Time-uniform concentration bounds for iterative algorithms

TL;DR

This paper introduces a versatile framework for deriving time-uniform concentration bounds applicable to stochastic algorithms, especially when traditional supermartingale methods are infeasible, with applications to PCA, SGD, and stochastic approximation.

Contribution

The authors present a new approach for time-uniform concentration bounds that extends classical Robbins-Siegmund Lemma, applicable to a broad class of stochastic algorithms.

Findings

Derived new optimal bounds for Oja's streaming PCA algorithm

Established bounds for stochastic gradient descent

Provided bounds for stochastic approximation methods

Abstract

We develop a new framework for deriving time-uniform concentration bounds for the output of stochastic sequential algorithms satisfying certain recursive inequalities akin to those defining the almost-supermartingale processes introduced by \cite{robbins1971convergence}. Our approach is of wide applicability, and can be deployed in settings in which exponential supermartingale processes, required by prevailing methodologies for anytime-valid concentration inequalities, are not readily available. Our results can be viewed as quantitative versions of the classical Robbins-Siegmund Lemma. We demonstrate the effectiveness of our method by providing new and optimal time-uniform concentration bounds for Oja's algorithm for streaming PCA, stochastic gradient descent, and stochastic approximations.