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| "text": "Graph Signal Sampling via Reinforcement Learning Oleksii Abramenko and Alexander Jung\nDepartment of Computer Science, Aalto University, Finland\nfirstname.lastname@aalto.fi Abstract—We formulate the problem of sampling and recov- Notation. The vector with all entries equal to zero is\nering clustered graph signal as a multi-armed bandit (MAB) denoted 0. Given a vector x with non-negative entries, we\nproblem. This formulation lends naturally to learning sampling denote by √x the vector whose entries are the square roots of\nstrategies using the well-known gradient MAB algorithm. Similarly, we denote the element-wise square particular, the sampling strategy is represented as a probability\ndistribution over the individual arms of the MAB and optimized of the vector as x2.\nusing gradient ascent. Some illustrative numerical experiments Outline. In Section II we formulate the problem of recov-2018 indicate that the sampling strategies based on the gradient MAB ering a clustered graph signal from its values on few nodes\nalgorithm outperform existing sampling methods. forming a sampling set as a convex optimization problem. Our\nmain contribution is in Section III where we introduce our Index Terms—machine learning, reinforcement learning, multiarmed bandit, graph signal processing, total variation, complex RL-based sampling method. The results of some numericalMay\nnetworks. experiments are presented in Section IV. We discuss our\n15 findings in Section V and finally conclude in Section VI.", |
| "paper_id": "1805.05827", |
| "title": "Graph Signal Sampling via Reinforcement Learning", |
| "authors": [ |
| "Oleksii Abramenko", |
| "Alexander Jung" |
| ], |
| "published_date": "2018-05-15", |
| "primary_category": "stat.ML", |
| "arxiv_url": "http://arxiv.org/abs/1805.05827v1", |
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| "text": "PROBLEM FORMULATION\nModern information processing systems generate massive We consider datasets which are represented by a data graph\ndatasets which are often strongly heterogeneous, e.g., par- G = (V, E). The data graph is an undirected connected\ntially labeled mixtures of different media (audio, video, text). graph (no self-loops and no multi-edges) with nodes V =\nA quite successful approach to such datasets is based on {1, . . . , N}, which are connected by edges {i, j} ∈E. Each\nrepresenting the data as networks or graphs. In particular, node i ∈V represents an individual data point and an edge[stat.ML] we represent datasets by graph signals defined over an un- {i, j} ∈E connects nodes representing similar data points.\nderlying graph, which reflects similarities between individual The distance dist(i, j) between two different nodes i, j ∈V\ndata points. The graph signal values encode label information is defined as the length of the shortest path between those\nwhich often conforms to a clustering hypothesis, i.e., the signal nodes. For a given node i ∈V, we define its neighbourhood\nvalues (labels) of close-by nodes (similar data points) are as\nsimilar. N(i) := {j ∈V : {i, j} ∈E}. Two core problems considered within graph signal processing (GSP) are (i) how to sample them, i.e., which signal values It will be handy to generalize the notion of neighbourhood\nprovide the most information about the entire dataset, and (ii) and define, for some r ∈N, the r-step neighbourhood of a\nhow to recover the entire graph signal from these few signal node i ∈V as N(i, r) := {j ∈V : dist(i, j) = r}. The 1-step\nvalues (samples). These problems have been studied in [1]–[6] neighbourhood coincides with the neighbourhood of a node,\nwhich discussed convex optimization methods for recovering a i.e., N(i, 1) = N(i).\ngraph signal from a small number of signal values observed on In many applications we can associate each data point i ∈V\nthe nodes belonging to a given (small) sampling set. Sufficient with a label x[i] ∈R. These labels induce a graph signal\nin [4], [7]. its values x[i] only for nodes i belonging to a sampling set\nContribution. We propose a novel approach to the graph\nM := {i1, . . . , iM} ⊆V.\nsignal sampling and recovery it by interpreting it as a reinforcement learning (RL) problem. In particular, we interpret Since acquiring signal values (i.e., labelling data points) is\nonline sampling algorithm as an artificial intelligence agent often expensive (requiring manual labor), the sampling set is\nwhich chooses the nodes to be sampled on-the-fly. The be- typically much smaller than the overall dataset, i.e., M =\nhavior of the sampling agent is represented by a probability |M| ≪N.", |
| "paper_id": "1805.05827", |
| "title": "Graph Signal Sampling via Reinforcement Learning", |
| "authors": [ |
| "Oleksii Abramenko", |
| "Alexander Jung" |
| ], |
| "published_date": "2018-05-15", |
| "primary_category": "stat.ML", |
| "arxiv_url": "http://arxiv.org/abs/1805.05827v1", |
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| "text": "For a fixed sampling set size (sampling budget)\ndistribution (\"policy\") over a discrete set of different actions M it is important to choose the sampling set such that the\nwhich are at the disposal of the sampling agent in order to signal samples {x[i]}i∈M carry maximal information about\nchoose the next node at which the graph signal is sampled. the entire graph signal. The ultimate goal is to learn a sampling policy which chooses The recovery of the entire graph signal from (few) signal\nsignal samples that allow for a small reconstruction error. samples {x[i]}i∈M is possible for clustered graph signals which do not vary too much over well-connected subset of N(it, 1) N(it, 2) N(it, 3)\nnodes (clusters) (cf. [4], [8]). We will quantify how well a\ngraph signal is aligned to the cluster structure using the total\nvariation (TV)\n∥x∥TV := X |x[j]−x[i]|. Recovering a graph signal based on the signal values x[i] for\nthe nodes i ∈M of the sampling set M can be accomplished\nby solving ˆxM ∈arg min ∥˜x∥TV s.t. ˜x[i]=x[i] for all i∈M. (1)\n˜x Fig. 1: The filled node represents the current location it of\nThis is a convex optimization problem with a non- the sampling agent at time t. We also indicate the 1-, 2- and\ndifferentiable objective function which precludes the use of 3-step neighbourhoods.\nsimple gradient descent methods. However, by applying the\nThe problem of optimally selecting actions at given time canprimal-dual method of Pock and Chambolle [9] to solve the\nbe formulated as a MAB problem. Each arm of the bandit isrecovery problem (1), an efficient sparse label propagation\nassociated with an action. In our setup, a sampling strategy (oralgorithm has been obtained in [8].\npolicy) amounts to specifying a probability distribution over\nA simple but useful model for clustered graph signals is:\nthe individual actions a ∈A. We parametrize this probability\nx = X aCtC, (2) distribution with a weight vector w = (w1, . . . , wH) ∈RH\nC∈F using the softmax rule:\nwith the cluster indicator signals ewa\nπ(w)(a) =\nP ewb ( 1, if i ∈C b∈A\ntC[i] =\n0 else. The weight vector w is tuned in the episodic manner with\nThe partition F underlying the signal model (2) can be chosen each episode amounting to selecting sampling set M based on\nthe policy π(w). At each timestep t the agent randomly drawsarbitrarily in principle.", |
| "paper_id": "1805.05827", |
| "title": "Graph Signal Sampling via Reinforcement Learning", |
| "authors": [ |
| "Oleksii Abramenko", |
| "Alexander Jung" |
| ], |
| "published_date": "2018-05-15", |
| "primary_category": "stat.ML", |
| "arxiv_url": "http://arxiv.org/abs/1805.05827v1", |
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| "text": "However, our methods are expected to\nan action at according to the distribution π(w) and performsbe most useful if the partition matches the intrinsic cluster\nstructure of the data graph G. The clustered graph signals transition to the next node it+1 which is selected uniformly\nat random from the at-step neighbourhood N(it, at). As wasof the form (2) conform with the network topology, in the\nsense of having small TV ∥x∥TV, if the underlying partition mentioned earlier, the node tt+1 is added to the sampling set,\nF = {C1, . . . , C|F|} consists of disjoint clusters Cl with i.e., M := M ∪{it+1}. We also record the action at and add\nit to the action list, i.e., L := L∪{at}.", |
| "paper_id": "1805.05827", |
| "title": "Graph Signal Sampling via Reinforcement Learning", |
| "authors": [ |
| "Oleksii Abramenko", |
| "Alexander Jung" |
| ], |
| "published_date": "2018-05-15", |
| "primary_category": "stat.ML", |
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| "text": "The process continuessmall cut-sizes. Relying on the clustered signal model (2),\nuntil we obtain a sampling set M with the prescribed size[7, Thm. 3] presents a sufficient condition on the choice of\nsampling set such that the solution ˆx of (1) coincides with (sampling budget) M. Our goal is to learn an optimal policy π(w) for the samplingthe true underlying clustered graph signal of the form (2).\nagent in order to obtain signal samples which allow recoveryThe condition presented in [7, Thm. 3] suggests to choose\nof the entire graph signal with minimum error. We assess thethe nodes in the sampling set preferably near the boundaries\nquality of the policy using the mean squared error (MSE)between the different clusters.\nincurred by the recovered signal ˆxM which is obtained via\n(1) using the sampling set M by following policy π(w):\nIII. SIGNAL SAMPLING AS REINFORCEMENT LEARNING\n−1 X (x[j] −ˆxM[j])2 The problem of selecting the sampling set M and recover- R := N\ning the entire graph signal x from the signal values x[i] can be j∈V\ninterpreted as a RL problem. Indeed, we consider the selection The obtained reward is associated with all actions/arms\nof the nodes to be sampled being carried out by an \"agent\" which contributed to picking samples into sampling set during\nwhich crawls over the data graph G. The set of actions our the episode. For example, if the sampling set has been obtained\nsampling agent may take is A = {1, . . . , H}. by pulling arms 1, 2 and 5, the obtained reward will be\nA specific action a ∈A refers to the number of hops the associated with all these arms, because we do not know what is\nsampling agent performs starting at the current node it to reach the exact contribution of the specific arm to the finally obtained\na new node it+1, which will be added to the sampling set, i.e., MSE. In particular, the new node it+1 is selected The key idea behind gradient MAB is to update weights w\nuniformly at random among the nodes which belong to its a- so that actions yielding higher rewards become more probable\nstep neighbourhood N(it, a) (see Figure 1). under π(w) [10, Chapter 2.8].", |
| "paper_id": "1805.05827", |
| "title": "Graph Signal Sampling via Reinforcement Learning", |
| "authors": [ |
| "Oleksii Abramenko", |
| "Alexander Jung" |
| ], |
| "published_date": "2018-05-15", |
| "primary_category": "stat.ML", |
| "arxiv_url": "http://arxiv.org/abs/1805.05827v1", |
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| "text": "According to the aforementioned of pseudocode (see Algorithm 1). Algorithm 1 Online Sampling and Reconstruction Input: data graph G, sampling budget M, batch size B, α\nInitialize: w := 0, M = {∅}, L = {∅}, ∇w = 0, g =\n0, ep = 0\n(a) (b)\n1: repeat\nFig. 2: Illustration of reward being conditioned on the position 2: select starting node i ∈V randomly\nof a sampling agent. In this picture: red node – current position 3: M := {i}\nof the sampling agent, blue region – nodes within distance 1 4: L = {∅}\nfrom the sampling agent. Node indices are shown inside the 5: for t := 1; t < M do\nnodes, signal values – outside. 6: a := SAMPLEACTION(π(w))\n7: inext := SAMPLENODE(G, N(i, a))\nbook weights update can be accomplished using gradient 8: M := M ∪{inext}\nascent algorithm: 9: L := APPENDTOLIST(L, a)\n( wa + αR(1 −π(a)), a = ak 10: i := inext\nwa := 11: end for\nwa −αRπ(a), ∀a ̸= ak (3)\n12: ˆx∈arg min ∥˜x∥TV\nfor k = 1..M −1, a ∈A, ak ∈L ˜x\n13: s.t. ˜x[i]=x[i] for all i∈M\nThe single difference between update rule (3) and one 14: R := −1N P (x[j] −ˆx[j])2\npresented in the book [10, Eq. 2.10] is that in our case j∈V\n15: for k := 1; k < M doweights update is performed in the end of each episode and\n16: for a := 1; a ⩽H donot after an arm pull. That is because we do not know\n(reward immediately after pulling an arm and should wait until ∇wa + R(1 −π(a)), if a = L[k] 17: ∇wa :=\nthe whole sampling set is collected and reward is observed. ∇wa −Rπ(a), otherwise\nThe intuition behind the update equation (3) is that for each 18: end for\narm which has participated in picking a node into sampling 19: end for\nset (a = at), the weight is increased, whereas weights of 20: ep := ep + 1\nremaining arms (∀a ̸= at) are decreased. In both cases 21: if ep mod B = 0 then\ndegree of weight increase/decrease is scaled by the reward 22: g := 0.9g + 0.1(∇w)2\nobtained with help of this arm as well as by the learning rate 23: w := w + α∇w/√g\nα. For faster convergence in our implementation, instead of 24: ∇w := 0\nstochastic gradient ascent we use mini-batch gradient ascent 25: end if\nin combination with RMSprop technique [11] (see Algorithm 26: until convergence is reached\n1 for implementation details). Output: π(w)\nChoice of the gradient MAB algorithm can be additionally\njustified by the study [12] which shows that in the environ- Obtained probability distribution π(w) represents sampling\nments with non-stationary rewards probabilistic MAB policy strategy which incurs the minimum reconstruction MSE when\ncan result in higher expected reward in comparison to single- using the convex recovery method (1).\nbest action policies. In our problem non-stationarity of reward\narises from the graph structure itself, i.e., reward distribution IV.", |
| "paper_id": "1805.05827", |
| "title": "Graph Signal Sampling via Reinforcement Learning", |
| "authors": [ |
| "Oleksii Abramenko", |
| "Alexander Jung" |
| ], |
| "published_date": "2018-05-15", |
| "primary_category": "stat.ML", |
| "arxiv_url": "http://arxiv.org/abs/1805.05827v1", |
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| "text": "NUMERICAL RESULTS\nfor a particular arm of a bandit depends on the location of\nthe sampling agent. Suppose sampling budget M is 2 and We now verify the effectiveness of the proposed sampling\nconsider example presented in Figure 2. In case (a) sampling set selection algorithm using synthetic data and compare it\nagent is initially located at node 4. By pulling arm #1 it can to two other existing approaches, i.e., random walk sampling\nonly pick node 3 which is in the other cluster. It is easy to (RWS) [13] and uniform random sampling (URS) [14, Section\nverify that by using recovery method (1) graph signal will be 2.3]. We define a random graph with 10 clusters where sizes\nperfectly reconstructed (MSE = 0). On the other hand, case of clusters are drawn from the geometric distribution with\n(b) shows the situation when the agent can only pick nodes 2 probability of success 8/100. In accordance to the stochastic\nor 3 belonging to the same cluster as currently sampled node, block model (SBM) [15] intra- and inter-cluster connection\nleading to non-zero reconstruction MSE. probabilities are parametrized as p = 7/10 and q = 1/100. The whole process of weight updates is repeated for suffi- We then generate a clustered graph signal according to (2) with\ncient number of episodes until convergence is reached and the signal coefficients aCl = l for l = 1, 2, ..., 10. Example of\nthe optimal stochastic policy is attained. Described above a typical instance of random graph with such parameters is\nlearning procedure can be efficiently summarized in the form shown in Figure 3. Fig. 3: Data graph obtained from the stochastic block model Fig. 5: Mean policy for the stochastic block model family G.\nwith p = 7/10 and q = 1/100. Fig. 6: Test set error obtained from graph signal recovery based\nFig. 4: Convergence of gradient MAB for one learning instance on different sampling strategies.", |
| "paper_id": "1805.05827", |
| "title": "Graph Signal Sampling via Reinforcement Learning", |
| "authors": [ |
| "Oleksii Abramenko", |
| "Alexander Jung" |
| ], |
| "published_date": "2018-05-15", |
| "primary_category": "stat.ML", |
| "arxiv_url": "http://arxiv.org/abs/1805.05827v1", |
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| "text": "Gi (showing first 3700 episodes). 1 Given the model we generate training data consisting of NMSE = X NMSEGi\nK = 500 random graphs and for each graph instance we 500 i=1\nrun Algorithm 1 for 10000 episodes, which is sufficient to\nWe perform similar measurements of the NMSE for ranreach convergence. It is interesting to note that the algorithm\ndom walk and random sampling algorithms under different\noutperforms RWS and URS strategies after 200 and 800\nsampling budgets and convert results to the logarithmic scale.\nepisodes respectively (see Figure 4). Convergence speed is\nThe Figure 6 shows that for relative sampling budget 0.2\nhigh at the initial stage and then substantially decreases after\nimprovement in NMSE amounts to 5 dBs in comparison to\napproximately 1000 episodes.\nrandom sampling and 10 dBs in comparison to random walk\nIn Figure 5 we illustrate the mean policy\napproach. This gap increases even more for the sampling\nK budget 0.4, to 8 dBs and 20 dBs respectively. The general\nπ(w) = X π(w)i (4) tendency suggests further increase of the gap for larger sam- K i=1 pling budgets. The finally obtained policy (4) is then evaluated by applying V.", |
| "paper_id": "1805.05827", |
| "title": "Graph Signal Sampling via Reinforcement Learning", |
| "authors": [ |
| "Oleksii Abramenko", |
| "Alexander Jung" |
| ], |
| "published_date": "2018-05-15", |
| "primary_category": "stat.ML", |
| "arxiv_url": "http://arxiv.org/abs/1805.05827v1", |
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| "text": "DISCUSSION\nit to 500 new i.i.d. realizations of the data graph, yielding\nWe now interpret the results and explain the poor perfor-the sampling sets M(i), i = 1, . . . , 500, and measuring the\nmance of RWS using a simple argument based on the proper-normalized mean squared error (NMSE) incurred by graph\nties of Markov chains. For simplicity we consider a graph withsignal recovery from those sampling sets:\nclusters C1 and C2 having sizes N1 and N2. The probability of\n∥ˆx(i) −x(i)∥22 having an edge between nodes in the same cluster is denoted\nNMSEGi = p, while the probability of having an edge between nodes in ∥x(i)∥22 different clusters is q. An elementary calculation yields the VI.", |
| "paper_id": "1805.05827", |
| "title": "Graph Signal Sampling via Reinforcement Learning", |
| "authors": [ |
| "Oleksii Abramenko", |
| "Alexander Jung" |
| ], |
| "published_date": "2018-05-15", |
| "primary_category": "stat.ML", |
| "arxiv_url": "http://arxiv.org/abs/1805.05827v1", |
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| "text": "CONCLUSIONS\nprobability of a random walk transitioning from C1 to C2 as: This paper proposes a novel approach for graph signal processing which is based on interpreting graph signal sampling\nqN2\np12 = and recovery as a reinforcement learning problem. Using the\nqN2 + p(N1 −1) lens of reinforcement learning lends naturally to an online\nsampling strategy which is based on determining an optimal\nLikewise, the probability of staying in the C1:\npolicy which minimizes MSE of graph signal recovery. The\np11 = 1 −p12 proposed approach has been tested on a synthetic dataset\ngenerated in accordance to the stochastic block model. ObWe note that qN2 is the expected number of edges between tained experimental results have confirmed effectiveness of\na particular node of C1 and C2 and p(N1 −1) is the expected the proposed sampling algorithm in the stochastic settings and\nnumber of edges between a particular node of C1 and the demonstrated its advantages over existing approaches.\nremaining nodes of C1.", |
| "paper_id": "1805.05827", |
| "title": "Graph Signal Sampling via Reinforcement Learning", |
| "authors": [ |
| "Oleksii Abramenko", |
| "Alexander Jung" |
| ], |
| "published_date": "2018-05-15", |
| "primary_category": "stat.ML", |
| "arxiv_url": "http://arxiv.org/abs/1805.05827v1", |
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