q-Paths: Generalizing the Geometric Annealing Path using Power Means

TL;DR

This paper introduces q-paths, a flexible family of annealing paths based on power means, which generalize geometric paths and improve Bayesian inference and generative model evaluation.

Contribution

It proposes q-paths derived from generalized means, providing a closed-form, flexible alternative to geometric paths with practical benefits in inference and evaluation.

Findings

Empirical gains in Bayesian inference using Sequential Monte Carlo.

Improved generative model evaluation with Annealed Importance Sampling.

Closed-form expression involving deformed logarithm functions.

Abstract

Many common machine learning methods involve the geometric annealing path, a sequence of intermediate densities between two distributions of interest constructed using the geometric average. While alternatives such as the moment-averaging path have demonstrated performance gains in some settings, their practical applicability remains limited by exponential family endpoint assumptions and a lack of closed form energy function. In this work, we introduce $q$-paths, a family of paths which is derived from a generalized notion of the mean, includes the geometric and arithmetic mixtures as special cases, and admits a simple closed form involving the deformed logarithm function from nonextensive thermodynamics. Following previous analysis of the geometric path, we interpret our $q$-paths as corresponding to a $q$-exponential family of distributions, and provide a variational representation of intermediate densities as minimizing a mixture of $\alpha$-divergences to the endpoints. We show that small deviations away from the geometric path yield empirical gains for Bayesian inference using Sequential Monte Carlo and generative model evaluation using Annealed Importance Sampling.