FIXING MAX-PRODUCT: A UNIFIED LOOK AT MESSAGE PASSING ALGORITHMS

Nicholas Ruozzi and Sekhar Tatikonda

Yale University

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]

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

PREVIOUS WORK

MAP MAP LPConcave

Dual

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

PREVIOUS WORK

THIS WORK

Focus on the primal problem:

Choose a reparameterization of the objective function

Reparameterization in terms of messages

Construct concave lower bounds from this reparameterization by exploiting concavity of min

Perform coordinate ascent on these lower bounds

MAPReparamet-erization

Concave Lower Bound

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

MAPReparamet-erization

Concave Lower Bound

OUTLINE

Background Min-sum Reparameterizations

“Splitting” Reparameterization Lower bounds

Message Updates

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)

CORRESPONDING GRAPH

2

1

3

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 ®

BELIEFS

Typically, we express the reparameterization in terms of beliefs (meant to represent min-marginals):

With this definition, we have:

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,

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

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

OUTLINE

Background Min-sum Reparameterizations

“Splitting” Reparameterization Lower bounds

Message Updates

“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

“SPLITTING” REPARAMETERIZATION

where ci, c® 0 and the beliefs are defined as:

“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]

“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

“SPLITTING” REPARAMETERIZATION

TRMP lower bound:

Max-Sum Diffusion lower bound:

OUTLINE

Background Min-sum Reparameterizations

“Splitting” Reparameterization Lower bounds

Message Updates

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

MAX-SUM DIFFUSION

Want

Solving for m®i gives

Do this for all ®2 i

SPLITTING UPDATE

Suppose all coefficients are positive and ci > 0

We want

Solving for m®i gives

Do this for all ®2 i

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?

QUESTIONS?