The Incremental Proximal Method: A Probabilistic Perspective
This work provides a theoretical link between optimization algorithms and probabilistic methods, which could aid in developing new procedures for large-scale or nonlinear problems, though it appears incremental in nature.
The paper connects the incremental proximal method to stochastic filters, showing that proximal operators correspond to Bayes updates and that the method can be realized by the Kalman filter for linear-quadratic problems, with implications for nonlinear optimization via the extended Kalman filter.
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.