Infinite-dimensional gradient-based descent for alpha-divergence minimisation

TL;DR

This paper proposes the $(\alpha, \Gamma)$-descent, a gradient-based algorithm for $\alpha$-divergence minimisation on measures, extending variational methods with convergence guarantees and applications to mixture models in high dimensions.

Contribution

It introduces the $(\alpha, \Gamma)$-descent algorithm, unifying and extending existing divergence minimisation methods with convergence proofs and broad applicability.

Findings

The algorithm guarantees systematic decrease in $\alpha$-divergence.

It recovers the Entropic Mirror Descent as a special case.

Empirical results show advantages over existing methods in high-dimensional settings.

Abstract

This paper introduces the $(\alpha, \Gamma)$-descent, an iterative algorithm which operates on measures and performs $\alpha$-divergence minimisation in a Bayesian framework. This gradient-based procedure extends the commonly-used variational approximation by adding a prior on the variational parameters in the form of a measure. We prove that for a rich family of functions $\Gamma$, this algorithm leads at each step to a systematic decrease in the $\alpha$-divergence and derive convergence results. Our framework recovers the Entropic Mirror Descent algorithm and provides an alternative algorithm that we call the Power Descent. Moreover, in its stochastic formulation, the $(\alpha, \Gamma)$-descent allows to optimise the mixture weights of any given mixture model without any information on the underlying distribution of the variational parameters. This renders our method compatible with many choices of parameters updates and applicable to a wide range of Machine Learning tasks. We demonstrate empirically on both toy and real-world examples the benefit of using the Power descent and going beyond the Entropic Mirror Descent framework, which fails as the dimension grows.