On statistical inference when fixed points of belief propagation are unstable

TL;DR

This paper investigates the limits of statistical inference in models like community detection and planted constraint satisfaction, showing that fixed points of belief propagation can be unstable even when detection and recovery are computationally feasible.

Contribution

It introduces a general framework for analyzing inference problems, computes the tractable regimes using the cavity method, and provides polynomial-time algorithms in these regimes.

Findings

Identifies regimes where detection and recovery are computationally feasible.

Provides polynomial-time algorithms for detection and recovery in the predicted regimes.

Shows that unstable fixed points of belief propagation do not necessarily imply computational hardness.

Abstract

Many statistical inference problems correspond to recovering the values of a set of hidden variables from sparse observations on them. For instance, in a planted constraint satisfaction problem such as planted 3-SAT, the clauses are sparse observations from which the hidden assignment is to be recovered. In the problem of community detection in a stochastic block model, the community labels are hidden variables that are to be recovered from the edges of the graph. Inspired by ideas from statistical physics, the presence of a stable fixed point for belief propogation has been widely conjectured to characterize the computational tractability of these problems. For community detection in stochastic block models, many of these predictions have been rigorously confirmed. In this work, we consider a general model of statistical inference problems that includes both community detection in stochastic block models, and all planted constraint satisfaction problems as special cases. We carry out the cavity method calculations from statistical physics to compute the regime of parameters where detection and recovery should be algorithmically tractable. At precisely the predicted tractable regime, we give: (i) a general polynomial-time algorithm for the problem of detection: distinguishing an input with a planted signal from one without; (ii) a general polynomial-time algorithm for the problem of recovery: outputting a vector that correlates with the hidden assignment significantly better than a random guess would.