q-Paths: Generalizing the Geometric Annealing Path using Power Means
This work addresses a foundational problem in machine learning for practitioners using annealing-based methods, offering a novel generalization that is incremental but provides specific performance improvements.
The paper tackles the problem of generalizing the geometric annealing path in machine learning by introducing q-paths, a family derived from power means that includes geometric and arithmetic mixtures as special cases and admits a closed form. The result shows that small deviations from the geometric path yield empirical gains, such as improved performance in Bayesian inference with Sequential Monte Carlo and generative model evaluation with Annealed Importance Sampling.
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 $α$-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.