Hardness of parameter estimation in graphical models

TL;DR

This paper proves that learning the canonical parameters of pairwise binary graphical models from mean parameters is computationally intractable in general, establishing a fundamental complexity barrier in statistical inference for these models.

Contribution

It provides the first proof that parameter estimation in graphical models is NP-hard, via a polynomial-time reduction from approximating the partition function of the hard-core model.

Findings

Parameter estimation is NP-hard for general pairwise binary graphical models.

The proof involves a reduction from the hard-core model partition function approximation.

The marginal polytope boundary has an inherent repulsive property, aiding the reduction.

Abstract

We consider the problem of learning the canonical parameters specifying an undirected graphical model (Markov random field) from the mean parameters. For graphical models representing a minimal exponential family, the canonical parameters are uniquely determined by the mean parameters, so the problem is feasible in principle. The goal of this paper is to investigate the computational feasibility of this statistical task. Our main result shows that parameter estimation is in general intractable: no algorithm can learn the canonical parameters of a generic pair-wise binary graphical model from the mean parameters in time bounded by a polynomial in the number of variables (unless RP = NP). Indeed, such a result has been believed to be true (see the monograph by Wainwright and Jordan (2008)) but no proof was known. Our proof gives a polynomial time reduction from approximating the partition function of the hard-core model, known to be hard, to learning approximate parameters. Our reduction entails showing that the marginal polytope boundary has an inherent repulsive property, which validates an optimization procedure over the polytope that does not use any knowledge of its structure (as required by the ellipsoid method and others).