Message Passing in the Presence of Erasures Nicholas Ruozzi - PowerPoint PPT Presentation

Message Passing in the Presence of Erasures Nicholas Ruozzi

Motivation • Real networks are dynamic and constrained • Messages are lost • Nodes join and leave • Nodes may be power constrained • Empirical studies suggest that belief propagation and its relatives continue to perform well over real networks • [Anker, Dolev, and Hod, 2008] • [Anker, Bickson, Dolev, and Hod, 2008] • Few theoretical guarantees

Convergent Message Passing • New classes of reweighted message passing algorithms guarantee convergence and a notion of correctness • e.g., MPLP, tree-reweighted max-product, norm-product, etc. • Need special updating schedules or central control • No guarantees if messages are lost or updated in the wrong order

Factorizations • A function, f, factorizes with respect to a graph G = (V, E) if Y Y f ( x 1 ; : : : ; x n ) = Á i ( x i ) Ã ij ( x i ; x j ) i 2 V ( i;j ) 2 E • Goal is to maximize the function, f • Max-product attempts to solve this problem by passing messages over the graph G

Reweighted Message Passing Q h i k 2N ( i ) m t ¡ 1 ki ( x i ) c ki m t à ij ( x i ; x j ) 1 =c ij Á i ( x i ) ij ( x j ) := max m t ¡ 1 ji ( x i ) x i • Messages passed from a node only depend on the messages received by that node at the previous time step • Generalization of max-product

Reweighted Message Passing Y b t m t ki ( x i ) c ki i ( x i ) = Á i ( x i ) k 2N ( i ) b t à ij ( x i ; x j ) 1 =c ij b t j ( x j ) i ( x i ) b t ij ( x i ; x j ) = m t m t ji ( x i ) ij ( x j ) • These “beliefs” provide an alternative factorization of the objective function b i ( x i ) (1 ¡ P Y Y k 2 @i c ik ) b ij ( x i ; x j ) c ij f ( x ) = i 2 V ( i;j ) 2 E

Reweighted Message Passing Y b t m t ki ( x i ) c ki i ( x i ) = Á i ( x i ) k 2N ( i ) b t à ij ( x i ; x j ) 1 =c ij b t j ( x j ) i ( x i ) b t ij ( x i ; x j ) = m t m t ji ( x i ) ij ( x j ) • Certain choices of the reweighting parameters produce natural convex upper bounds on the objective function x i b i ( x i ) (1 ¡ P Y k 2 @i c ik ) max f ( x ) · max x i 2 V Y x i ;x j b ij ( x i ; x j ) c ij ¢ max ( i;j ) 2 E

Reweighted Message Passing Q h i k 2N ( i ) m t ¡ 1 ki ( x i ) c ki m t à ij ( x i ; x j ) 1 =c ij Á i ( x i ) ij ( x j ) := max m t ¡ 1 ji ( x i ) x i • If each c < 1/max degree, then there is a simple, “asynchronous” coordinate descent scheme

Reweighted Message Passing • Convergence is guaranteed by performing coordinate descent on a convex upper bound • Can we extend our convergence guarantees to networks in which messages can be lost? • Delivered too slowly • Adversarially lost • Intentionally not sent • Lost independently with some fixed probability

Results • For pairwise MRFs: • Can modify the graph locally in order to guarantee convergence when there are message erasures • Yields a completely local message passing algorithm as a side effect • If no messages are lost, the convergence of the asynchronous algorithm implies convergence of the synchronous one

Extending Convergence • With a linear amount of additional state at each node of the network we can, again, guarantee convergence with erasures • Construct a new graphical model such that message passing on the new model can be simulated over the network • Update messages “internal” to each node in such a way as to guarantee convergence

Extending Convergence • Construct a new graphical model from the network: • Create a copy of node i for each one of i’s neighbors • Attach each copy to exactly one copy of each neighbor • Enforce equality among the copies of each node with equality constraints • Messages can only be lost between copies of different nodes (all other messages are internal to a node of the network)

Extending Convergence 1 = 1 2 2 = = = = = 3 4 = = 3 4 New graphical model Original network (dashed circles are the nodes of the network)

Extending Convergence Á 1 ( x 1 ; 1 ) 3 1 = Á 1 ( x 1 ) 2 2 = = = Á 1 ( x 1 ; 2 ) 3 Á 1 ( x 1 ; 3 ) 3 = = 3 4 = = 3 4 New graphical model Original network (dashed circles are the nodes of the network)

Extending Convergence • Convergence on the new network follows from the convergence of the asynchronous message passing algorithm • Works even in the presence of erasures • Requires no global knowledge of the network • Can convert any network into a equivalent 3-regular network

Other Extensions • Many different updating strategies can be used to guarantee convergence: • Solve the “internal” problem exactly • Complete graph versus single cycle • Don’t divide the potentials evenly • Other graph modifications?

Performance • The additional overhead may result in slower rates of convergence • In practice, there exist sequences of erasures for which either algorithm outperforms the other • However, the reweighted max-product algorithm always seems to converge in practice for appropriate choices of the parameters

Networks Without Erasures • Synchronous algorithm is an asynchronous algorithm on the bipartite 2-cover of the network 1 ’ 1 1 2 2’ 2 3’ 3 3 4 4’ 4 Bipartite 2-cover Original network

Networks Without Erasures • Synchronous algorithm is an asynchronous algorithm on the bipartite 2-cover of the network 1 ’ 1 1 2 2’ 2 3’ 3 3 4 4’ 4 Bipartite 2-cover Original network

Networks Without Erasures • Synchronous algorithm is an asynchronous algorithm on the bipartite 2-cover of the network 1 ’ 1 1 2 2’ 2 3’ 3 3 4 4’ 4 Bipartite 2-cover Original network

Networks Without Erasures • Synchronous algorithm is an asynchronous algorithm on the bipartite 2-cover of the network 1 ’ 1 1 2 2’ 2 3’ 3 3 4 4’ 4 Bipartite 2-cover Original network

Conclusions • Understanding the convergence behavior of BP-like algorithms on a network with errors is a challenging problem • Can engineer around the problem to achieve a purely local algorithm • May incur a performance penalty • What is the exact relationship between these algorithms? • Empirically, the reweighted algorithm on the original network appears to always converge • Prove it?