Efficient Peer-to-Peer Belief Propagation
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1 Efficient Peer-to-Peer Belief Propagation Roman Schmidt, Karl Aberer School of Computer and Communication Sciences Ecole Polytechnique Fédérale de Lausanne (EPFL) 14 th International Conference on Cooperative Information Systems Montpellier, France, November 1-3, 26 Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26
2 Motivation Users share (correlated) data in P2P systems currently mainly for retrieval but correlations hold hidden knowledge Profit by correlations for new services Distributed Knowledge Base (e.g., for bugs) Structure/cluster data (e.g., for better search results) Recommendation system (e.g., for data annotation) etc. Distributed Inference System on top of a P2P system Paper focus and contribution Message reduction Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 2
3 Outline Motivation Basic Concepts Belief Propagation The P-Grid Overlay P2P Belief Propagation Inference Architecture The Relaxation Algorithm Evaluation Conclusions Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 3
4 Belief Propagation Inference based on Bayesian networks models dependencies between variables True False Installed.2.8 OS1 Driver1 True False Installed.2.8 App1 OS1 Driver1 Runs Error T T.9.1 T F.4.6 F T. 1. F F. 1. Iterative message-passing algorithm compute marginal probabilities ( beliefs ) provably efficient on trees, works for arbitrary networks Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 4
5 The BP message-passing algorithm Sends messages across edges 2 messages per edge and if all messages from previous were received Beliefs are updated per algorithm terminates if beliefs stabilize Messages are vectors length corresponds to the number of node states Computation complexity grows exponentially with the number of states of nodes Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 5
6 The P-Grid Overlay Peers are organized in a binary trie structure one node for every common prefix trie is only virtual (exists only via routing tables) all nodes remain at the leaf-level (no hierarchy) Multiple peers per key space partition Multiple routing entries (random choice) per routing table level Logarithmic search complexity even for skewed data distributions Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 6
7 The Distributed Inference System P-Grid Bug reports, metadata, tags, etc. Bayesian network Variables (spread over P-Grid nodes) Dependencies between variables Distributive learning Belief Propagation Distributed inference Message-passing algorithm Identified problem high message cost Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 7
8 Spring Relaxation Bayesian network as spring network find minimum energy configuration (relax springs) energy is proportional to the distance between P-Grid nodes variables at the same node require no energy optimal: all variables at one node (load balancing) Decentralized algorithm nodes try to relax their springs move correlated variables close to each other optimally, at the same node (no physical message) considering load distribution Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 8
9 Spring relations in P-Grid * * 1* 1* * 11* A FF BB C D EE 1* : C, D 1* : B 1* : E 1* : B 1* : C, D * : F * : A, B 11* : E * : A, F 11* : E * : B, F * : D a -> h, t f -> o, r a -> h, t f -> o, r h -> a, m m -> h, u o -> f r -> f, t o -> f r -> f, t t -> a, r u -> m Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 9
10 The Relaxation Algorithm (relax variables) currentload = length(localvars); overload = currentload - avgload / 2; IF (overload <= ) return; ENDIF undirvars = variables having a tension only at one level; WHILE ((overload > ) AND (length(unidirvars) > )) move variable to a peer from the level with the tension; removefirst(unidirvars); overload = overload - 1; ENDWHILE Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26
11 The Relaxation Algorithm (balance load) multidirvars = vars having tensions at multiple levels; WHILE ((currentload > avgload) AND (length(multidirvars) > )) FOR i = routingtable.levels TO 1 IF (level i is underpopulated) cand = vars having a tension at level i; FOR j = 1 TO length(cand) IF (cand(j).tension(i) >= max(cand(j).tension)) move variable to a peer from level i; remove(multidirvars, cand(j)); currentload = currentload - 1; IF (currentload <= avgload) break; ENDIF; ENDIF; ENDFOR; ENDIF; ENDFOR; ENDWHILE Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 11
12 Evaluation Matlab implementation Diverse Bayesian networks random, binary trees, scale-free up to 248 Bayesian nodes up to 512 P-Grid nodes repetitions 2 evaluation criterions message reduction load balance Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 12
13 Random network 24 nodes, average node degree 4 degree variable Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 13
14 Binary tree network 23 nodes 3 degree variable Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 14
15 Scale-free network 24 nodes, average node degree 4 5 degree variable Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 15
16 Message reduction (random) / 24 / / 248 / messages [%] messages [%] messages [%] / 248 / 8 messages [%] / 248 / Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 16
17 Message reduction (binary tree) / / messages [%] messages [%] / messages [%] Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 17
18 Message reduction (scale-free) / / messages [%] messages [%] / messages [%] Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 18
19 Load balancing (random) / 24 / / 248 / variables/node variables/node / 248 / / 248 / 8 variables/node 15 variables/node Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 19
20 Load balancing (binary tree) / variables/node variables/node / variables/node / Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 2
21 Load balancing (scale-free) variables/node / / 248 variables/node variables/node / Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 21
22 Related Work Generalized Belief Propagation Clusters correlate nodes Joined probabilities of clustered nodes Exponential complexity increase (number of states nodes in the cluster ) No decentralized algorithm to cluster nodes Sensor network architecture Based on junction tree algorithm Requires a preformed spanning tree of nodes Evaluated with 54 sensor nodes in a local setup Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 22
23 Conclusions Decentralized relaxation algorithm Reduces message cost for Belief Propagation Considers load balance Several scenarios (Distributed Knowledge Base) First evaluation looks promising Intermediate steps are still missing Learning of Bayesian network Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26 23
24 Thank you! Questions? Distributed Information Systems Laboratory CoopIS'6, Montpellier, France November 1, 26
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