Online Convex Optimization with Switching Cost with Only One Single Gradient Evaluation

TL;DR

This paper develops online algorithms for convex optimization with switching costs using minimal information, achieving optimal competitive ratios in the noiseless case and analyzing performance under noise.

Contribution

It introduces algorithms that operate with only one gradient evaluation per step, achieving optimal competitive ratios for linear switching costs in the frugal information setting.

Findings

Optimal competitive ratios achieved for linear switching costs

Algorithms perform well with noiseless gradient information

Performance degrades quadratically with noise magnitude

Abstract

Online convex optimization with switching cost is considered under the frugal information setting where at time $t$, before action $x_t$ is taken, only a single function evaluation and a single gradient is available at the previously chosen action $x_{t-1}$ for either the current cost function $f_t$ or the most recent cost function $f_{t-1}$. When the switching cost is linear, online algorithms with optimal order-wise competitive ratios are derived for the frugal setting. When the gradient information is noisy, an online algorithm whose competitive ratio grows quadratically with the noise magnitude is derived.