1. Gossip Algorithms and Mixing Times Alex Dimakis based on collaborations with Florence Benezit, Kannan Ramchandran Anand Sarwate, Patrick Thiran Martin Vetterli, Martin Wainwright

2. 2 problem: distributed averaging Every node has a number (i.e. sensing temperature) Every node wants access to global average Want a randomized, distributed, localized, algorithm to compute averages. 2 2 3 5 12

3. 3 gossip algorithms for aggregation Start with initial measurement as an estimate for the average and update Each node interacts with a random neighbor and both compute pairwise average Converges to true average Fundamental building block for numerous problems (distributed detection, localization, randomized projections for compressive sensing) How many messages ? 2 2 3 5 2.5 12 3.75 7.87 ( Related work: Tsitsiklis, Boyd et al, Alanyali et al. Spanos et al, Sundaram et al., Nedich et al.)

4. 4 Standard gossip 2 2 3 5 12 2.5 1/20 00 01000 0 00 00010 00001 2 3 5 12 2 3 5 12 x(0) W(1) = 2.5 2 2.5 5 12 2.5 2 2.5 5 12 x(1) x( t )= W ( t ) x( t -1) = Π t W ( t ) x(0) W ( t ) iid random matrices

5. 5 How many messages ε -averaging time: First time where x(t) is ε -close to the normalized true average with probability greater than 1- ε. x(t)= W (t) x(t-1) = Π t W (t) x(0). Define W = E W (t) Theorem: ε -averaging time can be bounded using the spectral gap of W : (Boyd, Gosh, Prabhakar and Shah, IEEE Trans. On Information Theory, June 2006) Scaling laws ? or how does the number of messages scale if I double my network Scaling laws ? or how does the number of messages scale if I double my network

6. 6 cost of standard gossip Depends on graph and the gossip probabilities: Complete graph: T ave = Θ (n log1/ε) (Kempe et al. FOCS’03) Small World/Expander: T ave = Θ (n logn) Random Geometric Graphs: T ave = Θ (n 2 ) Note that flooding everything everywhere costs Θ (n 2 ) messages

7. 7 random geometric graphs Realistic sensor network model (Gupta & Kumar, Penrose): Random Geometric Graph G(n,r): n random points, connect if distance<r Connectivity threshold: r

8. 8 Standard Gossip algorithms require a lot of energy. (For realistic sensor network topologies) Why: useful information performs random walks, diffuses slowly Can we save energy with extra information? Idea: gossip in random directions, diffuse faster. Assume each node knows its location and locations of 1-hop neighbors. cost of standard Gossip

9. 9 Random Target Routing: How to find an (almost) random node Node picks a random location (=“target”) Greedy routing towards the target Probability to receive ~ Voronoi cell area

10. 10 Geographic Gossip Nodes use random routing to gossip with nodes far away in the network Each interaction costs But faster mixing Number of messages 2 3 2.5 (D, Sarwate, and Wainwright. IPSN’06, IEEE Trans. on Signal Processing)

11. 11 Recent related work Wenjun, Huaiyu - O( n 1.5 log1/ε ) using partial location lifting (L.A.D.A. ISIT’2007) Shah, Jung, Shin - O( n 1.5 log1/ε) by expander liftings (related to liftings of Markov chains for faster mixing) Different target distribution? (say ~1/r 2 ) ? Rabbat (Statistical Signal Processing ’07) showed it still scales like O( n 1.5 log1/ε) Can n 1.5 be improved ?

12. 12 Why not average on the path? Averaging on the routed path? The routed packet computes the sum of all the nodes it visits, and a hop-count. The average is propagated backwards to all the nodes on the path. 1 2 4 1 2 2 2 2 2

13. 13 Path averaging is optimal Theorem : Geographic gossip with path averaging on G(n, r) requires expected number of messages T ave = Θ ( n log1/ε) (with high probability over random graphs) Optimal number of messages (since averaging n numbers). Analysis bounds eigenvalues of a random matrix through the Poincare inequality. (Benezit, D., Thiran, and Vetterli, Allerton 2007)

14. 14 Proof ingredients: appetizers ε -averaging time: First time where x(k) is ε -close to the normalized true average with probability greater than 1- ε. x(t)= W (t) x(t-1) = Π t W (t) x(0). Define W = E W (t) The ε -averaging time can be bounded using the spectral gap of W:

15. 15 Proof ingredients:Poincare inequality Capacity of an edge e=(i,j),C(e)= π(i)Pij Each pair of states x,y has demand D(x,y)= π(x) π(y) A flow is a way to route D(x,y) units from x to y, for all pairs. Cost of a flow is ρ (f)= max e f(e)/ C(e). Length of a flow is the length of the longest flow-carrying path. Poincare inequality: i j Diaconis, Strook, Geometric bounds on eigenvalues, Ann. of Applied Probability, 1991 Sinclair, Improved bounds for mixing rates of markov chains and multicommodity flow. STOC’91, Combinatorics, Probability and Computing, volume 1, 1992

16. 16 Proof ingredients: Counting paths W (i,j)= 1/length of path * Prob(i,j) averaged together) Key lemma: Typical pairs of nodes i,j are averaged in Ω(n) paths. W (i,j)=Ω(n 1.5 ) if d(i,j)<= const n 0.5

17. 17 Mixing ingredients For each pair of nodes (i,j) we construct a flow that does not overload any edge. Can be done by going through one intermediate node for each pair i,j D(x,y)= 1/n 2, C(e)  1/n 1/n 1.5 =1/n 2.5 (for e typical) Cost= ρ (f)=n 1/2, (f)=2 Poincare coefficient O( n 1/2 ) Averaging time O(n 1/2 ), but each averaging round costs O(n 1/2 ) messages (= typical path length).

18. gossip and mobility 18

19. gossip and mobility 19 Refresh:Every node randomly and independently chooses a new site

20. gossip and mobility 20 Grid with no mobility T ave = Θ ( n 2 log1/ε) Grid with full mobility T ave = Θ ( n log1/ε) What if only horizontal mobility ?

21. gossip and mobility 21 Grid- horizontal mobility T ave = Θ ( n 2 log1/ε) (useless) Bidirectional mobility T ave = Θ ( n log1/ε) (as good as full) Adding m mobile agents: cuts time by m factor Sarwate,D: The impact of mobility on gossip algorithms (Infocom 2009)

22. gossip and mobility 22

23. Open problem: gossip and mobility 23 Random walk: Every node does random walk step on grid.

24. Open problem: gossip and mobility 24

25. Open problem: gossip and mobility 25 The random walk model tends to the refresh model if enough moving steps m are allowed between each gossip step Conjecture: the convergence is monotonically decreasing in the number m What is the scaling of messages for m=1 ? More realistic mobility models (e.g. random waypoint)? Gossip algorithms are especially interesting for scenarios with mobility and unknown topology

26. 26 Research directions Showed that random path averaging is optimal. Random path idea can be useful for message passing/distributed optimization algorithms more generally. This scheduling avoids diffusive nature of random walks-spreads spatial information fast. Impact of mobility? Exploiting the physical layer ? Impact of quantization, message errors ? Scaling laws for distributed optimization problems ?

27. 27 fin

28. 28 Poincare inequality: an example Capacity C(e)= π(i)Pij= 1/n 1/2 D(x,y)= π(x) π(y)= 1/n 2 A flow is a way to route D(x,y) units from x to y, for all pairs. (no choice!) Cost of a flow is ρ (f)= max e f(e)/ C(e) ρ (f)=1 / (1/2n)=2n Length of a flow is the length of the longest flow- carrying path. (f)=n Poincare inequality: 1/n 2

29. Statistical inference t1t1 t2t2 F t4t4 t3t3 x1x1 x5x5 x2x2 x3x3 x4x4  12 22 Consider an undirected graphical model with potential functions  i,  ij The probability of an observation (x 1..5 ) is What is the distribution of x 5 given some (noisy) observations?

30. Inference in sensor networks through message passing x1x1 x9x9 x7x7 x4x4 x8x8 y1y1 x2x2 y2y2 x5x5 y5y5 x6x6 y6y6 x3x3 y8y8

31. Message passing for sensor networks Optimal mappings from nodes to motes are NP- hard. There exist graphical models which require routing. Theorem: Any graphical model can be modified so that no routing is required. Reweighted belief propagation is very robust and practical. (packet drops, node failures, dynamic changes) (Schiff, Antonelli, D, Chu, and Wainwright. IPSN’07)

32. Deployment Two space heaters for temperature gradient Empirical model to generate potential functions Sample at red dots, predict at blue dots

33. Deployment Results Estimate temperature as mean of marginal Good accuracy –Within 3 degrees! NodeActual TempMean of Marginal 824.5ºC23.2ºC 1143.6ºC41.4ºC 2227.5ºC25.3ºC 81122

34. Simulation results Errors in Sensor Readings

35. Resilience to Communication Failures (simulation results) Dead Motes

36. Resilience to Communication Failure II Undirected Links Directed Links

37. 37 Poincare inequality: an example Capacity C(e)= π(i)Pij= 1/n 1/2 D(x,y)= π(x) π(y)= 1/n 2 A flow is a way to route D(x,y) units from x to y, for all pairs. (no choice!) Cost of a flow is ρ (f)= max e f(e)/ C(e) ρ (f)=1 / (1/2n)=2n Length of a flow is the length of the longest flow- carrying path. (f)=n Poincare inequality: 1/n 2