Stochastic Online Optimization using Kalman Recursion

TL;DR

This paper analyzes the Extended Kalman Filter as a stochastic optimization method, providing high probability bounds and a two-phase convergence analysis for various regression models, highlighting its efficiency and parameter-free nature.

Contribution

It offers a novel Bayesian perspective on EKF for stochastic optimization, with explicit convergence bounds and a two-phase analysis avoiding projection steps.

Findings

High probability bounds on excess risk for linear and logistic regressions.

Explicit convergence time bounds for the local phase.

EKF as a parameter-free algorithm with optimal complexity for certain problems.

Abstract

We study the Extended Kalman Filter in constant dynamics, offering a bayesian perspective of stochastic optimization. We obtain high probability bounds on the cumulative excess risk in an unconstrained setting. In order to avoid any projection step we propose a two-phase analysis. First, for linear and logistic regressions, we prove that the algorithm enters a local phase where the estimate stays in a small region around the optimum. We provide explicit bounds with high probability on this convergence time. Second, for generalized linear regressions, we provide a martingale analysis of the excess risk in the local phase, improving existing ones in bounded stochastic optimization. The EKF appears as a parameter-free online algorithm with O(d^2) cost per iteration that optimally solves some unconstrained optimization problems.