The Incremental Proximal Method: A Probabilistic Perspective

TL;DR

This paper reveals a connection between the incremental proximal method and stochastic filters, showing that for certain problems, the method can be realized through Bayesian updates like Kalman filters, with implications for nonlinear optimization.

Contribution

It establishes a probabilistic interpretation of the incremental proximal method, linking it to Bayesian filtering techniques such as Kalman and extended Kalman filters.

Findings

Proximal operators coincide with Bayes updates for Gaussian models.

Incremental proximal method can be realized by Kalman filter in linear-quadratic cases.

Extended Kalman filter offers a systematic approach for nonlinear optimization problems.

Abstract

In this work, we highlight a connection between the incremental proximal method and stochastic filters. We begin by showing that the proximal operators coincide, and hence can be realized with, Bayes updates. We give the explicit form of the updates for the linear regression problem and show that there is a one-to-one correspondence between the proximal operator of the least-squares regression and the Bayes update when the prior and the likelihood are Gaussian. We then carry out this observation to a general sequential setting: We consider the incremental proximal method, which is an algorithm for large-scale optimization, and show that, for a linear-quadratic cost function, it can naturally be realized by the Kalman filter. We then discuss the implications of this idea for nonlinear optimization problems where proximal operators are in general not realizable. In such settings, we argue that the extended Kalman filter can provide a systematic way for the derivation of practical procedures.