Fixing Max-Product: A Unified Look at Message Passing Algorithms

1. Fixing Max-Product: A Unified Look at Message Passing Algorithms Nicholas Ruozzi and Sekhar Tatikonda Yale University

2. Previous Work • Recent work related to max-product has focused on convergent and correct message passing schemes: • TRMP [Wainwright et al. 2005] • MPLP [Globerson and Jaakkola 2007] • Max-Sum Diffusion [Werner 2007] • “Splitting” Max-product [Ruozzi and Tatikonda 2010]

3. Previous Work • Typical approach: focus on a dual formulation of the MAP LP • Message passing scheme is derived as a coordinate ascent scheme on a concave dual: • MPLP • TRMP • Max-Sum Diffusion MAP MAP LP Concave Dual

4. Previous Work • Many of these algorithms can be seen as maximizing a specific lower bound [Sontag and Jaakkola 2009] • The maximization is performed over reparameterizations of the objective function that satisfy specific constraints • Different constraints correspond to different dual formulations

5. This Work • Focus on the primal problem: • Choose a reparameterization of the objective function • Reparameterizationin terms of messages • Construct concave lower bounds from this reparameterization by exploiting concavity of min • Perform coordinate ascent on these lower bounds MAP Reparamet-erization Concave Lower Bound

6. This Work • Many of the common message passing schemes can be captured by the “splitting” family of reparameterizations • Many possible lower bounds of interest • Produces an unconstrained concave optimization problem MAP Reparamet-erization Concave Lower Bound

7. Outline • Background • Min-sum • Reparameterizations • “Splitting” Reparameterization • Lower bounds • Message Updates

8. Min-Sum • Minimize an objective function that factorizes as a sum of potentials (assume f is bounded from below) • (some multiset whose elements are subsets of the variables)

9. Corresponding Graph 1 2 3

10. Reparameterizations • We can rewrite the objective function as • This does not change the objective function as long as the messages are finite valued at each x • The objective function is reparameterized in terms of the messages • No dependence on messages passed from i to ®

11. Beliefs • Typically, we express the reparameterization in terms of beliefs (meant to represent min-marginals): • With this definition, we have:

12. Min-Sum • The min-sum algorithm updates ensure that, after updating m®i, • In other words, • Can estimate an assignment from a collection of messages by choosing • Upon convergence,

13. Correctness Guarantees • The min-sum algorithm does not guarantee the correctness of this estimate upon convergence • Assignments that minimize bi need not minimize f: • Notable exceptions: trees, single cycles, singly connected graphs

14. Lower Bounds • Can derive lower bounds that are concave in the messages from reparameterizations: • Lower bound is a concave function of the messages (and beliefs) • We want to find the choice of messages that maximizes the lower bound • This lower bound may not be tight

15. Outline • Background • Min-sum • Reparameterizations • “Splitting” Reparameterization • Lower bounds • Message Updates

16. “Good” Reparameterizations • Many possible reparameterizations • How do we choose reparameterizations that produce “nice” lower bounds? • Want estimates corresponding to the optimal choice of messages to minimize the objective function • Want the bound to be concave in the messages • Want the coordinate ascent scheme to remain local

17. “Splitting” reparameterization where ci, c®0 and the beliefs are defined as:

18. “Splitting” reparameterization • TRMP: • is a collection of spanning trees in the factor graph and ¹ is a probability distribution on spanning trees • Choose c® = ¹® • Can extend this to a collection of singly connected subgraphs[Ruozzi and Tatikonda 2010]

19. “Splitting” reparameterization • Min-sum, TRMP, MPLP, and Max-Sum Diffusion can all be characterized as splitting reparameterizations • One possible lower bound: • We could choose c such that f can be written as a nonnegative combination of the beliefs

20. “Splitting” Reparameterization • TRMP lower bound: • Max-Sum Diffusion lower bound:

21. Outline • Background • Min-sum • Reparameterizations • “Splitting” Reparameterization • Lower bounds • Message Updates

22. From Lower bounds to Message Updates • We can construct the message updates by ensuring that we perform coordinate ascent on our lower bounds • Can perform block updates over trees [Meltzer et al. 2009] [Kolmogorov 2009] [Sontag and Jaakkola 2009] • Key observation: • Equality iff there is an x that simultaneously minimizes both functions

23. Max-Sum Diffusion • Want • Solving for m®i gives • Do this for all ®2i

24. Splitting Update • Suppose all coefficients are positive and ci > 0 • We want • Solving for m®i gives • Do this for all ®2i

25. Conclusion • MPLP, TRMP, and Max-Sum Diffusion are all instances of the splitting reparameterization for specific choices of the constants and lower bound • Different lower bounds produce different unconstrained concave optimization problems • Choice of lower bound corresponds to choosing different dual formulations • Maximization is performed with respect to the messages, not the beliefs • Many more reparameterizations and lower bounds are possible • Is there a reparameterization in which the lower bounds are strictly concave?

26. Questions?