Faster Stochastic Optimization with Arbitrary Delays via Asynchronous Mini-Batching

TL;DR

This paper introduces a new asynchronous stochastic optimization method that adapts to arbitrary delays, providing improved convergence guarantees for non-convex and convex problems by leveraging quantile-based delay analysis.

Contribution

It presents a procedure transforming standard stochastic methods into asynchronous ones with delay-dependent convergence guarantees, including an adaptive approach that handles all delay quantiles without prior delay knowledge.

Findings

Achieves convergence rates depending on the $q$-quantile delay.

Provides an adaptive method that automatically adjusts to delays.

Improves upon existing results by generalizing delay dependence.

Abstract

We consider the problem of asynchronous stochastic optimization, where an optimization algorithm makes updates based on stale stochastic gradients of the objective that are subject to an arbitrary (possibly adversarial) sequence of delays. We present a procedure which, for any given $q \in (0,1]$, transforms any standard stochastic first-order method to an asynchronous method with convergence guarantee depending on the $q$-quantile delay of the sequence. This approach leads to convergence rates of the form $O(\tau_q/qT+\sigma/\sqrt{qT})$ for non-convex and $O(\tau_q^2/(q T)^2+\sigma/\sqrt{qT})$ for convex smooth problems, where $\tau_q$ is the $q$-quantile delay, generalizing and improving on existing results that depend on the average delay. We further show a method that automatically adapts to all quantiles simultaneously, without any prior knowledge of the delays, achieving convergence rates of the form $O(\inf_{q} \tau_q/qT+\sigma/\sqrt{qT})$ for non-convex and $O(\inf_{q} \tau_q^2/(q T)^2+\sigma/\sqrt{qT})$ for convex smooth problems. Our technique is based on asynchronous mini-batching with a careful batch-size selection and filtering of stale gradients.