Exploiting Curvature in Online Convex Optimization with Delayed Feedback

Hao Qiu, Emmanuel Esposito, Mengxiao Zhang

Keywords: online convex optimization, curved losses, delayed feedback

TL;DR: We study the online convex optimization problem with curved losses and delayed feedback

Abstract:

In this work, we study the online convex optimization problem with curved losses and delayed feedback.
When losses are strongly convex, existing approaches
obtain regret bounds of order $d_{\max} \ln T$, where $d_{\max}$ is the maximum delay and $T$ is the time horizon.
However, in many cases, this guarantee can be much worse than $\sqrt{d_{\mathrm{tot}}}$ as obtained by a delayed version of online gradient descent, where $d_{\mathrm{tot}}$ is the total delay.
We bridge this gap by proposing a variant of follow-the-regularized-leader that obtains regret of order $\min\\{\sigma_{\max}\ln T, \sqrt{d_{\mathrm{tot}}}\\}$, where $\sigma_{\max}$ is the maximum number of missing observations.
We then consider exp-concave losses and extend the Online Newton Step algorithm to handle delays with an adaptive learning rate tuning, achieving regret $\min\\{d_{\max} n\ln T, \sqrt{d_{\mathrm{tot}}}\\}$ where $n$ is the dimension.
To our knowledge, this is the first algorithm to achieve such a regret bound for exp-concave losses.
We further consider the problem of unconstrained online linear regression and achieve a similar guarantee by designing a variant of the Vovk-Azoury-Warmuth forecaster with a clipping trick.
Finally, we implement our algorithms and conduct experiments under various types of delay and losses, showing an improved performance over existing methods.

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Track: Fast Track: published work

Publication Link: https://icml.cc/virtual/2025/poster/44622

Submission Number: 124