Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM

TL;DR

This paper extends mirror descent to measure spaces with relative smoothness, providing convergence proofs for algorithms like Sinkhorn and EM in infinite-dimensional settings, with applications to optimal transport and signal processing.

Contribution

It introduces a convergence analysis of mirror descent in measure spaces using relative smoothness, applying it to Sinkhorn's algorithm and EM, with new proofs and convergence rates.

Findings

Sinkhorn iterations correspond to mirror descent in continuous optimal transport.

The paper proves sublinear convergence rates for EM when optimizing on latent distributions.

Provides a new proof of Sinkhorn's linear convergence in the continuous setting.

Abstract

Many problems in machine learning can be formulated as optimizing a convex functional over a vector space of measures. This paper studies the convergence of the mirror descent algorithm in this infinite-dimensional setting. Defining Bregman divergences through directional derivatives, we derive the convergence of the scheme for relatively smooth and convex pairs of functionals. Such assumptions allow to handle non-smooth functionals such as the Kullback--Leibler (KL) divergence. Applying our result to joint distributions and KL, we show that Sinkhorn's primal iterations for entropic optimal transport in the continuous setting correspond to a mirror descent, and we obtain a new proof of its (sub)linear convergence. We also show that Expectation Maximization (EM) can always formally be written as a mirror descent. When optimizing only on the latent distribution while fixing the mixtures parameters -- which corresponds to the Richardson--Lucy deconvolution scheme in signal processing -- we derive sublinear rates of convergence.