Non-asymptotic Error Bounds For Constant Stepsize Stochastic Approximation For Tracking Mobile Agents

TL;DR

This paper derives the first non-asymptotic, time-uniform error bounds for constant stepsize stochastic approximation algorithms used in tracking moving targets, highlighting their dependence on problem parameters.

Contribution

It introduces a novel non-asymptotic analysis using Alekseev's formula, providing bounds valid for all time without relying on vanishing stepsize assumptions.

Findings

First non-asymptotic bound for entire time axis

Explicit dependence on problem parameters and dimension

Bound applicable to tracking slowly moving targets

Abstract

This work revisits the constant stepsize stochastic approximation algorithm for tracking a slowly moving target and obtains a bound for the tracking error that is valid for the entire time axis, using the Alekseev non-linear variation of constants formula. It is the first non-asymptptic bound for the entire time axis in the sense that it is not based on the vanishing stepsize limit and associated limit theorems unlike prior works, and captures clearly the dependence on problem parameters and the dimension.