Convergent Message-Passing Algorithms for Inference over General Graphs with Convex Free Energies Tamir Hazan, Amnon Shashua School of Computer Science and Engineering Hebrew University of Jerusalem

2 Input: Graphical Models - Background Output:

3 Input: Output: Graphical Models - Background

4 Input: Output: Belief Propagation: Graphical Models - Background

5 BP - Variational Methods Bethe free energy: Approximating the marginals

6 BP - Variational Methods Stationary Bethe free energy = BP fixed points (Yedidia, Freeman, Weiss ’01) When the factor graph has no cycles Bethe is convex over the marginalization constraints and BP is exact. Bethe free energy: Approximating the marginals Factor graph has cycles: Bethe is non-convex and BP might not converge.

7 BP - Variational Methods TRW free energy (Wainwright, Jaakkola, Willsky ’02): Weighted number of spanning trees through an edge α TRW free energy is convex over the marginalization constraints. Convergent message passing algorithm for cliques of size 2 (Globerson & Jaakkola, ’07)

8 “convex free energy” is convex (over the marginalization constraints) for all factor graphs if there exists such that and Claim: (Pakzad and Anantharam ‘02, Heskes ’04, Weiss et al. ’07) Convex Free Energies

9 “convex free energy” is convex (over the marginalization constraints) for all factor graphs if there exists such that and Claim: (Pakzad and Anantharam ‘02, Heskes ’04, Weiss et al. ’06) Convex Free Energies

10 Background - Summary Bethe free energy TRW free energyconvex free energy non-convex for general graphs specific convexification of free energy general form of convexification with parameters Contributions: 1) Convergent message passing algorithm for convex free energy 2) Heuristic for choosing good convex free energy. Belief PropagationGloberson Jaakkola ’07 message passing algorithm for |α|=2 ? message passing algorithm for |α|≥2

11 strictly convex & differentiable Strictly convex & proper ( for some b) Convex Belief Propagation

12 strictly convex & differentiable Strictly convex & proper ( for some b) marginals of x i agree Convex Belief Propagation

13 strictly convex & differentiable Strictly convex & proper ( for some b) marginals of x i agree Whenever marginals of x i agree Otherwise

14 Output: For t=1,2,... For i=1,2,...,n Convex Message Passing * We also have a similar parallel message passing algorithm

15 Output: For t=1,2,... For i=1,2,...,n Convex Message Passing * We also have a similar parallel message passing algorithm

16 Primer on Fenchel Duality convex conjugate of h(b) concave conjugate of g(b) PrimalDual

17 Primer on Fenchel Duality More Conveniently:

18 Sequential message-passing Fenchel duality theorem

19 Generalized Fenchel duality theorem Sequential message-passing Fenchel duality theorem

20 Generalized Fenchel duality theorem Sequential message-passing Fenchel duality theorem Block update approach: Optimize Set

21 Generalized Fenchel duality theorem Sequential message-passing Fenchel duality theorem Block update approach: Optimize Set

22 Generalized Fenchel duality theorem Sequential message-passing Fenchel duality theorem Block update approach: Optimize Set

23 Generalized Fenchel duality theorem Sequential message-passing Fenchel duality theorem Algorithm: Repeat until convergence Output: Block update approach: Optimize Set

24 Sequential message-passing Bregman’s successive projection algorithm (Hildreth, Dykstra, Csiszar, Tseng) Output: For t=1,2,... For i=1,2,...,n

25 Convex Belief Propagation marginals of x i agree Output: For t=1,2,... For i=1,2,...,n

26 Convex Belief Propagation Our algorithm: Closed form solution for Belief propagation:

27 Convex Belief Propagation Our algorithm: Closed form solution for Belief propagation:

28 3) For general and (non-convex!) our algorithm 1) when f(),h i () are non-convex if the algorithm converges then it is to a stationary point. 2) Set (Bethe free energy, non-convex!) our message passing algorithm = belief propagation. Convex Belief Propagation Interesting Notes: Open question =? “Fractional BP” (Wiegerinck, Heskes ‘03)

29 Determine Convex Energy Heuristic: Find convex energy while is close to 1 as possible Motivation: - Bethe approximation

30 Experiments – Ising Model grid Every variable has two values variables field interaction attractive mixed

31 Experiments – Efficiency Grid size Seconds

32 Approximating Marginals – Ising model convex free energies Vs. Bethe free energy Marginal Error Interaction strength

33 convex free energies Vs. Bethe free energy Marginal Error Interaction strength Approximating Marginals – Ising model

34 Approximating Marginals – Random Graph 10 vertices. Edge density 30% (almost tree), 50% (far from tree) TRW free energy Vs. L2-convex free energy

35 Novel perspective of message passing as primal-dual optimization for the general class of Convex free energy mapped to Heuristic for setting convex free energy by “convexifying” Bethe free energy Algorithm applies to general clique sizes; BP is a particular case of our algorithm Region Graphs energies (Kikuchi) Future work: Theoretical foundation on our heuristic for setting the convex free energy. Summary