Exploiting Curvature in Online Convex Optimization with Delayed Feedback

Hao Qiu, Emmanuel Esposito, Mengxiao Zhang
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:50448-50479, 2025.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-qiu25a, title = {Exploiting Curvature in Online Convex Optimization with Delayed Feedback}, author = {Qiu, Hao and Esposito, Emmanuel and Zhang, Mengxiao}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {50448--50479}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/qiu25a/qiu25a.pdf}, url = {https://proceedings.mlr.press/v267/qiu25a.html}, 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.} }
Endnote
%0 Conference Paper %T Exploiting Curvature in Online Convex Optimization with Delayed Feedback %A Hao Qiu %A Emmanuel Esposito %A Mengxiao Zhang %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-qiu25a %I PMLR %P 50448--50479 %U https://proceedings.mlr.press/v267/qiu25a.html %V 267 %X 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.
APA
Qiu, H., Esposito, E. & Zhang, M.. (2025). Exploiting Curvature in Online Convex Optimization with Delayed Feedback. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:50448-50479 Available from https://proceedings.mlr.press/v267/qiu25a.html.

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