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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf,len=1305
+page_content='SoftTreeMax: Exponential Variance Reduction in Policy Gradient  via Tree Search  Gal Dalal * Assaf Hallak * Gugan Thoppe Shie Mannor Gal Chechik  Abstract  Despite the popularity of policy gradient meth-  ods, they are known to suffer from large vari-  ance and high sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' To mitigate  this, we introduce SoftTreeMax – a generaliza-  tion of softmax that takes planning into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  In SoftTreeMax, we extend the traditional logits  with the multi-step discounted cumulative reward,  topped with the logits of future states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We con-  sider two variants of SoftTreeMax, one for cumu-  lative reward and one for exponentiated reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  For both, we analyze the gradient variance and  reveal for the first time the role of a tree expan-  sion policy in mitigating this variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We prove  that the resulting variance decays exponentially  with the planning horizon as a function of the  expansion policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Specifically, we show that the  closer the resulting state transitions are to uni-  form, the faster the decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In a practical imple-  mentation, we utilize a parallelized GPU-based  simulator for fast and efficient tree search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Our  differentiable tree-based policy leverages all gra-  dients at the tree leaves in each environment step  instead of the traditional single-sample-based gra-  dient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We then show in simulation how the vari-  ance of the gradient is reduced by three orders  of magnitude, leading to better sample complex-  ity compared to the standard policy gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' On  Atari, SoftTreeMax demonstrates up to 5x better  performance in a faster run time compared to dis-  tributed PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Lastly, we demonstrate that high  reward correlates with lower variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Introduction  Policy Gradient (PG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' 1999) methods for Re-  inforcement Learning (RL) are often the first choice for  environments that allow numerous interactions at a fast pace  (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Their success is attributed to several  Equal contribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Correspondence to:  Gal Dalal  <gdalal@nvidia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='com>, Assaf Hallak <ahallak@nvidia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='com>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Preperint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  factors, including that they are easy-to-distribute to multiple  workers, require no assumptions on an underlying value  function, and have both on-policy and off-policy variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Despite their popularity, PG algorithms are also notoriously  unstable since they compute gradients over entire trajec-  tories (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' As a result, PG  algorithms tend to be highly inefficient in terms of sample  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Several solutions were proposed to mitigate the  instability of PG methods, including baseline subtraction  (Greensmith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Weaver & Tao, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Thomas &  Brunskill, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2018), anchor-point averaging  (Papini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2018), and other variance reduction techniques  (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  A second family of algorithms that achieved state-of-the-art  results in several domains is based on planning (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=',  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Planning is exercised primarily in the  context of value-based RL and is usually implemented using  a Tree Search (TS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Coulom 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Silver 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In this work,  we combine PG with TS by introducing a parameterized dif-  ferentiable policy that incorporates tree expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Namely,  our SoftTreeMax policy replaces the standard policy logits  of a state and action, with the expected value of trajectories  that originate from these state and action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Combining TS into PG suite should be done with care given  the biggest hurdle of PG – its high gradient variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This  raises prominent actionable questions that were ignored un-  til this work: How does the tree-expansion policy affect  the PG variance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' And, can we design tree-expansion that  is guaranteed to strongly reduces that variance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Here, we  analyze the gradient variance of SoftTreeMax, and provide  a practical methodology to choose the expansion policy to  minimize the resulting variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Our main result shows that  a desirable expansion policy is one that induces transitions  as close to uniform as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' More generally, we show  that the gradient variance of SoftTreeMax decays at an expo-  nential rate of |λ2|d, where d is the tree depth and λ2 is the  second eigenvalue of the transition matrix induced by the  tree expansion policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This paper is the first to prove such a  relation between PG variance and TS expansion policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Common practices for expanding the tree rely on a value  estimate, using UCT (Kocsis & Szepesv´ari, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Browne  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2012), or based on some prior knowledge such as  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='13236v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='LG]  30 Jan 2023  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  human-collected trajectories (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Our work  raises the question of whether optimal variance reduction  corresponds to the appealing regret properties of UCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  To verify our results, we implemented a practical version of  SoftTreeMax that exhaustively searches the entire tree and  applies a neural network on its leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We test our algorithm  in the Atari domain, where it is possible to span the whole  tree of the (nearly) deterministic Atari environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Hence,  the gradient variance has no sampling component, and our  variance calculations indeed match the empirical gradient  variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Our search mechanism uses a GPU simulator that  allows multiple copies of the environment to be run in par-  allel (Dalton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' To enable a tractable deep search,  up to depth eight, we also introduce a pruning technique  that limits the width of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We do so by sampling only  the most promising nodes at each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  We integrate our SoftTreeMax GPU implementation into  the popular PPO (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2017) and compare it to  the flat distributed variant of PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' For a fair comparison,  we also run the distributed PPO baseline with the parallel  GPU emulator by Dalton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In all tested Atari  games, our results outperform the baseline and obtain up  to 5x more reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We further show in Section 6 that the  associated gradient variance is smaller by three orders of  magnitude in all games, demonstrating the relation between  low gradient variance and high reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  We summarize our key contributions:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We explore the relation between two seemingly unre-  lated families of SoTA approaches: PG and TS, and  show how they can be combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We introduce SoftTreeMax: A novel parametric policy  that generalizes softmax to planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We propose both  cumulative and exponentiated reward variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We prove that the gradient variance of SoftTreeMax in  its two variants decays exponentially with its TS depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Our analysis sheds new light on the choice of tree  expansion policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' It raises the question of optimality in  terms of variance versus the previously studied regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We implement a differentiable deep version of  SoftTreeMax that employs a parallelized GPU TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We  demonstrate how its gradient variance is reduced by  three orders of magnitude over PPO while obtaining  up to 5x reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Preliminaries  We follow the standard notation by (Puterman, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Con-  sider a discounted Markov Decision Process (MDP) M =  (S, A, P, r, γ), where S is a finite state space of size S, A  is a finite action space of size A, r : S × A → [0, 1] is  the reward function, P : S × A → ∆S is the transition  function, and γ ∈ (0, 1) is the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In vector  form, denote the transition matrix starting from state s by  [Ps]a,s′ = Pr(s′|a, s) ∈ [0, 1]A×S, and the corresponding  reward vector by Rs = r(s, ·) ∈ RA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Let π : S → ∆A be a stationary policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We define the in-  duced transition matrix P π(s′|s) = �  a π(a|s) Pr(s′|s, a)  and reward function Rπ(s) = �  a π(a|s)r(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  De-  note by µπ ∈ RS the stationary distribution of P π, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  µ⊤  π P π = P π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Also, let V π ∈ RS be the value function of  π defined by V π(s) = Eπ [�∞  t=0 γtr (st, π(st)) | s0 = s],  and let Qπ ∈ RS×A be the Q-function such that Qπ(s, a) =  Eπ [r(s, a) + γV π(s′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Our goal is to find an optimal policy π⋆ such that  V ⋆(s) ≡ V π⋆(s) = max  π  V π(s),  ∀s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Lastly, for the analysis in Section 4, we introduce the follow-  ing vector notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Denote by Θ ∈ RS the vector represen-  tation of θ(s) ∀s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' For a vector u, denote by exp(u) the  coordinate-wise exponent of u and by D(u) the diagonal  square matrix with u in its diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' For matrix A, denote  its i-th eigenvalue by λi(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Denote the k-dimensional iden-  tity matrix and all-ones vector by Ik and 1k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  We denote the trace operator by Tr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Finally, We treat all  vectors as column vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Policy Gradient  PG schemes seek to maximize the cumulative reward as a  function of the parameterized policy πθ(a|s) by perform-  ing gradient steps on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The celebrated Policy Gradient  Theorem (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 1999) states that  ∂  ∂θ  �  µ⊤  πθV πθ�  = Es∼µπθ ,a∼πθ(·|s) [∇θ log πθ(a|s)Qπθ(s, a)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  The variance of the gradient is thus  Vars∼µπθ ,a∼πθ(·|s) (∇θ log πθ(a|s)Qπθ(s, a)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (1)  In the notation above, we denote the variance of a vector  random variable X by:  Varx (X) = Tr  �  Ex  �  (X − ExX)⊤ (X − ExX]  ��  ,  similarly as in (Greensmith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' From here on, we  drop the subscript from Var in (1) for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  When the action space is discrete, a commonly used param-  eterized policy is softmax:  πθ(a|s) ∝ exp (θ(s, a)) ,  where θ : S × A → R is a state-action parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' SoftTreeMax: Exponent of trajectories  We introduce a new family of policies called SoftTreeMax,  which are a model-based generalization of the popu-  lar softmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  We propose two variants:  Cumulative  (C-SoftTreeMax) and Exponentiated (E-SoftTreeMax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In  both variants, we replace the generic softmax logits θ(s, a)  with the score of a trajectory of horizon d starting from  s, a, generated by applying a behavior policy πb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  In  C-SoftTreeMax, we exponentiate the expectation of the log-  its.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In E-SoftTreeMax, we first exponentiate the logits, and  only then compute their expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Let the SoftTreeMax logit ℓs,a(d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' θ) be a random  variable depicting the score of a trajectory of horizon d  starting from s, a and following the policy πb :  ℓs,a(d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' θ) =  d−1  �  t=0  γtrt + γdθ(sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (2)  Namely, s0 = s, a0 = a, at ∼ πb(·|st) ∀t ≥ 1, and  rt ≡ r (st, at) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' For brevity of the analysis, we let the para-  metric score θ in (2) be state-based, similarly to a value  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Instead, one could use a state-action input analo-  gous to a Q-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This freedom allows easy integration  of SoftTreeMax to the two types of RL algorithm imple-  mentations in standard packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  C-SoftTreeMax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Given an inverse temperature parameter  β, let C-SoftTreeMax be  πC  d,θ(a|s) ∝ exp [βEπbℓs,a(d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' θ)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (3)  C-SoftTreeMax gives higher weight for actions that result  in higher expected returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' While standard softmax relies  entirely on parametrization θ, C-SoftTreeMax also interpo-  lates a Monte-Carlo portion of the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Using the monotone convergence theorem (since rewards  are non-negative), it follows that when d → ∞,  πC  d→∞,θ(a|s) ∝ exp [βQπb(s, a)] ,  corresponding to Boltzmann exploration (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 1999)  using the behavior policy πb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  E-SoftTreeMax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' A second natural operator to consider is  E-SoftTreeMax, in which the expectation is taken outside  the exponent:  πE  d,θ(a|s) ∝ Eπb exp [(βℓs,a(d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' θ))] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (4)  This objective corresponds to the exponentiated reward ob-  jective which is often used for risk-sensitive RL (Howard  & Matheson, 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Fei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Noorani & Baras,  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The common risk-sensitive objective is of the form  log E[exp(δR)], where δ is the risk parameter and R is the  cumulative reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Similarly to that literature, the exponent  in (4) emphasizes the most promising trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  SoftTreeMax properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' SoftTreeMax is a natural model-  based generalization of softmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' For d = 0, both variants  above coincide, since (2) becomes deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In that  case and for a state-action parametrization, they reduce  to standard softmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' When β → 0, both variants again  coincide and sample actions uniformly (exploration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' When  β → ∞, the policies become deterministic and greedily  optimize for the best trajectory (exploitation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The best  trajectory is in expectation in the case of C-SoftTreeMax,  and in terms of best sample-path for E-SoftTreeMax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  SoftTreeMax convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Under regularity conditions,  for any parametric policy, PG converges to local optima  (Bhatnagar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2009), and thus also SoftTreeMax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Specif-  ically for softmax PG, asymptotic (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2021) and  rate results (Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2020b) were recently obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' A fu-  ture direction would be to extend those for the convergence  properties of SoftTreeMax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  SoftTreeMax gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The two variants of SoftTreeMax  involve an expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This expectation is taken over Sd  many trajectories from the root state s and are weighted  according to their probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Thus, during the PG train-  ing process, the gradient ∇θ log πθ is calculated using a  weighted sum of gradients over all reachable states starting  from s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Our method exploits the exponential number of tra-  jectories to reduce the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Indeed, in the next section  we prove that the gradient variance of SoftTreeMax decays  exponentially fast as a function of the behavior policy πb and  trajectory length d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In the experiments in Section 6, we also  show how the practical version of SoftTreeMax achieves  a significant reduction in the noise of the PG process and  leads to faster convergence and higher reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Theoretical Analysis  In this section, we bound the variance of PG when using  SoftTreeMax policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Specifically, we show that the variance  decreases exponentially with the tree depth, where the rate  is determined by the second eigenvalue of the transition  kernel induced by πb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We analyze the gradient variance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  state-action frequencies, as a function of problem param-  eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Other types of analyses could have instead focused  on the estimation aspect in the context of sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Indeed,  in our implementation in Section 5, we manage to avoid  sampling and directly compute the expectations in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' (3)  and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' As we show later, we do so by leveraging efficient  parallel simulation on the GPU in feasible run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In our  application, due to the nature of the finite action space and  quasi-deterministic Atari dynamics (Bellemare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2013),  our expectation estimator is noiseless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We encourage future  work to account for the finite-sample variance component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  We begin with a general variance bound that holds for any  parametric policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We defer all the proofs in this section to  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1 (Bound on the policy gradient variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' For  any parametric policy πθ and function Qπθ : S × A → R,  Var (∇θ log πθ(a|s)Qπθ(s, a))  ≤ max  s,a [Qπθ(s, a)]2 max  s  ||∇θ log πθ(·|s)||2  F ,  where ∇θ log πθ(·|s) ∈ RA×dim(θ) is a matrix whose a-th  row is ∇θ log πθ(a|s)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Hence, to bound (1), it is sufficient to bound the Frobenius  norm of the policy gradient ∇θ log πθ(·|s) for any s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  A common assumption in the RL literature (Szepesv´ari,  2010) that we adopt for the remainder of the section is that  the transition matrix P πb, induced by the behavior policy  πb, is irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Subsequently, its second  highest eigenvalue holds: |λ2(P πb)| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  From here on, we split the variance results for the  two variants of SoftTreeMax to two subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  For  C-SoftTreeMax, the analysis is simpler and we provide an  exact bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The case of E-SoftTreeMax is more involved  and we provide for it a more general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In both cases,  we show that the variance decays exponentially with the  planning horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Variance of C-SoftTreeMax  We express C-SoftTreeMax in vector form as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='2 (Vector form of C-SoftTreeMax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' For d ≥ 1,  (3) is given by  πC  d,θ(·|s) =  exp  �  β  �  Cs,d + γdPs (P πb)d−1 Θ  ��  1⊤  A exp  �  β  �  Cs,d + γdPs (P πb)d−1 Θ  ��,  (5)  where  Cs,d = Rs + Ps  �d−1  �  h=1  γh (P πb)h−1  �  Rπb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (6)  The matrix Cs,d ∈ RA×S represents the cumulative dis-  counted reward in expectation along the trajectory of hori-  zon d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Starting from the state s, the reward Rs is collected  and a transition occurs according to Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Then, the policy  πb is applied to obtain the reward Rπb and transition, and  the process repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' When depth d is reached, we apply the  score function on the last state as depicted in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Next, we express the policy gradient of C-SoftTreeMax  Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='3  (Gradient  of  C-SoftTreeMax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  The  C-SoftTreeMax gradient of dimension A × S is given by  ∇θ log πC  d,θ = βγd �  IA − 1A(πC  d,θ)⊤�  Ps (P πb)d−1 ,  where for brevity, we drop the s index in the policy above,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', πC  d,θ ≡ πC  d,θ(·|s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  We are now ready to present our first main result:  Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='4  (Exponential  variance  decay  of  C-SoftTreeMax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' For every Q  :  S × A  →  R, the  C-SoftTreeMax policy gradient is bounded by  Var  �  ∇θ log πC  d,θ(a|s)Q(s, a)  �  ≤ 2 A2S2β2  (1 − γ)2 γ2d|λ2(P πb)|2(d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Although we provide a rigorous proof in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='4,  since the proof relatively accessible, we briefly outline its  essence here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Proof outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1 allows us to bound the variance  using a direct bound on the gradient norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The gradient is  given in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='3 as a product of three matrices, which  we now study from right to left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The matrix P πb is a row-  stochastic matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Because the associated Markov chain is  irreducible and aperiodic, it has a unique stationary distribu-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This implies that P πb has one and only one eigenvalue  equal to 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' all others have magnitude strictly less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Let  us suppose that all these other eigenvalues have multiplicity  1 (the general case with repeated eigenvalues can be handled  via Jordan decompositions as in (Pelletier, 1998, Lemma1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Then, P πb has the spectral decomposition  P πb = 1Sµ⊤  πb +  S  �  i=2  λiviu⊤  i ,  where λi is the i-th eigenvalue of P πb (ordered in descend-  ing order according to their magnitude) and ui and vi are  the corresponding left and right eigenvectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Therefore,  (P πb)d−1 = 1Sµ⊤  πb +  S  �  i=2  λd−1  i  viu⊤  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (7)  The second matrix in the gradient relation in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='3, Ps,  is a rectangular transition matrix that translates the vector  of all ones from dimension S to A : Ps1S = 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Lastly,  the first matrix  �  IA − 1A(πC  d,θ)⊤�  is a pro-  jection whose null-space includes the vector 1A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=',  �  IA − 1A(πC  d,θ)⊤�  1A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Combining the three properties above when multiplying  the three matrices of the gradient, it is easy to see that  the first term in (7) gets canceled, and we are left with  bounded summands scaled by λi(P πb)d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Recalling that  |λi(P πb)| < 1 and that |λ2| > |λ3| > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' for i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , S,  we obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  2  4  6  8  10  Depth d  10  31  10  26  10  21  10  16  10  11  10  6  10  1  SoftTreeMax  Gradient variance  Permutation: True variance  Permutation: Variance bound  Random: True variance  Random: Variance bound  Uniform: True variance  Uniform: Variance bound  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' A comparison of the analytical PG variance and our  bound for E-SoftTreeMax on randomly drawn MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We present  three cases for P πb : (i) close to uniform, (ii) drawn randomly, and  (iii) close to a permutation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This experiment verifies the  optimal and worse-case rate decay cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The variance bounds here  are taken from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='7 where we substitute α = |λ2(P πb)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='4 guarantees that the variance of the gradient  decays exponentially with d, regardless of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' It also pro-  vides a novel insight that drives us to choose the behavior  policy πb as the policy that minimizes the absolute second  eigenvalue of the P πb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Indeed, the second eigenvalue of a  Markov chain has known connections to its connectivity and  its rate of convergence to the stationary distribution (Levin  & Peres, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Optimal variance decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' To achieve the best reduction  in variance, the behavior policy πb should be chosen to  achieve uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' That is, that transitions induced by  the interaction of πb with the environment are uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In  that case, P πb is a rank one matrix of the form 1Sµ⊤  πb, and  λ2(P πb) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Then, Var (∇θ log πθ(a|s)Q(s, a)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' As  we show in Section 5, we choose our tree expansion policy  accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Worst-case variance decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In contrast, and somewhat  surprisingly, when πb is chosen so that the dynamics is  deterministic, there is no guarantee that it will decay expo-  nentially fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' For example, if P πb is a permutation matrix,  then λ2(P πb) = 1, and advancing the tree amounts to only  updating the gradient of one state for every action, as in the  basic softmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Variance of E-SoftTreeMax  The proof of the variance bound for E-SoftTreeMax is sim-  ilar to that of C-SoftTreeMax, but more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' It also  requires the assumption that the reward depends only on the  state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' r(s, a) ≡ r(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This is indeed the case in most  standard RL environments such as Atari and Mujoco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  We begin with expressing E-SoftTreeMax in vector form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='5 (Vector form of E-SoftTreeMax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' For d ≥ 1,  (4) is given by  πE  d,θ(·|s) =  Es,d exp(βγdΘ)  1⊤  AEs,d exp(βγdΘ),  (8)  where  Es,d = Ps  d−1  �  h=1  �  D  �  exp(βγhR)  �  P πb�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (9)  The vector R above is the S-dimensional vector whose s-th  coordinate is r(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  The matrix Es,d ∈ RA×S has a similar role to Cs,d from (6),  but it represents the exponentiated cumulative discounted  reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Accordingly, it is a product of d matrices as opposed  to a sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' It captures the expected reward sequence starting  from s and then iteratively following P πb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' After d steps, we  apply the score function on the last state as in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='6  (Gradient  of  E-SoftTreeMax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  The  E-SoftTreeMax gradient of dimension A × S is given by  ∇θ log πE  d,θ =  βγd �  IA − 1A(πE  d,θ)⊤� D  �  πE  d,θ  �−1  Es,dD(exp(βγdΘ))  1⊤  AEs,d exp(βγdΘ)  ,  where for brevity, we drop the s index in the policy above,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', πE  d,θ ≡ πE  d,θ(·|s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  This gradient structure is harder to handle than that of  C-SoftTreeMax in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='3, but here we also prove an  exponential variance decay nonetheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='7  (Exponential  variance  decay  of  E-SoftTreeMax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' There exists α  ∈  (0, 1) such that,  for any function Q : S × A → R,  Var  �  ∇θ log πE  d,θ(a|s)Q(s, a)  �  ∈ O  �  β2γ2dα2d�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  If all rewards are equal (r ≡ const), then α = |λ2(P πb)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  The proof structure is similar in spirit to that of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='4,  but several new technical arguments are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We give it  in full in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='4, but briefly outline it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Proof outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Recall that thanks to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1, we can  bound the PG variance using a direct bound on the gradient  norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The definition of the induced norm is  ∥∇θ log πE  d,θ∥ = max  z:∥z∥=1 ∥∇θ log πE  d,θz∥,  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  with ∇θ log πE  d,θ given in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Let z ∈ RS be an  arbitrary vector such that ∥z∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Then, z = �S  i=1 cizi,  where ci are scalar coefficients and zi are vectors spanning  the S-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In the full proof, we show our  specific choice of zi and prove they are linearly independent  given that choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We do note that z1 = 1S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  The first part of the proof relies on the fact that  (∇θ log πE  d,θ)z1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This is easy to verify using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='6  together with (8), and because  �  IA − 1A(πE  d,θ)⊤�  is a pro-  jection matrix whose null-space is spanned by 1S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Thus,  ∇θ log πE  d,θz = ∇θ log πE  d,θ  S  �  i=2  cizi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  In the second part of the proof, we focus on Es,d from (9),  which appears within ∇θ log πE  d,θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Notice that Es,d consists  of the product �d−1  h=1  �  D  �  exp(βγhR  �  P πb�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Even though  the elements in this product are not stochastic matrices, in  the full proof we show how to normalize each of them to a  stochastic matrix Bh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We thus obtain that  Es,d = PsD(M1)  d−1  �  h=1  Bh,  where M1 ∈ RS is some strictly positive vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Then,  we can apply a result by Mathkar & Borkar (2016), which  itself builds on (Chatterjee & Seneta, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The result  states that the product of stochastic matrices �d−1  h=1 Bh of  our particular form converges exponentially fast to a matrix  of the form 1Sµ⊤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' ∥1Sµ⊤ − �d−1  h=1 Bh∥ ≤ Cαd for  some constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Lastly, 1Sµ⊤  πb gets canceled due to our choice of zi, i =  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This observation along with the above fact that the  remainder decays then shows that ∇θ log πE  d,θ  �S  i=2 zi =  O(αd), which gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Although our proof guarantees that α = |λ2(P πb)| only  in the constant-reward case, we conjecture that this is also  true in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' To demonstrate this, we run the  following simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We drew a random finite MDP, pa-  rameter vector Θ ∈ RS  +, and behavior policy πb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We then  analytically computed the PG variance of E-SoftTreeMax  as given in (1) and compared it to |λ2(P πb)|d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' As seen, the  true variance and our bound matched almost identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  This suggests that indeed α = |λ2(P πb)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We repeat this  experiment three times for different P πb : (i) close to uni-  form, (ii) drawn randomly, and (iii) close to a permutation  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The three cases match our takeaways on the opti-  mal and worst-case rate decay cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We ran multiple such  experiments and in all of them the lines match closely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' we  give here one such instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' To account the for constants,  we match the values for the first point in d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  𝑊(𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  $,$ )  logits  for  logits  for  logits  for  𝑎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "$  ($)  Policy  network  𝑎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "$  (()  𝑎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "$  ($)  𝑎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "$  (()  𝑎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "$  ($)  𝑎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "$  (()  𝑎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='")  ($)  𝑎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='")  (#)  𝑎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='")  (()  𝑎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  (%)  𝑎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  (\')  𝑎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  (()  𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='")  𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "$  ($)  𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "$  (#)  𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "$  (()  𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  ($,$)  𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  (#,$)  𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  ($,()  𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  (#,()  𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  ((,$)  𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  ((,()  𝑊(𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  $,( )  𝑊(𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  #,$ )  𝑊(𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  #,( )  𝑊(𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  (,$ )  𝑊(𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' "#  (,( )  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' SoftTreeMax policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Our exhaustive parallel TS ex-  pands all actions at each state up to depth d (= 2 here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The leaf  state of every trajectory is used as input to the policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  The output is then added to the trajectory’s cumulative reward as  described in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', instead of the standard softmax logits, we  add the cumulative discounted reward to the policy network output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  This policy is differentiable and can be easily integrated into any  PG algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In this work, we build on PPO and use its loss  function to train the policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' SoftTreeMax: Deep Parallel  Implementation  Following the success of deep RL (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2015), deep  neural networks are used nowadays almost exclusively in  practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Depending on the RL algorithm, a loss function  is defined and gradients on the network weights can be  calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In PG methods, the scoring function used in the  softmax is commonly replaced by a neural network Wθ:  πθ(a|s) ∝ exp (Wθ(s, a)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Similarly, we implement SoftTreeMax by replacing θ(s) in  (2) with a neural network Wθ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Although both variants of  SoftTreeMax from Section 3 involve computing an expecta-  tion, this can be hard in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' One approach to handle it is  with sampling, though these introduce estimation variance  into the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We leave the question of sample-based  theory and algorithmic implementations for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Instead, in finite action space environments such as Atari,  we compute the exact expectation in SoftTreeMax with an  exhaustive TS of depth d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Despite the exponential computa-  tional cost of spanning the entire tree, recent advancements  in parallel GPU-based simulation allow efficient expansion  of all nodes at the same depth simultaneously (Dalal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Rosenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This is possible when a simu-  lator is implemented on GPU (Dalton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Makoviy-  chuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Freeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2021), or when a forward  model is learned (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Ha & Schmidhuber,  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' To reduce the complexity to be linear in depth, we  apply tree pruning to a limited width in all levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We do so  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  by sampling only the most promising actions at each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  To summarize, in the practical SoftTreeMax algorithm we  perform an exhaustive TS to obtain all trajectories up to  depth d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We expand the tree by exhaustively expanding  all actions, which corresponds to a uniform tree expansion  policy πb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We apply a neural network on the leaf states,  and accumulate the result with the rewards along each tra-  jectory to obtain the logits in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Finally, we aggregate  the results using C-SoftTreeMax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We leave experiments  E-SoftTreeMax for future work on risk-averse RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' During  training, the gradient propagates to the NN weights of Wθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  When the gradient ∇θ log πd,θ is calculated at each time  step, it updates Wθ for all leaf states, similarly to Siamese  networks (Bertinetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' An illustration of the  policy is given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Experiments  We conduct our experiments on multiple games from the  Atari simulation suite (Bellemare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' As a baseline,  we train a PPO (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2017) agent with 256  workers in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In a hyperparameter search, we found  this number of workers to be the best in terms of run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  The environment engine is the highly efficient Atari-CuLE  (Dalton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2020), a CUDA-based version of Atari that  runs on GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Similarly, we use Atari-CuLE for the GPU-  based breadth-first TS as done in (Dalal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We  then train SoftTreeMax for depths d = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' 8, with a single  worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We use five seeds for each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  For the implementation, we extend Stable-Baselines3 (Raf-  fin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2019) with all parameters taken as default from the  original PPO paper (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We will release  the code upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' For depths d ≥ 3, we limited  the tree to a maximum width of 1024 nodes and pruned  non-promising trajectories in terms of estimated weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Since the distributed PPO baseline advances significantly  faster in terms of environment steps, for a fair comparison,  we ran all experiments for one week on the same machine  and use the wall-clock time as the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We use Intel(R)  Xeon(R) CPU E5-2698 v4 @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='20GHz equipped with one  NVIDIA Tesla V100 32GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  In Figure 3, we plot the reward and variance of SoftTreeMax  for each game, as a function of depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The dashed lines are  the results for PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Each value is taken after convergence,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', the average over the last 20% of the run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The numbers  represent the average over five seeds per game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We choose  to exclude the standard deviation to avoid excessive clutter  in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The plot conveys three intriguing conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  First, in all cases, SoftTreeMax achieves significantly higher  reward than PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Its gradient variance is also orders of  magnitude lower than that of PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Second, the reward and  variance are negatively correlated – they mirror each other  in almost all of the games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This phenomenon demonstrates  how crucial it is to lower the variance of PG for improving  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' And specifically, it highlights the benefits of  SoftTreeMax over “flat” PG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The third conclusion is that  each game has a different sweet-spot in terms of optimal TS  depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Recall that we limit the run-time in all experiments  to one week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The deeper the TS, the slower each step and  less steps are finished by the end of the run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This type of  comparison also explains why there is no reason to expect  monotone variance reduction as a function of depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  We also provide the training curves in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' For brevity,  we exclude a few of the depths from the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' As seen, there  is a clear benefit for SoftTreeMax over distributed PPO with  the standard softmax policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In most games, PPO with the  SoftTreeMax policy shows very high sample efficiency: it  achieves higher episodic reward even though it observes  much less episodes, for the same running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Related Work  Our work intersects several fields of the RL literature:  Softmax Operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The softmax policy became a canonical  part of PG to the point where theoretical results of PG focus  specifically on it (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Even though we focus on a tree extension to the softmax  policy, the methodology we propose is general and can be  easily applied to other discrete or continuous parameterized  policies as in (Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Miahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Tree Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Planning with a TS is the process of using a  forward model to consider possible future trajectories and  decide on the best action at the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' One famous such algo-  rithm is Monte-Carlo TS (MCTS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Browne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' 2012) used  in AlphaGo (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2016) and MuZero (Schrittwieser  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Other principal algorithms such as Value Itera-  tion, Policy Iteration and DQN were also shown to give an  improved performance with a tree search extensions (Efroni  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Dalal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Risk Aversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Many works considered an exponential  utility function for risk aversion (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Garcıa  & Fern´andez, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Fei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This utility function  is the same as E-SoftTreeMax formulation from (4), but we  have it directly in the policy instead of the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Reward-free RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We showed that the gradient variance is  minimized when the transitions induced by the behavior  policy πb are uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This is expressed by the second  eigenvalue of the transition matrix P πb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This notion of  uniform exploration is common to the reward-free RL setup  (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Several such works considered the same  second eigenvalue in their analysis (Liu & Brunskill, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Tarbouriech & Lazaric, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Reward and Gradient variance: GPU SoftTreeMax (single worker) vs PPO (256 GPU workers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The blue reward plots  show the average of 50 evaluation episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The red variance plots show the average gradient variance of the corresponding training runs,  averaged over five seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The dashed lines represent the same for PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Note theat the variance y-axis is in log-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The reward and  variance are negatively correlated and mirror each other in almost all games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This demonstrates the necessity to lower the variance of PG  for improving performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We limit the training run-time in all experiments to one week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The deeper the TS, the slower each step and  less steps are finished by the end of the training run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This explains the non-monotone performance and variance as a function of depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Training curves: GPU SoftTreeMax (single worker) vs PPO (256 GPU workers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The plots show average reward and  standard deviation over five seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The x-axis is the wall-clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The runs ended after a maximum of 200M time-steps, and after no  longer than one week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The standard PPO finished in less than one week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The training curves correspond to the evaluation runs in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
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+page_content=' An improved convergence  analysis of stochastic variance-reduced policy gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
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+page_content='  PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
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+page_content=' Mas-  tering atari games with limited data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Advances in Neural  Information Processing Systems, 34:25476–25488, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
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+page_content=', Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' On the  convergence and sample efficiency of variance-reduced  policy gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Advances in Neural Information  Processing Systems, 34:2228–2240, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Proofs  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' PROOF OF LEMMA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1 – BOUND ON THE POLICY GRADIENT VARIANCE  For any parametric policy πθ and function Q : S × A → R,  Var (∇θ log πθ(a|s)Q(s, a)) ≤ max  s,a [Q(s, a)]2 max  s  ||∇θ log πθ(·|s)||2  F ,  where ∇θ log πθ(·|s) ∈ RA×dim(θ) is a matrix whose a-th row is ∇θ log πθ(a|s)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The variance for a parametric policy πθ is given as follows:  Var (∇θ log πθ(a|s)Q(a, s)) =Es∼µπθ ,a∼πθ(·|s)  �  ∇θ log πθ(a|s)⊤∇θ log πθ(a|s)Q(s, a)2�  −  Es∼ρπθ ,a∼πθ(·|s) [∇θ log πθ(a|s)Q(s, a)]⊤ Es∼µπθ ,a∼πθ(·|s) [∇θ log πθ(a|s)Q(s, a)] ,  where Q(s, a) is the currently estimated Q-function and µπθ is the stationary distribution induced by following the policy  πθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Since the second term we subtract is always positive (it is of quadratic form v⊤v) we can bound the variance by the first  term:  Var (∇θ log πθ(a|s)Q(a, s)) ≤Es∼µπθ ,a∼πθ(·|s)  �  ∇θ log πθ(a|s)⊤∇θ log πθ(a|s)Q(s, a)2�  =  �  s  µπθ(s)  �  a  πθ(a|s)∇θ log πθ(a|s)⊤∇θ log πθ(a|s)Q(s, a)2  ≤ max  s,a  �  [Q(s, a)]2 πθ(a|s)  � �  s  µπθ(s)  �  a  ∇θ log πθ(a|s)⊤∇θ log πθ(a|s)  ≤ max  s,a [Q(s, a)]2 max  s  �  a  ∇θ log πθ(a|s)⊤∇θ log πθ(a|s)  = max  s,a [Q(s, a)]2 max  s  ||∇θ log πθ(·|s)||2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' PROOF OF LEMMA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='2 – VECTOR FORM OF C-SOFTTREEMAX  In vector form, (3) is given by  πC  d,θ(·|s) =  exp  �  β  �  Cs,d + γdPs (P πb)d−1 Θ  ��  1⊤  A exp  �  β  �  Cs,d + γdPs (P πb)d−1 Θ  ��,  (10)  where  Cs,d = Rs + Ps  �d−1  �  h=1  γh (P πb)h−1  �  Rπb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (11)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Consider the vector ℓs,· ∈ R|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Its expectation satisfies  Eπbℓs,·(d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' θ) = Eπb  �d−1  �  t=0  γtrt + γdθ(sd)  �  = Rs +  d−1  �  t=1  γtPs(P πb)t−1Rπb + γdPs(P πb)d−1Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  As required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' PROOF OF LEMMA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='3 – GRADIENT OF C-SOFTTREEMAX  The C-SoftTreeMax gradient of dimension A × S is given by  ∇θ log πC  d,θ = βγd �  IA − 1A(πC  d,θ)⊤�  Ps (P πb)d−1 ,  where for brevity, we drop the s index in the policy above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', πC  d,θ ≡ πC  d,θ(·|s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The (j, k)-th entry of ∇θ log πC  d,θ satisifes  [∇θ log πC  d,θ]j,k =  ∂ log(πC  d,θ(aj|s))  ∂θ(sk)  = βγd[Ps(P πb)d−1]j,k −  �  a  �  exp  �  β  �  Cs,d + γdPs (P πb)d−1 Θ  ���  a βγd �  Ps(P πb)d−1�  a,k  1⊤  A exp  �  β  �  Cs,d + γdPs (P πb)d−1 Θ  ��  = βγd[Ps(P πb)d−1]j,k − βγd �  a  πC  d,θ(a|s)  �  Ps(P πb)d−1�  a,k  = βγd[Ps(P πb)d−1]j,k − βγd �  (πC  d,θ)⊤Ps(P πb)d−1�  k  = βγd[Ps(P πb)d−1]j,k − βγd �  1A(πC  d,θ)⊤Ps(P πb)d−1�  j,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Now, moving back to matrix form, we obtain the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' PROOF OF THEOREM 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='4 – EXPONENTIAL VARIANCE DECAY OF C-SOFTTREEMAX  The C-SoftTreeMax policy gradient is bounded by  Var  �  ∇θ log πC  d,θ(a|s)Q(s, a)  �  ≤ 2 A2S2β2  (1 − γ)2 γ2d|λ2(P πb)|2(d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1 directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' First of all, it is know that when the reward is bounded in [0, 1], the maximal value of  the Q-function is  1  1−γ as the sum as infinite discounted rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Next, we bound the Frobenius norm of the term achieved in  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='3, by applying the eigen-decomposition on P πb:  P πb = 1Sµ⊤ +  S  �  i=2  λiuiv⊤  i ,  (12)  where µ is the stationary distribution of P πb, and ui and vi are left and right eigenvectors correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  ||βγd �  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤�  Ps(P πb)d−1||F = βγd||  �  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤�  Ps  �  1Sµ⊤ +  S  �  i=2  λd−1  i  uiv⊤  i  �  ||F  (Ps is stochastic)  = βγd||  �  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤�  �  1Aµ⊤ +  S  �  i=2  λd−1  i  Psuiv⊤  i  �  ||F  (projection nullifies 1Aµ⊤)  = βγd||  �  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤�  � S  �  i=2  λd−1  i  Psuiv⊤  i  �  ||F  (triangle inequality)  ≤ βγd  S  �  i=2  ||  �  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤� �  λd−1  i  Psuiv⊤  i  �  ||F  (matrix norm sub-multiplicativity)  ≤ βγd|λd−1  2  |  S  �  i=2  ||IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤||F ||Ps||F ||uiv⊤  i ||F  = βγd|λd−1  2  |(S − 1)||IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤||F ||Ps||F  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  Now we can bound the norm ||IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤||F by direct calculation:  ||IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤||2  F = Tr  ��  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤� �  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤�⊤�  (13)  = Tr  �  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤ − π1⊤  A + π⊤π1A1⊤  A  �  (14)  = A − 1 − 1 + Aπ⊤π  (15)  ≤ 2A  (16)  And from the Cauchy-Schwartz inequality:  ||Ps||2  F =  �  a  �  s  [[Ps]a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='s]2 =  �  a  ||[Ps]a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='·||2  2 ≤  �  a  ||[Ps]a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='·||1||[Ps]a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='·||∞ ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  So:  Var  �  ∇θ log πC  d,θ(a|s)Q(s, a)  �  ≤ max  s,a [Q(s, a)]2 max  s  ||∇θ log πC  d,θ(·|s)||2  F  ≤  1  (1 − γ)2 ||βγd �  IA,A − 1Aπ⊤�  Ps(P πb)d−1||2  F  ≤  1  (1 − γ)2 β2γ2d|λ2(P πb)|2(d−1)S2(2A2)  Which obtains the desired bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' A LOWER BOUND ON C-SOFTTREEMAX GRADIENT (RESULT NOT IN THE PAPER)  For completeness we also supply a lower bound on the Frobenius norm of the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Note that this result does not  translate to the a lower bound on the variance since we have no lower bound equivalence of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The Frobenius norm on the gradient of the policy is lower-bounded by:  ||∇θ log πC  d,θ(·|s)||F ≥ C · βγd|λ2(P πb)|(d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (17)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We begin by moving to the induced l2 norm by norm-equivalence:  ||βγd �  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤�  Ps(P πb)d−1||F ≥ ||βγd �  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤�  Ps(P πb)d−1||2  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' taking the vector u to be the eigenvector of the second eigenvalue of P πb:  ||βγd �  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤�  Ps(P πb)d−1||2 ≥ ||βγd �  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤�  Ps(P πb)d−1u||2  = βγd||  �  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤�  Psu||2  = βγd|λ2(P πb)|(d−1)||  �  IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='A − 1Aπ⊤�  Psu||2  Note that even though Psu can be 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' that is not the common case since we can freely change πb (and therefore the  eigenvectors of P πb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' PROOF OF LEMMA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='5 – VECTOR FORM OF E-SOFTTREEMAX  For d ≥ 1, (4) is given by  πE  d,θ(·|s) =  Es,d exp(βγdΘ)  1⊤  AEs,d exp(βγdΘ),  (18)  where  Es,d = Ps  d−1  �  h=1  �  D  �  exp[βγhR]  �  P πb�  (19)  with R being the |S|-dimensional vector whose s-th coordinate is r(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Recall that  ℓs,a(d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' θ) = r(s) +  d−1  �  t=1  γtr(st) + γdθ(sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (20)  and, hence,  exp[βℓs,a(d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' θ)] = exp  �  β  �  r(s) +  d−1  �  t=1  γtr(st) + γdθ(sd)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (21)  Therefore,  E[exp βℓs,a(d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' θ)] = E  �  exp  �  β  �  r(s) +  d−1  �  t=1  γtr(st)  ��  E  �  exp  �  β  �  γdθ(sd)  ����s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , sd−1  �  �  (22)  = E  �  exp  �  β  �  r(s) +  d−1  �  t=1  γtr(st)  ��  P πb(·|sd−1)  �  exp(βγdΘ)  (23)  = E  �  exp  �  β  �  r(s) +  d−2  �  t=1  γtr(st)  ��  exp[βγd−1r(sd−1)]P πb(·|sd−1)  �  exp(βγdΘ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (24)  By repeatedly using iterative conditioning as above, the desired result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Note that exp(βr(s)) does not depend on the  action and is therefore cancelled out with the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' PROOF OF LEMMA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='6 – GRADIENT OF E-SOFTTREEMAX  The E-SoftTreeMax gradient of dimension A × S is given by  ∇θ log πE  d,θ = βγd �  IA − 1A(πE  d,θ)⊤� D  �  πE  d,θ  �−1  Es,dD(exp(βγdΘ))  1⊤  AEs,d exp(βγdΘ)  ,  where for brevity, we drop the s index in the policy above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', πE  d,θ ≡ πE  d,θ(·|s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The (j, k)-th entry of ∇θ log πE  d,θ satisfies  [∇θ log πE  d,θ]j,k =  ∂ log(πE  d,θ(aj|s))  ∂θ(sk)  =  ∂  ∂θ(sk)  �  log[(Es,d)⊤  j exp(βγdΘ)] − log[1⊤  AEs,d exp(βγdΘ)]  �  = βγd(Es,d)j,k exp(βγdθ(sk))  (Es,d)⊤  j exp(βγdΘ)  − βγd1⊤  AEs,dek exp(βγdθ(sk))  1⊤  AEs,d exp(βγdΘ)  = βγd(Es,dek exp(βγdθ(sk)))j  (Es,d)⊤  j exp(βγdΘ)  − βγd1⊤  AEs,dek exp(βγdθ(sk))  1⊤  AEs,d exp(βγdΘ)  = βγd  �  e⊤  j  e⊤  j Es,d exp(βγdΘ) −  1⊤  A  1⊤  AEs,d exp(βγdΘ)  �  Es,dek exp(βγdθ(sk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Hence,  [∇θ log πE  d,θ]·,k = βγd �  D(Es,d exp(βγdΘ))−1 − (1⊤  AEs,d exp(βγdΘ))−11A1⊤  A  �  Es,dek exp(βγdθ(sk))  From this, it follows that  ∇θ log πE  d,θ = βγd �  D  �  πE  d,θ  �−1 − 1A1⊤  A  � Es,dD(exp(βγdΘ))  1⊤  AEs,d exp(βγdΘ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (25)  The desired result is now easy to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' PROOF OF THEOREM 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='7 — EXPONENTIAL VARIANCE DECAY OF E-SOFTTREEMAX  There exists α ∈ (0, 1) such that, for any function Q : S × A → R,  Var  �  ∇θ log πE  d,θ(a|s)Q(s, a)  �  ∈ O  �  β2γ2dα2d�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  If all rewards are equal (r ≡ const), then α = |λ2(P πb)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Let d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Recall that  Es,d = Ps  d−1  �  h=1  �  D  �  exp[βγhR]  �  P πb�  ,  (26)  and that R refers to the S-dimensional vector whose s-th coordinate is r(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Define  Bi =  �  P πb  if i = d − 1,  D−1(P πbMi+1)P πbD(Mi+1)  if i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , d − 2,  (27)  and the vector  Mi =  �  exp(βγd−1R)  if i = d,  exp(βγiR) ◦ P πbMi+1  if i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , d − 2,  (28)  where ◦ denotes the element-wise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Then,  Es,d = PsD(M1)  d−1  �  i=1  Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (29)  It is easy to see that each Bi is a row-stochastic matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=', all entries are non-negative and Bi1S = 1S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Next, we prove that all non-zeros entries of Bi are bounded away from 0 by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This is necessary to apply the next  result from (Chatterjee & Seneta, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' The j-th coordinate of Mi satisfies  (Mi)j = exp[βγiRj]  �  k  [P πb]j,k(Mi+1)k ≤ ∥ exp[βγiR]∥∞∥Mi+1∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (30)  Separately, observe that ∥Md−1∥∞ ≤ ∥ exp(βγd−1R)∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Plugging these relations in (28) gives  ∥M1∥∞ ≤  d−1  �  h=1  ∥ exp[βγhR]∥∞ =  d−1  �  h=1  ∥ exp[βR]∥γh  ∞ = ∥ exp[βR]∥  �d−1  h=1 γh  ∞  ≤ ∥ exp[βR]∥  1  1−γ  ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (31)  Similarly, for every 1 ≤ i ≤ d − 1, we have that  ∥Mi∥∞ ≤  d−1  �  h=i  ∥ exp[βR]∥γh  ∞ ≤ ∥ exp[βR]∥  1  1−γ  ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (32)  The jk-th entry of Bi = D−1(P πbMi+1)P πbD(Mi+1) is  (Bi)jk =  P πb  jk [Mi+1]k  �|S|  ℓ=1 P πb  jℓ [Mi+1]ℓ  ≥  P πb  jk  �|S|  ℓ=1 P πb  jℓ [Mi+1]ℓ  ≥  P πb  jk  ∥ exp[βR]∥  1  1−γ  ∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (33)  Hence, for non-zero P πb  jk , the entries are bounded away from zero by the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We can now proceed with applying the  following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Now, by (Chatterjee & Seneta, 1977, Theorem 5) (see also (14) in (Mathkar & Borkar, 2016)), limd→∞  �d−1  i=1 Bi exists and  is of the form 1Sµ⊤ for some probability vector µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Furthermore, there is some α ∈ (0, 1) such that ε(d) :=  ��d−1  i=1 Bi  �  −  1S µ⊤ satisfies  ∥ε(d)∥ = O(αd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (34)  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  Pick linearly independent vectors w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , wS such that  µ⊤wi = 0 for i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (35)  Since �S  i=2 αiwi is perpendicular to µ for any α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' αS and because µ⊤ exp(βγdΘ) > 0, there exists no choice of  α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , αS such that �S  i=2 αiwi = exp(βγdΘ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Hence, if we let z1 = 1S and zi = D(exp(βγdΘ))−1wi for i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , S,  then it follows that {z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , zS} is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' In particular, it implies that {z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , zS} spans RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Now consider an arbitrary unit norm vector z := �S  i=1 cizi ∈ RS s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' ∥z∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  ∇θ log πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θz = ∇θ log πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ  S  �  i=2  cizi  (36)  = βγd �  IA − 1A(πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ)⊤� D  �  πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ  �−1  Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='dD(exp(βγdΘ))  1⊤  AEs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='d exp(βγdΘ)  S  �  i=2  cizi  (37)  = βγd �  IA − 1A(πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ)⊤� D  �  πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ  �−1  Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='d  1⊤  AEs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='d exp(βγdΘ)  S  �  i=2  ciwi  (38)  = βγd �  IA − 1A(πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ)⊤� D  �  πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ  �−1 �  1Sµ⊤ + ε(d)  �  1⊤  AEs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='d exp(βγdΘ)  S  �  i=2  ciwi  (39)  = βγd �  IA − 1A(πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ)⊤� D  �  πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ  �−1  ε(d)  1⊤  AEs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='d exp(βγdΘ)  S  �  i=2  ciwi  (40)  = βγd �  IA − 1A(πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ)⊤� D  �  πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ  �−1  ε(d)D(exp(βγdΘ))  1⊤  AEs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='d exp(βγdΘ)  (z − c11S),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (41)  where (36) follows from the fact that ∇θ log πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θz1 = ∇θ log πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ1S = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' (37) follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='6, (38) holds  since zi = D(exp(βγdΘ))−1wi, (40) because µ is perpendicular wi for each i, while (41) follows by reusing zi =  D(exp(βγdΘ))−1wi relation along with the fact that z1 = 1S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  From (41),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' it follows that  ∥∇θ log πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θz∥ ≤ βγd∥ε(d)∥  �������  �  IA − 1A(πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ)⊤�  D  �  πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ  �−1  1⊤  AEs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='d exp(βγdΘ)  �������  ∥D(exp(βγdΘ))∥ ∥z − c11S∥  (42)  ≤ βγdαd(∥IA∥ + ∥1A(πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ)⊤∥)  �������  D  �  πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ  �−1  1⊤  AEs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='d exp(βγdΘ)  �������  exp(βγd max  s  θ(s))∥z − c11S∥  (43)  ≤ βγdαd(1 +  √  A)  �������  D  �  πE  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='θ  �−1  1⊤  AEs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='d exp(βγdΘ)  �������  exp(βγd max  s  θ(s))∥z − c11S∥  (44)  ≤ βγdαd(1 +  √  A)  ��D−1(Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='d exp(βγdΘ))  �� exp(βγd max  s  θ(s))∥z − c11S∥  (45)  ≤ βγdαd(1 +  √  A)  1  mins[Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='d exp(βγdΘ]s  exp(βγd max  s  θ(s))∥z − c11S∥  (46)  ≤ βγdαd(1 +  √  A)  exp(βγd maxs θ(s))  exp(βγd mins θ(s)) mins |M1|∥z − c11S∥  (47)  ≤ βγdαd(1 +  √  A)  exp(βγd maxs θ(s))  exp(βγd mins θ(s)) exp(β mins r(s))∥z − c11S∥  (48)  ≤ βγdαd(1 +  √  A) exp(β[max  s  θ(s) − min  s  θ(s) − min  s  r(s)])∥z − c11S∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (49)  Lastly, we prove that ∥z−c11S∥ is bounded independently of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' First, denote by c = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , cS)⊤ and ˜c = (0, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , cS)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Also, denote by Z the matrix with zi as its i-th column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Now,  ∥z − c11S∥ = ∥  S  �  i=2  cizi∥  (50)  = ∥Z˜c∥  (51)  ≤ ∥Z∥∥˜c∥  (52)  ≤ ∥Z∥∥c∥  (53)  = ∥Z∥∥Z−1z∥  (54)  ≤ ∥Z∥∥Z−1∥,  (55)  where the last relation is due to z being a unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' All matrix norms here are l2-induced norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  Next, denote by W the matrix with wi in its i-th column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Recall that in (35) we only defined w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , wS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We now set  w1 = exp(βγdΘ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Note that w1 is linearly independent of {w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' , wS} because of (35) together with the fact that  µ⊤w1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' We can now express the relation between Z and W by Z = D−1(exp(βγdΘ))W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Substituting this in (55), we  have  ∥z − c11S∥ ≤ ∥D−1(exp(βγdΘ))W∥∥W −1D(exp(βγdΘ))∥  (56)  ≤ ∥W∥∥W −1∥∥D(exp(βγdΘ))∥∥D−1(exp(βγdΘ))∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (57)  It further holds that  ∥D(exp(βγdΘ))∥ ≤ max  s  exp  �  βγdθ(s)  �  ≤ max{1, exp[β max  s  θ(s)])},  (58)  where the last relation equals 1 if θ(s) < 0 for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' Similarly,  ∥D−1(exp(βγdΘ))∥ ≤  1  mins exp (βγdθ(s)) ≤  1  min{1, exp[β mins θ(s)])}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (59)  SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search  Furthermore, by the properties of the l2-induced norm,  ∥W∥2 ≤  √  S∥W∥1  (60)  =  √  S max  1≤i≤S ∥wi∥1  (61)  =  √  S max{exp(βγdΘ), max  2≤i≤S ∥wi∥1}  (62)  ≤  √  S max{1, exp[β max  s  θ(s)], max  2≤i≤S ∥wi∥1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (63)  Lastly,  ∥W −1∥ =  1  σmin(W)  (64)  ≤  �S−1  �  i=1  σmax(W)  σi(W)  �  1  σmin(W)  (65)  = (σmax(W))S−1  �S  i=1 σi(W)  (66)  = ∥W∥S−1  | det(W)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (67)  The determinant of W is a sum of products involving its entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' To upper bound (67) independently of d, we lower bound  its determinant by upper and lower bounds on the entries [W]i,1 that are independent of d, depending on their sign:  min{1, exp[β min  s  θ(s)])} ≤ [W]i,1 ≤ max{1, exp[β max  s  θ(s)])}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content='  (68)  Using this, together with (55), (57), (58), (59), and (63), we showed that ∥z − c11S∥ is upper bounded by a constant  independent of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}
+page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQfEDVH/content/2301.13236v1.pdf'}