk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The update for xk requires also C gradient steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In the prediction pass, the filter includes a process to determine any prediction Jk`1, computing its derivatives, and P gradient steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Considering a n-dimensional state, and letting g, h, j, dj be the computational effort to determine the gradient, Hessian, predicted gradient, and its derivatives, then the overall computational complexity of TV-EKF is OpCph ` gq ` Ppdj ` gq ` n3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We will study the empirical performance of TV-EKF in Section 6, but we close here with some remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 5 Algorithm 1 An extended Kalman filter (TV-EKF) Input: Initialize: x1|0 “ 0, P1|0 “ In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Number of prediction and correction steps P, C, sampling time h, step sizes α, β, covariance matrices Rk, Qk for all k, as well as prediction strategy J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Output: A sequence pxkqkPN 1: for k P N, k ě 1 do 2: Receive yk 3: Compute Hk as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 4: Correction step: Kk “ Pk|k´1HkpHkPk|k´1HT k ` Rkq´1 xk “ xk|k´1 ` KkpΨpxk|k´1, ykqq Pk “ pI ´ KkHkqPk|k´1 5: Compute Fk as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 6: Prediction step: xk`1|k “ Φkpxkq, Pk`1|k “ FkPkFT k ` Qk 7: end for Remark 1 (Prediction-Correction methods) We can see how xk is updated as xk “ Φkpxk´1q ` KkpΨpΦkpxk´1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ykqq “ pIn ´ Kkq rI ´ αJkp‚qs˝P xk´1 looooooooooomooooooooooon prediction ` ` Kk rI ´ β∇xfp‚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ykqs˝C ˝ rI ´ αJkp‚qs˝P xk´1 looooooooooooooooooooooooooomooooooooooooooooooooooooooon correcting the prediction , (16) and if we let Kk ” In, then we obtain back the standard prediction-correction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Remark 2 The TV-EKF algorithms that is presented here could be extended to equality-constrained opti- mization problems, once formulated as saddle-points, but we leave this for future endeavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The TV-EKF algorithm that we have presented offers several advantages, and above all the ease of implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' However, from an optimization perspective, it is lacking in good convergence guarantees2, and from a noise perspective, we do not have a good intuition or recipe on how to set the covariances Rk, Qk in meaningful ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' This, combined with the fact that TV-EKF will not work for non-smooth costs, pushes us to look beyond to a more general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' However, before moving on, we give some intuition on why the TV-EKF does perform well empirically in the numerical settings that are presented in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Proposition 1 (Equivalence to a damped Newton’s step) When the noise on the measurement is neg- ligible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', Rk « 0, and we take only one step of correction C “ 1, then TV-EKF is a damped Newton’s method, with update xk “ xk|k´1 ´ βr∇xxfpxk|k´1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ykqs´1∇xfpxk|k´1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ykq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ Proposition 1 implies that TV-EKF includes second-order information and could be thought of as a stochastic quasi-Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In this sense, if the noise covariances are well estimated, or small, then TV-EKF is expected to do better than standard prediction-correction methods that only use first-order information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' This is not surprising, but the connection Kalman-Newton in optimization is interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2Besides the fact that the prediction and correction steps represent contractive operators for α ă 2µ{L2, β ă 2{L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 6 4 The general case The standard extended Kalman filter can be easily derived when the cost is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We generalize now our filter design to the case in which one has also the term gpxq in the cost, modeling constraints and non-smooth regularizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Reconsider the dynamical system (11) in par with the measurement equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Under Assumption 1, from standard operator theory, we know that if α and β are chosen small enough, and in particular3 α ă 2µ{L2, β ă 2{L, both the prediction and the correction represent contraction operators, which converge to their respectives unique fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In particular, their contraction factors are [29]: ρp “ a 1 ´ 2αµ ` α2L2, ρc “ maxt|1 ´ βµ|, |1 ´ βL|u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (17) for prediction and correction operators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 A static gain filter With our dynamical model (11) and measurement equation (12), we are now ready to build our filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Here, since the model and the measurements are non-differentiable equations, we will focus on static gain filters, that can be computed off-line, before running the time-varying algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We let Ψ1 gpx, yq “ Ψgpx, yq ` x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Therefore, we start by considering the update equation xk`1 “ Φk,gpxkq´KpΦk,gpxkq´Ψ1 gpΦk,gpxkq, yk`1qq “ pIn´KqΦk,gpxkq`KΨ1 gpΦk,gpxkq, yk`1q, k P N (18) consisting of running a prediction Φk,gpxkq and then correcting it via the correction Ψ1 gpΦk,gpxkq, yk`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' As mentioned in Remark 1, by setting K ” In, we obtain back the standard prediction-correction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Algorithm 2 makes the update explicit along with all the involved computations, for a generic choice of J and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' As we can see the computational complexity here does not involve matrix inversions, but it adds proximal mapping computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' If the proximal step is easy to perform compared to the other computations, then the complexity is OpCph ` gq ` Ppdj ` gqq, which is better than TV-EKF, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Algorithm 2 A static contractive filter (TV-CONTRACT) Input: Initialize: x1|0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Number of prediction and correction steps P, C, sampling time h, step sizes α, β, prediction strategy J, as well as a filter gain K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Output: A sequence pxkqkPN 1: for k P N, k ě 1 do 2: Receive yk 3: Correction step: xk “ pIn ´ Kqxk|k´1 ` KΨ1 gpxk|k´1, ykq 4: Compute Jk`1pxkq 5: Prediction step: Compute xk`1|k “ Φk,gpxkq 6: end for As mentioned before, in this paper, we are interested in designing K in such a way to reduce the tracking error of the sequence txkukPN, and in particular: which K would deliver the smallest lim supkÑ8 Er}xk´ˆx‹ k}s?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 Scalar, worst-case convergence results We start by analyzing the easier case of determining the best scalar gain χ P r0, 1s, for the update, xk`1 “ p1 ´ χqΦk,gpxkq ` χΨ1 gpΦk,gpxkq, yk`1q, k P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (19) While this is restrictive in practice, it will give us some intuition on the general problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Also, by restricting χ in r0, 1s, we are considering all convex combinations of prediction and correction phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The latter is important if g represents a feasible set and we want the sequence txkukPN to be feasible for every k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We have the following Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 3This is due to the fact that Jk is µ-strongly monotone and L-Lipschitz, but not a gradient per se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Sharper conditions can be derived if Jk would be a gradient, like in the correction case, for which we can choose α ă 2{L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 7 Theorem 1 Let Assumptions 1-2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Assume furthermore that the optimizer trajectory is bounded as, }ˆx‹ k`1 ´ ˆx‹ k} ď ∆ ă 8, @k P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (20) Choose α ă 2µ{L2, β ă 2{L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Consider Algorithm 2 with the selection of K “ χIn, χ P r0, 1s, leading to the update (19), and its sequence txkukPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Define functions ζ and ξ as: ζℓ,ρ “ t1 if ℓ “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ρℓ otherwise u, ξℓ,ρ “ t0 if ℓ “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 1 ` ρℓ otherwise u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Recall the contraction parameters (17) and choose the number of prediction and correction steps P and C such that ζC,ρcζP,ρp ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Then, by calling ϱχ “ p1 ´ χqζP,ρp ` χζC,ρcζP,ρp, the asymptotic error is upper bounded as, lim sup kÑ8 Er}xk ´ ˆx‹ k}s “ 1 1 ´ ϱχ ” p1 ´ χq “ ζP,ρp∆ ` ξP,ρpτµ ‰ ` χ “ ζC,ρcrζP,ρp∆ ` ξP,ρpτµs ` σc ‰ ı , (21) with τµ “ τ{µ, and σc “ βσ{p1 ´ ρcq Finally, under the setting of Example 1, ∆ “ C0h{µ, τµ “ pC0Ch2 ` 3C0Σq{µ and σ “ C0Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ Theorem 1 captures the asymptotic tracking error of the proposed TV-CONTRACT algorithm, when K “ χIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The requirement (20) is standard, assuring that the trajectory is regular enough to be tracked, see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The condition ζC,ρcζP,ρp ă 1 is verified, whenever P ` C ě 1, since ρp, ρc P r0, 1q with the choice of α, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For χ “ 1, Theorem 1 extends [27, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1] to stochastic settings and [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7] to f ` g settings with multiple prediction and correction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The question we have for filter design is how to tune χ to lower the asymptotical error, given all the rest fixed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 Tuning χ The filter design problem can be now formulated as, min χPr0,1s (21) (22) Problem (22) is a linear-fractional programming, that can be solved by transforming it into a linear program, once all the coefficients are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We do not give the details of this, since easily found in standard books [30, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Nonetheless we report an interesting fact on the nature of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Proposition 2 The solution of the tuning problem (22) is either χ‹ “ 1, χ‹ “ 0, or in a special case, any χ P r0, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ What Proposition 2 says is that from a worst-case perspective (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', from an asymptotic tracking error) we are better off to either just do predictions, or just do prediction-correction (and in a very special case, we can take any choice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The choice is made a priori, from the size of the prediction or correction errors (see the proof for exact conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' This is hardly satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We see next how to extend the above to a generic matrix gain K, via dissipativity theory, which has a richer behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 5 Dissipativity theory and filter design We move now to the general case of designing a matrix K in an optimal fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We will be using recent tools from dissipativity theory applied to optimization algorithm design, and we were particularly inspired by [18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 8 � ��� 0 B I 0 � ��� ΦΦg ΨΨ′ g xx ww uu � ��� 0 B I 0 � ��� T 1 T 2 Qqq Rrr abstraction xx ww uu Figure 1: The algorithmic choice (26) is an abstraction of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The matrix block indicates the algorithmic update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 Filter design To start our filter design, we need to recast Algorithm 2 as a block diagram, where the optimization algorith- mic updates are interpreted as nonlinear blocks and modeled as quadratic constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Consider then noisy algorithmic update, $ & % wk`1 “ ¯wk`1 ` Qqk`1, ¯wk`1 “ T 1 k`1pxkq uk`1 “ ¯uk`1 ` Rrk`1, ¯uk`1 “ T 2 k`1p ¯wk`1q xk`1 “ pI ´ Kqwk`1 ` Kuk`1 (23) where qk`1 and rk`1 are noise terms, whose expected norm is bounded, as we will see shortly, and Q, R are tuning matrices that can model the relative amount of error or correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The algorithmic choice (26) is an abstraction of Algorithm 2, as we can see in Figure 1, where T 1 k`1 and T 2 k`1 are the ideal operators representing the ideal prediction and correction steps, respectively, and both with fixed point ˆx‹ k`1 “ ¯w‹ k`1 “ ¯u‹ k`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The fact that, technically, one should consider ¯uk`1 “ T 2 k`1pwk`1q, and the noise terms qk`1 and rk`1 are in fact correlated, since rk`1 should depend on a noisy prediction, can be ignored here, since we will only look at worst-case performance guarantees, from bounded errors to bounded output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Under Assumptions 1 and 2, with the same notation of Theorem 1 and from the proof of [27, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1] with Er}τk}s ď τµ, we can bound, }wk`1 ´ ¯w‹ k`1} “ }xk`1|k ´ ˆx‹ k`1} ď ζP,ρp}xk ´ ˆx‹ k`1} ` ξP,ρpτk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (24) On the other hand, }wk`1 ´ ¯w‹ k`1} “ } ¯wk`1 ´ ¯w‹ k`1 ` Qqk`1} ď ζP,ρp}xk ´ ˆx‹ k`1} ` }Qqk`1}, (25) so we can look at bounded errors as }Qqk`1} ď ξP,ρpτk and Er}Qqk`1}s ď ξP,ρpτµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Similarly for the error term Rrk, we can look only for bounded errors as Er}Rrk}s ď ζC,ρcξP,ρpτµ ` σc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In particular, we consider Er}qk}s ď 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2, Er}rk}s ď 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2 and model the matrices Q, R to have their largest singular value at ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2pξP,ρpτµq and ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2pζC,ρcξP,ρpτµ ` σcq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For the sake of ease of notation, in (26), we define matrices B, Be, and the nominal input and error signal ¯z, e as, B “ rpIn ´ Kq, Ks, Be “ rpIn ´ KqQ, KRs, (26) ¯z “ r ¯wT, ¯uTsT, e “ rqT, rTsT, xk`1 “ B¯zk`1 ` Beek`1, (27) and Er}ek}s ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We also introduce point-wise-in-time quadratic constraints for T 1 and T 2 as, „ xk ´ ˆx‹ k`1 ¯wk`1 ´ ¯w‹ k`1 ȷT„ ω2 1In 0 0 ´In ȷ„ xk ´ ˆx‹ k`1 ¯wk`1 ´ ¯w‹ k`1 ȷ ě 0, (28) „ xk ´ ˆx‹ k`1 ¯uk`1 ´ ¯u‹ k`1 ȷT„ ω2 1ω2 2In 0 0 ´In ȷ„ xk ´ ˆx‹ k`1 ¯uk`1 ´ ¯u‹ k`1 ȷ ě 0, (29) with ω1 “ ζP,ρp and ω2 “ ζC,ρcζP,ρp, which are due to the contracting properties of T 1 and T 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' With this in place, convergence and asymptotic tracking error performance can be formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 9 Theorem 2 Consider Algorithm 2 and its abstraction (27), to find and track the filtered optimizer trajectory ˆx‹ptq of the time-varying stochastic optimization problem minxPRn Erfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqqs ` gpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Assume that the optimizer trajectory varies in a bounded way as }ˆx‹ptk`1q ´ ˆx‹ptkq} ď ∆, for all k P N and ∆ ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Let Assumptions 1-2 hold as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Introduce matrix W P Rnˆn and X P Rnˆn, X ą 0, scalars λ1 ě 0, λ2 ě 0, in addition to supporting scalars γ1, γ2, and consider the tuning matrix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For any fixed scalar ρ P p0, 1q, by solving the convex problem4 minimize X ą 0, λ1 ě 0, λ2 ě 0, W, γ2 1, γ2 2 γ2 1ρ2∆2 ` γ2 2 (30a) subject to ρ2X ľ pλ1ω2 1 ` λ2ω2 1ω2 2qIn, (30b) X ľ In, X ĺ γ2 1In (30c) » ————– ´λ1In 0 ´λ2In 02nˆ2n X ´ WT WT ´γ2 2I2n QpX ´ WTq RWT ‹ ´X fi ffiffiffiffifl ĺ 0, (30d) then Algorithm 2 with K “ WX´1 generates a sequence txkukPN that converges as, AE :“ lim sup kÑ8 Er}xk ´ ˆx‹ k}s ď 1 1 ´ ρ pγ1ρ∆ ` γ2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (31) Furthermore, for any fixed ρ, solving Problem (30) diminishes the asymptotical error AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ Theorem 2 describes how to best tune the matrix K to minimize the asymptotic tracking error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In particular, doing a grid search on ρ P p0, 1q, one can identify the best K that minimizes the worst-case asymptotical error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Two remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' First the convex problem (30) grows linearly in the dimension of the problem n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' However, if the matrices R and Q have some particular structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', diagonal, block diagonal), we can reduce the problem size considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In the case of Q “ qIn, R “ rIn, then choosing X “ pIn, W “ wIn, the problem becomes independent of the problem size n, as it happens in other examples [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Second, the matrices R and Q are gateways through which one can include variable correlations and dependency, which is not present in a standard prediction-correction algorithm, nor in the scalar worst-case convergence tuning χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 LPV filter design The presented filter design, captured by the conditions in Theorem 2, could suffer from conservatism, since the matrices Q and R are selected as worst cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For example, we have modeled Q at having its largest singular value at ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2pξP,ρpτµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In practice, one could wish to model these matrices as parameter-varying, and directly dependent on how the data changes in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We propose here an LPV filter design that accomplishes this task and reduce conservatism of the design process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We focus on a simplified setting to convey the basic ideas, and the reader is left to generalize the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Consider the change in time of the data ∇typtq, and let θ P r0, 1s be a normalized parameter that capture this change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For example, we let θ “ p}∇typtq}8q{pmaxt }∇typtq}8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We then consider Q as an affine parameter-varying matrix: Qpθq “ Q0 ` θQ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We consider here a static R for simplicity, thereby assuming that the change in the data only affects the prediction accuracy, which is reasonable to assume (and easy to lift if needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In parallel, we will be looking for an affine parameter-varying Lyapunov matrix Xpθq “ X0 ` θX1, and a parameter-varying filter gain matrix Kpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The following theorem is then in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 4For readability, we indicate in blue the decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 10 Theorem 3 Consider Algorithm 2 and its abstraction (27), to find and track the filtered optimizer trajectory ˆx‹ptq of the time-varying stochastic optimization problem minxPRn Erfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqqs ` gpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Assume that the optimizer trajectory varies in a bounded way as }ˆx‹ptk`1q ´ ˆx‹ptkq} ď ∆, for all k P N and ∆ ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Let Assumptions 1-2 hold as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Introduce matrix function Wpθq : r0, 1s Ñ Rnˆn “ W0 ` θW1 and Xpθq : r0, 1s Ñ Rnˆn “ X0 ` θX1, Xpθq ą 0, scalars λ1 ě 0, λ2 ě 0, in addition to supporting scalars γ1, γ2, and consider the tuning matrix function Kpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Assume that Qpθq “ Q0 ` θQ1 and that the value of θ at subsequent time step is upper bounded as |θs`1 ´ θs| ď ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For any fixed scalar ρ P p0, 1q, by solving the problem minimize X0, X1, λ1 ě 0, λ2 ě 0, W0, W1, γ2 1, γ2 2 γ2 1ρ2∆2 ` γ2 2 (32a) subject to ρ2rX0 ` X1s ľ pλ1ω2 1 ` λ2ω2 1ω2 2qIn, (32b) rX0 ` X1s ľ In, X0 ĺ γ2 1In (32c) X1 ĺ 0 (32d) » ————– ´λ1In 0 ´λ2In 02nˆ2n Ypθq ´ WpθqT WT ´γ2 2I2n QpθqpYpθq ´ WpθqTq RWpθqT ‹ ´Ypθq fi ffiffiffiffifl ĺ 0, Ypθq “ X0 ´ νX1 ` θX1 , / / / / / / / / .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' / / / / / / / / @θ P r0, 1s(32e) then Algorithm 2 with Kpθq “ WpθqYpθq´1 generates a sequence txkukPN that converges as, AE :“ lim sup kÑ8 Er}xk ´ ˆx‹ k}s ď 1 1 ´ ρ pγ1ρ∆ ` γ2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (33) Furthermore, for any fixed ρ, solving Problem (30) diminishes the asymptotical error AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ Theorem 3 describes parametric conditions for Algortihm 2 to converge to the optimizer trajectory in expectation and within an error ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We remark here the extra constraint X1 ĺ 0, which requires some explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' As one can see in the proof of Theorem 3, the key step in proving convergence of the algorithm is to ensure that a Lyapunov function decreases at subsequent times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The Lyapunov function that we consider is Lkpθkq “ pxk ´ ˆx‹ kqJXpθkqpxk ´ ˆx‹ kq, and therefore we have to deal with matrices Xpθq at subsequent times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' By imposing X1 ĺ 0, we can write however, Xpθs`1q “ Xpθsq ` pθs`1 ´ θsqX1 ĺ Xpθsq ´ |θs`1 ´ θs|X1 “ Ypθsq, from which we can derive Theorem 3 and this renders the derivation of a solvable program easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The constraint X1 ĺ 0 induces conservatism in the design, but in all our numerical simulations we found it to be redundant (meaning that the optimal X‹ 1 was ĺ 0 with or without the constraint), and therefore not conservative in our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We remark that a more correct approach would be to consider both extremes (`ν and ´ν) as done in [25] and remove X1 ĺ 0, but this would lead to an ambiguous definition of Kpθq and an harder problem to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Another possible approach is to remove X1 ĺ 0 and to consider only slowly changing parameters, for which ν !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Since in our simulations ν « 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4, we have preferred to focus on the former approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Finally, note that considering a X1 ĺ 0 is not totally unreasonable, since we can assume that increasing θ, the convergence performance would be negatively affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For fixed ρ, the problem to be solved is infinite dimensional (yet convex once θ is fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We recall that the constraints in (32e) are quadratic in θ, due to the product with Qpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' A possible way to solve problem (32) is to discretize the domain with a uniform grid Θ :“ t0, θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' , θq, 1u and impose (32e) for all the points of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Another possible way is to introduce another variable η “ θ2 P r0, 1s, and render affine (32e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' A more sophisticated way to proceed is to introduce a more conservative yet convex condition in θ, hinging on the concept of matrix sum of squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 11 Definition 1 Let Ppθq be a symmetric matrix of polynomials of degree up to 2d P N in the variable θ P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' A matrix is sum of squares (MSOS) if there exists a finite number l of symmetric matrices of polynomials Πipθq such that Ppθq “ lÿ i“1 ΠipθqTΠipθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The decomposition implies that Ppθq ľ 0 for all θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The constraint “Ppθq is MSOS” is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We are now ready for the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Theorem 4 Consider the matrix multiplicator Λ P R5nˆ5n, Λ ľ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In Theorem 3, condition (32e) can be substituted with the more conservative, yet convex and finite dimensional condition: ´ » ————– ´λ1In 0 ´λ2In 02nˆ2n Ypθq ´ WpθqT WT ´γ2 2I2n QpθqpYpθq ´ WpθqTq RWpθqT ‹ ´Ypθq fi ffiffiffiffifl ´ Λθp1 ´ θq is MSOS, (34a) Λ ľ 0, Ypθq “ X0 ´ νX1 ` θX1 (34b) where Λ is now a new decision variable in the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ Theorems 3 and 4 describe an LPV design strategy for our filter gain design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' This is a particular choice due to the affine parameter-varying Qpθq and static R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' More complex choices can be made (relatively) straightforwardly following the same pattern of the presented theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We now look at some numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 6 Numerical simulations We focus now on showcasing the performance of the proposed algorithms on a real dataset and problem stemming from ride-hailing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We obtain trips data from the New York City dataset5 for the yellow taxi cab in the month of November 2019, totalling over 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='8 millions trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We group the trips in 5 minutes intervals and divide the trip requests among n “ 5 different ride-hailing companies (such as taxi cab, Uber, Lyft, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The trip requests are divided randomly among the companies in a way that different companies do not have the same number of requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In the modern context of mobility as a service with ride-hailing orchestration [31], it makes sense for the city to provide software platforms to decide caps on the number of vehicles that each company can put on the streets depending on trading-off satisfying the demand and limiting traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' A natural optimization problem that a city can formulate is ˆx‹ptq “ arg min xPrx,xsn n ÿ i“1 ´ E ”1 2}xi ´ ciyiptq}2ı ` logp1 ` κ exppxiqq ` ς 2 n ÿ j“1 }xi ´ xj}2¯ , (35) where xi for each i represents the upper limit on vehicles on the roads for company i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The constraint rx, xsn represents box constraints on the number of allowed vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The term ci ą 0 multiplies the trip requests for company i at time t, yiptq to be able to match most of the requests as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The logistic term with κ ą 0 is a regularization to make the cost non-quadratic but still convex and favor a smaller number of vehicles on the roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Finally, the coupling term }xi ´xj}2 is set to have a similar regulation among different companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For the sake of the simulations, we take ci “ 1, κ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='02, ς “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1, and we sample Problem (35) at every 5 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We simulate also the ground truth considering a smoothed version of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 5Open data from the NYC Taxi and Limousine Commission Data Hub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 Unconstrained case We start by considering the unconstrained case, where x P R5, and we look at different noise regimes and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' As for the algorithms, we consider our TV-EKF (Algorithm 1), a standard prediction-correction [27], and the stochastic prediction-correction version of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For the latter, with their optimal choice of window size and weights for two-point evaluation, we can see that it is equivalent to the AGT algorithm of [32], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', exact prediction with Hessian inversion, but with a finite difference evaluation of ∇txf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For the three algorithms, we consider different choices of prediction steps P and correction steps C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For the stochastic prediction-correction version of [8], prediction is exact, so P is not a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We simulate three noise regimes: ‚ The case of very good prediction Q{R « 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For this, J generates as predictive signal a random signal around the ground truth with variance 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We use for the correction the true data stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' This case represents a realistic scenario of having a very good predictor (based on accurate historical data and, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', on a periodic Kernel method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' This could be typical in ride-hailing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ‚ The case of poor prediction R{Q « 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For this, J generates as predictive signal a random signal around the ground truth with variance 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We use for the correction a convex combination of the true data stream and the ground truth (weighting the true data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' This case represents a potential scenario of situation in which the system transitions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', a lock-down happens) and we have poor prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' This is in general less typical, since prediction can be built online on current data, based, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', on extrapolation, but still interesting to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ‚ The case of J generated by the true data based on an extrapolation-based prediction [27], and correction also based on true data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We label this case Q « R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The error between the true data and the ground truth has variance « 50, and the prediction « 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We design Qk and Rk accordingly, also taking into account that the data streams are correlated, and therefore Qk, Rk are full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Table 1 displays the results obtained in these settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' As we can see, Algorithm 1 performs the best by a significant margin, when prediction is accurate (Q{R « 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' When prediction is poor (R{Q « 0), then Algorithm 1 behaves close to a damped Newton’s method, which significantly outperforms a standard prediction-correction algorithm, but it is in par with its stochastic version (since the latter still uses the very accurate data stream to built its exact prediction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Finally, for the setting of Q « R, then all the three methods perform very similarly, which is almost expected since taking prediction, or prediction and then correction incur the same “error”, so any combination could achieve similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In this case, Algorithm 1 chooses a Kk « In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The results in Table 1 support Algorithm 1 as an algorithm that can automatically tune prediction and correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' based on this tuning it can be significantly better than the competitors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' and in the worst case it performs as state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We finally remark that the good prediction case is considered to be typical in this application scenario, and that the method from [8] could also be updated by designing an EKF for it, in the same way we did for the standard prediction-correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 Constrained case We analyze now the constrained case, for which we set x “ 100 and x “ 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' These constraints are not overly restrictive, but the point here is to see how our BMI-based method performs with respect to a standard prediction-correction method in different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Here, the stochastic variant [8] cannot be applied, but one could use the methods in [5] with exact prediction and finite-difference computations for ∇txf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We do not look into that, since typically these methods are more computationally demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The settings we investigate are similar to the unconstrained case, with the difference that we use our second algorithm TV-CONTRACT with a static K generated via Problem (30), and uniform grid-search on ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We also consider two cases for Q « R, the first has Q « 200In and R « 50In, the second Q « 67In and R « 50In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In both cases, as before, Q, R are full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In Table 2, we displays the results obtained in these settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' As we can see, Algorithm 2 has the best advantage when prediction is accurate and K can be chosen different from In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In general, the performance of Algorithm 2 is comparable with the competition, unless there is a clear advantage to choose a different from In gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In this latter case, the performance gain can be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In Table 2, we have also added the selected best K, which is full whenever we use the « sign, with diagonal elements close to the indicated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 13 Table 1: Performance of the considered algorithm in an unconstrained setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For each row, the first line represents the average error }xk ´ ˆx‹ k}, the second line the 25% percentile, and the third line the 75% percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In bold, the smallest error for the selected case and parameter choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' With ˚ we indicate a ą 10% error reduction with respect to the closest competitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Regime Algorithm Extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' P-C, [27], pP, Cq Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' P-C [8], C TV-EKF: Algortihm 1, pP, Cq p1, 1q p5, 1q p1, 5q p5, 5q 1 5 p1, 1q p5, 1q p1, 5q p5, 5q Q R « 0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='0 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='9 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1˚ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='8˚ 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2˚ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='0˚ 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7 44.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='6 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='6 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='0 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='0 Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 25 Mo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 26 Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 27 We.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 28 Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 28 Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 29 Sa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 30 0 200 400 600 800 1000 Allowed Vehicles Solutions for a week in November 2019 Trip count P-C method TV-Contract True solution Figure 2: Display of selected trajectories in Thanksgiving week of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The setting is Q « Rp2q and P “ C “ 5, and the trajectories are for one of the five companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The results in Table 2 support Algorithm 2 as an algorithm that can automatically tune prediction and correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' based on this tuning it can be better than the competitors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' and in the worst case it performs in par with state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We finally remark that the good prediction case is considered to be typical in this application scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For display purposes, Figure 2 illustrates the different trajectories for one of the five companies during Thanksgiving week of the selected month of November 2019, for the case Q « Rp2q and P “ C “ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 Variable K case We finish the simulation assessment showing the tracking results obtained solving Problem (32) for an affine parametric-varying Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In particular, we let Q0 “ Q{5 and Q1 “ 4Q{5, where Q is numerically defined as before, and run Algorithm 2 on all the four cases that we have looked at in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We solve Problem (32) by uniform gridding with 4 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' As mentioned, in our case ν « 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In Table 3, we report the results for the Q « R cases, since we do not observe any substantial difference for the other cases of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We indicate in bold if we have a gain w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' a static gain, and with a dagger, if we have also a gain w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We also report how the maximal element of the diagonal of K changes in time in a selected week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' As we can see, the results are very similar to the static results, but the gains can be important in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' As we see, the filter gain does not change over a wide range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' However, it appears that even these small changes are enough to reduce the asymptotical error in selected scenarios, and behaving in par with the static approach in the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' As one can infer, parametric-varying gain design does depend on the modelling choices for Qpθq and Xpθq, and one could expect possibly more performant results in the case of more complex dependencies on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We leave this analysis for future endeavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 14 Table 2: Performance of the considered algorithm in a constrained setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For each row, the first line represents the average error }xk ´ ˆx‹ k}, the second line the 25% percentile, and the third line the 75% percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In bold, the smallest error for the selected case and parameter choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' With ˚ we indicate a ą 10% error reduction with respect to the closest competitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Regime Algorithm Kp5,5q Extrapolation P-C, [27], pP, Cq TV-CONTRACT: Algorithm 2, pP, Cq p1, 1q p5, 1q p1, 5q p5, 5q p1, 1q p5, 1q p1, 5q p5, 5q Q R « 0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3˚ 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='0˚ 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='6˚ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='9 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='5 K « 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='24In 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4 R Q « 0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='6 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='5 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='6 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='5 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='9 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 K “ In 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='6 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='6 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 Q « R 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 p1q 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='9 K « In 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='8 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='6 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='8 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='8 Q « R 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7˚ p2q 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 K « 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='86In 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='0 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='6 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='0 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 Table 3: Performance of the considered algorithm in a constrained setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For each row, the first line represents the average error }xk ´ ˆx‹ k}, the second line the 25% percentile, and the third line the 75% percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We indicate in bold if we have a gain w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' a static gain of Table 2, and with a dagger, if we have also a gain w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' the state of the art of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Finally, with ˚ we indicate a ą 10% error reduction with respect to the closest competitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Regime Algorithm TV-CONTRACT-LPV: Algorithm 2, pP, Cq p1, 1q p5, 1q p1, 5q p5, 5q Q « R 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='9 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3: 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='8 Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 25 Mo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 26Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 27We.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 28Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 28 Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 29 Sa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='00 Max ( Diag (K) ) K (1,5) K (5,5) p1q 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='6 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 Q « R 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='9 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='9 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4:˚ 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3: Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 25 Mo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 26Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 27We.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 28Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 28 Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 29 Sa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='90 Max ( Diag (K) ) K (1,5) K (5,5) p2q 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='6 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='5 15 7 Conclusions We have discussed several methods to generalize time-varying optimization algorithms to the case of noisy data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The methods are rooted in the intuition that prediction and correction can be seen as a nonlin- ear dynamical system and a nonlinear measurement equation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' This leads to extended Kalman filter formulations as well as contractive filters based on bilinear matrix inequalities (BMI’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Numerical results are promising, even when using possibly conservative BMI conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' A Proofs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 An additional example Example 2 (Deterministic example) Consider a deterministic method with an extrapolation predictor [27], meaning: Jk`1pxq “ 2∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ykq ´ ∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yk´1q, where now yptq is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Assume fpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq is strongly convex and smooth, uniformly in y, assume that the Hessian of fpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq does not depend on y, and assume the following bounds on data and mixed derivatives: maxt}∇typtq} , }∇ttyptq} , }∇yxfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq} , }∇yyxfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq}u ď C, @x, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (36) Then we have that }Jk`1px‹ k`1q ´ ∇xfpx‹ k`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yk`1q} ď pC2 ` C3qh2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ Proof: From [27, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='5], we know that there exists a τ P rtk´1, tk`1s such that ››Jk`1px‹ k`1q ´ ∇xfpx‹ k`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yk`1q ›› ď ››∇tt∇xfpx‹ k`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ypτqq ›› h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (37) Let the i-th component of ∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq be Dipx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' By using the higher-derivatives Fa`a di Bruno’s chain rule: r∇tt∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqqsi “ B2 Bt2 Dipx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq “ ÿ j ˆ B Byj Dipx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq B2yjptq Bt2 ˙ ` ÿ j,ℓ ˆ B2 ByjByℓ Dipx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq Byjptq Bt Byℓptq Bt ˙ , (38) from which the thesis follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ♦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='2 Derivations for Example 1 We choose to write x‹ “ x‹ k`1 as a short-hand notation, in this proof only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' By linearity of ∇xfpx‹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yptqq with respect to the parameter yptq, and the linearity of the expectation, we can write EwPYk,zPYk´1r}∇xfpx‹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2w ´ zq ´ EyPYk`1r∇xfpx‹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs}s “ EwPYk,zPYk´1r}∇xfpx‹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2w ´ z ´ EyPYk`1rysq}s “ EwPYk,zPYk´1r}∇xfpx‹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2w ´ z ´ ¯yk`1q}s “ Ee1PN p0,Σkq,e2PN p0,Σk´1qr}∇xfpx‹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2¯yk ´ ¯yk´1 ´ ¯yk`1q ` ∇xfpx‹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2e1 ´ e2q}s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We now use the Triangle inequality, the result (38), and the mean value theorem for the nominal trajectory, to upper bound the last inequality as Ee1PN p0,Σkq,e2PN p0,Σk´1qr}∇xfpx‹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2¯yk ´ ¯yk´1 ´ ¯yk`1q} ` }∇xfpx‹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2e1 ´ e2q}s “ }∇xfpx‹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2¯yk ´ ¯yk´1 ´ ¯yk`1q} ` Ee1PN p0,Σkq,e2PN p0,Σk´1qr}∇xfpx‹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2e1 ´ e2q}s ď C0Ch2 ` C0 Ee1PN p0,Σkq,e2PN p0,Σk´1qr}2e1 ´ e2}s ď C0Ch2 ` 3C0Σ, from which the first claim is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 16 For the second, EyPYk`1r}∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yq ´ EyPYk`1r∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs}s “ EyPYk`1r}∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yq ´ ∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ¯yk`1qs}s “ EePN p0,Σk`1qr}∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' eqs}s ď C0 EePN p0,Σk`1qr}e}s ď C0Σ, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='3 Proof of Proposition 1 Consider C “ 1 in Algorithm 1, as well as a negligible Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Then, we can simplify the Kalman gain as: Kk “ Pk|k´1HkrHkPk|k´1HT ks´1 “ H´1 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Therefore, the state update reads xk “ xk|k´1 ` KkpΨpxk|k´1, ykqq “ xk|k´1 ` H´1 k p´xk|k´1 ` xk|k´1 ´ β∇xfpxk|k´1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ykqq “ xk|k´1 ´ βr∇xxfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ykqs´1∇xfpxk|k´1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ykq, from which the thesis is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='4 Supporting results for Theorem 1 Lemma 1 Let Assumptions 1-2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Choose α ă 2µ{L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Let xf k`1 be the fixed point of the prediction “pseudo”-dynamical model: xf k`1 “ Φk,gpxf k`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Then the distance between xf k`1 and the optimizer trajectory is bounded in expectation as, Er}xf k`1 ´ ˆx‹ k`1}s ď 1 µEr}Jk`1pˆx‹ k`1q ´ EyPYk`1r∇xfpˆx‹ k`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs}s ď τ µ “: τµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Furthermore, let the setting of Example 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Then, Er}xf k`1 ´ ˆx‹ k`1}s ď C0Ch2{µ ` 3C0Σ{µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ Proof: Choosing α ă 2µ{L2 and under Assumption 1, we know that the prediction is a contractive operator and its fixed point exists and it is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' By implicit function theorems, see for instance [33, Theorem 2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='9] and [27, Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1], being careful to J being strongly monotone and not generally the gradient of a strongly convex function, then, }xf k`1 ´ ˆx‹ k`1} ď 1 µ }Jk`1pˆx‹ k`1q ´ EyPYk`1r∇xfpˆx‹ k`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs} loooooooooooooooooooooooooomoooooooooooooooooooooooooon p˛q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (39) Passing in expectations, and by using Assumption 2, the first thesis follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' As for the second statement, it follows from the derivations of Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ♦ Lemma 2 Let Assumptions 1-2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Choose α ă 2µ{L2, β ă 2{L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Consider the prediction update xk`1|k “ Φk,gpxkq with P prediction steps, and the correction update x1 k “ Ψ1 gpxk`1|k, yk`1q with C correction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Let the contraction factors ρp, ρc be defined as in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Then, the following error bounds are in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Er}xk`1|k ´ ˆx‹ k`1}s ď ρP p Er}xk ´ xf k`1}s ` τµ (40) Er}x1 k ´ ˆx‹ k`1}s ď ρC c Er}xk`1|k ´ ˆx‹ k`1}s ` σc, (41) where σc “ β σ 1´ρc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Furthermore, under the setting of Example 1, τµ “ pC0Ch2 ` 3C0Σq{µ and σ “ C0Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ 17 Proof: Choosing α ă 2µ{L2, β ă 2{L and under Assumption 1, we know that the prediction and correction are contractive operators and their fixed points are unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For the prediction part, by using Equation (39), we obtain, }xk`1|k ´ ˆx‹ k`1} “ }xk`1|k ˘ xf k`1 ´ ˆx‹ k`1} ď }rproxαgpI ´ αJk`1p‚qqs˝P xk ´ xf k`1} ` 1 µp˛q ď ρP p }xk ´ xf k`1} ` 1 µp˛q, (42) and passing in expectation with Assumption 2 the claim is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For the second claim, we can write }x1 k ´ ˆx‹ k`1} ď }rproxβgpI ´ β∇xfp‚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yk`1q ˘ βEyPYk`1r∇xfp‚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqsqs˝Cxk`1|k ´ ˆx‹ k`1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (43) Call ϵk`1pxq :“ ∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yk`1q ´ EyPYk`1r∇xfpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' yqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Then each proximal gradient step will incur in an additive }ϵk`1pxq} error, where x will be different at each step: }x1 k ´ ˆx‹ k`1} ď ρc}xC´1 k`1|k ´ ˆx‹ k`1} ` β}ϵk`1pxC´1 k`1|kq} ď ρC c }xk`1|k ´ ˆx‹ k`1} ` β C ÿ c“1 ρC´c c }ϵk`1pxC´c k`1|kq} looooooooooooooomooooooooooooooon p˛˛q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (44) Passing in expectation, with Assumption 2 and the sum of geometric series, the second claim is also proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ♦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='5 Proof of Theorem 1 The proof follows the one of [27, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1], combining Lemma 1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Start by considering χ “ 1, so a classical prediction-correction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We can use [27, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='1], with Erτks “ τµ, and the correction with an additional error term to say, }xk`1 ´ ˆx‹ k`1} ď ζC,ρc ´ ζP,ρp}xk ´ ˆx‹ k} ` ζP,ρp∆ ` ξP,ρpτk ¯ ` p˛˛q “: E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (45) Looking at prediction only, χ “ 0, we obtain instead, }xk`1 ´ ˆx‹ k`1} ď ´ ζP,ρp}xk ´ ˆx‹ k} ` ζP,ρp∆ ` ξP,ρpτk ¯ “: E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (46) For a generic χ P r0, 1s, we can combine the errors as }xk`1 ´ ˆx‹ k`1} ď p1 ´ χqE2 ` χE1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (47) Then, we can recursively compute the error via geometric series summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' By passing through expectations, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For Example 1, with ∇yxf bounded, by implicit function theorems [5], we have that ∆ “ C0h{µ, from which the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='6 Proof of Proposition 2 For Problem (22), we look at the minimum of the curve, min χPr0,1s aχ ` b cχ ` d “: Fpχq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (48) For our problem cχ ` d “ 1 ´ ζP,ρp ` χpζP,ρp ´ ζP,ρpζC,ρcq ą 0, b ��� ζP,ρp∆ ` ξP,ρpτµ ą 0, while a “ pζC,ρc ´ 1qpζP,ρp∆ ` ξP,ρpτµq ` ζC,ρcσc can be positive, negative, or zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Since function Fpχq is a linear- fractional function in one dimension, for χ ě 0, function Fpχq is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In particular, for a{c ă pąqb{d the function is decreasing (increasing), leading to the optimal choices of χ‹ “ 1p0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The condition means, pζC,ρc ´ 1qpζP,ρp∆ ` ξP,ρpτµq ` ζC,ρcσc ζP,ρp ´ ζP,ρpζC,ρc ă pąqζP,ρp∆ ` ξP,ρpτµ 1 ´ ζP,ρp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' For the special case a{c “ b{d, Fpχq ” 1 and any χ is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='7 Proof of Theorem 2 To impose convergence and performance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' we look at the following matrix condition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' featuring semidefinite matrix X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' the scalar ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' λ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' λ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' γ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' and matrix K which is implicit in B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Be: p‚qT „ X 0 0 ´X ȷ „ 0 B Be ρIn 012 012 ȷ ` λ1p‚qT „ ω2 1In 0 0 ´In ȷ „ In 0 0 012 0 In 0 012 ȷ ` ` λ2p‚qT „ ω2 1ω2 2In 0 0 ´In ȷ „ In 0 0 012 0 0 In 012 ȷ ` » – 0 022 ´γ2 2In fi fl ĺ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (49) where p‚qT means that what is post-multiplied is also pre-multiplied transposed and 0 “ 0nˆn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 0ij “ 0inˆjn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Condition (49) combines the system, the quadratic constraints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=', the contractivity) via an S-procedure, and the performance criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We now develop the multiplications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' we let } ¨ }2 X :“ p¨qTXp¨q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' and pre and post multiply with the vector rpxk ´ ˆx‹ k`1qT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' p ¯wk`1 ´ ˆx‹ k`1qT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' p¯uk`1 ´ ˆx‹ k`1qT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' eT k`1sT and we obtain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ´ ρ2}xk ´ ˆx‹ k`1}2 X ` }xk`1 ´ ˆx�� k`1}2 X ď ´ λ1 “ ω2 1}xk ´ ˆx‹ k`1}2 ´ } ¯wk`1 ´ ˆx‹ k`1}2‰ loooooooooooooooooooooooooomoooooooooooooooooooooooooon ě0 ` ´ λ2 “ ω2 1ω2 2}xk ´ ˆx‹ k`1}2 ´ }¯uk`1 ´ ˆx‹ k`1}2‰ looooooooooooooooooooooooooomooooooooooooooooooooooooooon ě0 `γ2}ek`1}2 ď γ2}ek`1}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (50) Define the error Ei :“ xi ´ ˆx‹ i and the drift δk “ ˆx‹ k`1 ´ ˆx‹ k, then }Ek`1}2 X ď ρ2}Ek ´ δk}2 X ` γ2}ek`1}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (51) Taking the square root of both sides, since ě 0 }Ek`1}X ď b ρ2}Ek ´ δk}2 X ` γ2}ek`1}2 ď ρ}Ek}X ` ρ}δk}X ` γ}ek`1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (52) Let }δk} ď ∆, also note that Er}ek`1}s ď 1 since Er}qk`1}s ď 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2, Er}rk`1}s ď 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Since we impose X ľ In without loss of generality (since the problem remains unchanged for any scalar scaling), then }Ek`1}X ě }Ek`1} and }Ek}X ď }X1{2}}Ek} “ γ1}Ek}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Here the equality sign is due to the fact that we minimize over γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Similarly, ρ}δk}X ď ργ1}δk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Then, iterating on k, and taking the expectations, we obtain, Er}Ek}s ď γ1ρkEr}E0}s ` 1 1 ´ ρ pγ1ρ∆ ` γ2q , (53) lim sup kÑ8 Er}Ek}s ď 1 1 ´ ρ pγ1ρ∆ ` γ2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (54) As such, for any fixed ρ, minimizing γ1ρ∆ ` γ2 minimizes the asymptotic tracking error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Furthermore, since }x}1 ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='n}x}2 for x P Rn, we know that a γ2 1ρ2∆2 ` γ2 2 ě pγ1ρ∆ ` γ2q{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' So our cost majorizes the asymptotical error γ1ρ∆`γ2 and therefore by minimizing our cost, we diminish the latter (notice, we do not minimize the latter, in general, since we have a constrained problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' To finish the proof, we need to transform (49) into (30d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We develop the matrix multiplications, and we observe that the resulting matrix is block diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The first block is ρ2X ľ pλ1ω2 1 `λ2ω2 1ω2 2qIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' The second block is p‚qTX rB Bes ` λ1 » – ´In 0 012 0 012 022 fi fl ` λ2 » – 0 0 012 ´In 012 022 fi fl ` » – 0 0 012 0 012 ´γ2I2n fi fl ĺ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (55) Expand X into XX´1X and introduce the variable W “ KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Then, taking the Schur’s complement, we obtain (30d), from which the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ 19 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='8 Proof of Theorem 3 The proof follows the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We focus here on the different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (49), we adapt the first term to: p‚qT „ Xpθs`1q 0 0 ´Xpθsq ȷ „ 0 Bpθsq Bepθsq ρIn 012 012 ȷ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' (56) The bottom diagonal leads to conditions ρ2rX0 ` θsX1s ľ pλ1ω2 1 ` λ2ω2 1ω2 2qIn, (57) rX0 ` θsX1s ľ In, rX0 ` θsX1s ĺ γ2 1In (58) which needs to be valid for the extreme points θs “ 0, 1, since affine in θs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' However, given the constraint X1 ĺ 0, the above simplify into (32b) and (32c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Bu using again the constraint X1 ĺ 0, the upper diagonal can be upper bounded as, p‚qTXpθs`1qrBpθsq Bepθsqs “ p‚qTrX0 ` θs`1X1srBpθsq Bepθsqs ĺ p‚qTrX0 ´ νX1 ` θsX1srBpθsq Bepθsqs “ p‚qTYpθsqrBpθsq Bepθsqs, (59) so imposing a ĺ condition on the latter, would imply a condition on the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' In particular, adapting (55), the condition p‚qTYpθsqrBpθsq Bepθsqs`λ1 » – ´In 0 012 0 012 022 fi fl`λ2 » – 0 0 012 ´In 012 022 fi fl` » – 0 0 012 0 012 ´γ2I2n fi fl ĺ 0 (60) would imply a similar condition on Xpθs`1q, thus the upper diagonal of (56), and for proof of Theorem 2, convergence of the algorithm as indicated in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Condition (60) leads to condition (32e), by Schur complement and dropping the now-redundant subscript s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' We remark here the importance of the constraint X1 ĺ 0, without which we would need two upper bounds in (59), one for ´ν and one for `ν, which would render the substitution Wpθq “ KpθqYpθq ambiguous, and determining Kpθq harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content='9 Proof of Theorem 4 The condition θ P r0, 1s is equivalent to θp1 ´ θq ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Then we apply the generalized S-procedure as in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' ■ References [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Hamam and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfyjjm/content/2301.13646v1.pdf'} +page_content=' Romberg, “Streaming solutions for time-varying 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