The Stochastic Proximal Distance Algorithm

TL;DR

This paper introduces a stochastic version of the proximal distance algorithm, providing theoretical convergence guarantees, finite error bounds, and demonstrating superior empirical performance over batch methods in machine learning tasks.

Contribution

It develops and analyzes a stochastic proximal distance algorithm, establishing convergence, error bounds, and practical heuristics validation.

Findings

Proposed stochastic proximal distance algorithm converges with finite error bounds.

The method outperforms batch algorithms on standard learning tasks.

Theoretical analysis links penalty parameter to learning rate, justifying heuristics.

Abstract

Stochastic versions of proximal methods have gained much attention in statistics and machine learning. These algorithms tend to admit simple, scalable forms, and enjoy numerical stability via implicit updates. In this work, we propose and analyze a stochastic version of the recently proposed proximal distance algorithm, a class of iterative optimization methods that recover a desired constrained estimation problem as a penalty parameter $\rho \rightarrow \infty$. By uncovering connections to related stochastic proximal methods and interpreting the penalty parameter as the learning rate, we justify heuristics used in practical manifestations of the proximal distance method, establishing their convergence guarantees for the first time. Moreover, we extend recent theoretical devices to establish finite error bounds and a complete characterization of convergence rates regimes. We validate our analysis via a thorough empirical study, also showing that unsurprisingly, the proposed method outpaces batch versions on popular learning tasks.