Provable benefits of annealing for estimating normalizing constants: Importance Sampling, Noise-Contrastive Estimation, and beyond

TL;DR

This paper analyzes annealing-based Monte Carlo methods for estimating normalization constants, showing how different choices in estimator type and path of distributions affect efficiency, and proposing a new two-step estimator for optimal path approximation.

Contribution

It provides a theoretical comparison of importance sampling and noise-contrastive estimation, and introduces a two-step estimator for better path selection.

Findings

NCE is more efficient than importance sampling, but the difference diminishes with infinitesimal steps.

Geometric paths reduce estimation error from exponential to polynomial.

Arithmetic paths can be optimal, especially in certain limits.

Abstract

Recent research has developed several Monte Carlo methods for estimating the normalization constant (partition function) based on the idea of annealing. This means sampling successively from a path of distributions that interpolate between a tractable "proposal" distribution and the unnormalized "target" distribution. Prominent estimators in this family include annealed importance sampling and annealed noise-contrastive estimation (NCE). Such methods hinge on a number of design choices: which estimator to use, which path of distributions to use and whether to use a path at all; so far, there is no definitive theory on which choices are efficient. Here, we evaluate each design choice by the asymptotic estimation error it produces. First, we show that using NCE is more efficient than the importance sampling estimator, but in the limit of infinitesimal path steps, the difference vanishes. Second, we find that using the geometric path brings down the estimation error from an exponential to a polynomial function of the parameter distance between the target and proposal distributions. Third, we find that the arithmetic path, while rarely used, can offer optimality properties over the universally-used geometric path. In fact, in a particular limit, the optimal path is arithmetic. Based on this theory, we finally propose a two-step estimator to approximate the optimal path in an efficient way.