# A probabilistic incremental proximal gradient method

## TL;DR

This paper introduces the probabilistic incremental proximal gradient (PIPG) method, which interprets the algorithm probabilistically, allowing uncertainty propagation and enabling the use of Bayesian filters for large-scale optimization.

## Contribution

It develops a novel probabilistic interpretation of the incremental proximal gradient algorithm and integrates Bayesian filtering techniques for improved uncertainty management.

## Key findings

- Enables uncertainty propagation in optimization iterations
- Allows use of Kalman and extended Kalman filters in optimization
- Facilitates large-scale regularized problem solving

## Abstract

In this paper, we propose a probabilistic optimization method, named probabilistic incremental proximal gradient (PIPG) method, by developing a probabilistic interpretation of the incremental proximal gradient algorithm. We explicitly model the update rules of the incremental proximal gradient method and develop a systematic approach to propagate the uncertainty of the solution estimate over iterations. The PIPG algorithm takes the form of Bayesian filtering updates for a state-space model constructed by using the cost function. Our framework makes it possible to utilize well-known exact or approximate Bayesian filters, such as Kalman or extended Kalman filters, to solve large-scale regularized optimization problems.