Adaptive Bandit Convex Optimization with Heterogeneous Curvature

Haipeng Luo, Mengxiao Zhang, Peng Zhao
Proceedings of Thirty Fifth Conference on Learning Theory, PMLR 178:1576-1612, 2022.

Abstract

We consider the problem of adversarial bandit convex optimization, that is, online learning over a sequence of arbitrary convex loss functions with only one function evaluation for each of them. While all previous works assume known and homogeneous curvature on these loss functions, we study a heterogeneous setting where each function has its own curvature that is only revealed after the learner makes a decision. We develop an efficient algorithm that is able to adapt to the curvature on the fly. Specifically, our algorithm not only recovers or \emph{even improves} existing results for several homogeneous settings, but also leads to surprising results for some heterogeneous settings — for example, while Hazan and Levy (2014) showed that $\tilde{O}(d^{\frac{3}{2}}\sqrt{T})$ regret is achievable for a sequence of $T$ smooth and strongly convex $d$-dimensional functions, our algorithm reveals that the same is achievable even if $T^{\frac{3}{4}}$ of them are not strongly convex, and sometimes even if a constant fraction of them are not strongly convex. Our approach is inspired by the framework of Bartlett et al. (2007) who studied a similar heterogeneous setting but with stronger gradient feedback. Extending their framework to the bandit feedback setting requires novel ideas such as lifting the feasible domain and using a logarithmically homogeneous self-concordant barrier regularizer.

Cite this Paper

BibTeX
@InProceedings{pmlr-v178-luo22a, title = {Adaptive Bandit Convex Optimization with Heterogeneous Curvature}, author = {Luo, Haipeng and Zhang, Mengxiao and Zhao, Peng}, booktitle = {Proceedings of Thirty Fifth Conference on Learning Theory}, pages = {1576--1612}, year = {2022}, editor = {Loh, Po-Ling and Raginsky, Maxim}, volume = {178}, series = {Proceedings of Machine Learning Research}, month = {02--05 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v178/luo22a/luo22a.pdf}, url = {https://proceedings.mlr.press/v178/luo22a.html}, abstract = {We consider the problem of adversarial bandit convex optimization, that is, online learning over a sequence of arbitrary convex loss functions with only one function evaluation for each of them. While all previous works assume known and homogeneous curvature on these loss functions, we study a heterogeneous setting where each function has its own curvature that is only revealed after the learner makes a decision. We develop an efficient algorithm that is able to adapt to the curvature on the fly. Specifically, our algorithm not only recovers or \emph{even improves} existing results for several homogeneous settings, but also leads to surprising results for some heterogeneous settings — for example, while Hazan and Levy (2014) showed that $\tilde{O}(d^{\frac{3}{2}}\sqrt{T})$ regret is achievable for a sequence of $T$ smooth and strongly convex $d$-dimensional functions, our algorithm reveals that the same is achievable even if $T^{\frac{3}{4}}$ of them are not strongly convex, and sometimes even if a constant fraction of them are not strongly convex. Our approach is inspired by the framework of Bartlett et al. (2007) who studied a similar heterogeneous setting but with stronger gradient feedback. Extending their framework to the bandit feedback setting requires novel ideas such as lifting the feasible domain and using a logarithmically homogeneous self-concordant barrier regularizer.} }
Endnote
%0 Conference Paper %T Adaptive Bandit Convex Optimization with Heterogeneous Curvature %A Haipeng Luo %A Mengxiao Zhang %A Peng Zhao %B Proceedings of Thirty Fifth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2022 %E Po-Ling Loh %E Maxim Raginsky %F pmlr-v178-luo22a %I PMLR %P 1576--1612 %U https://proceedings.mlr.press/v178/luo22a.html %V 178 %X We consider the problem of adversarial bandit convex optimization, that is, online learning over a sequence of arbitrary convex loss functions with only one function evaluation for each of them. While all previous works assume known and homogeneous curvature on these loss functions, we study a heterogeneous setting where each function has its own curvature that is only revealed after the learner makes a decision. We develop an efficient algorithm that is able to adapt to the curvature on the fly. Specifically, our algorithm not only recovers or \emph{even improves} existing results for several homogeneous settings, but also leads to surprising results for some heterogeneous settings — for example, while Hazan and Levy (2014) showed that $\tilde{O}(d^{\frac{3}{2}}\sqrt{T})$ regret is achievable for a sequence of $T$ smooth and strongly convex $d$-dimensional functions, our algorithm reveals that the same is achievable even if $T^{\frac{3}{4}}$ of them are not strongly convex, and sometimes even if a constant fraction of them are not strongly convex. Our approach is inspired by the framework of Bartlett et al. (2007) who studied a similar heterogeneous setting but with stronger gradient feedback. Extending their framework to the bandit feedback setting requires novel ideas such as lifting the feasible domain and using a logarithmically homogeneous self-concordant barrier regularizer.
APA
Luo, H., Zhang, M. & Zhao, P.. (2022). Adaptive Bandit Convex Optimization with Heterogeneous Curvature. Proceedings of Thirty Fifth Conference on Learning Theory, in Proceedings of Machine Learning Research 178:1576-1612 Available from https://proceedings.mlr.press/v178/luo22a.html.

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