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arXiv 2202.06150cs.LGstat.ML
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具有非均匀曲率的自适应Bandit凸优化

Adaptive Bandit Convex Optimization with Heterogeneous Curvature

University of Southern California(南加利福尼亚大学);National Key Laboratory for Novel Software Technology, Nanjing University(南京大学新型软件技术国家重点实验室)

Haipeng Luo, Mengxiao Zhang, Peng Zhao

摘要:

我们考虑对抗性的凸优化问题,即在一系列任意凸损失函数上的在线学习,其中每个损失函数只有一个函数评价。虽然所有以前的作品假设这些损失函数的已知和均匀的曲率,我们研究了一个异构的设置,每个函数都有自己的曲率,只有在学习者做出决定后才显示。我们开发了一种有效的算法,能够适应飞行中的曲率。具体来说,我们的算法不仅恢复或\emph{even improves}现有的结果为几个同质设置,但也导致了令人惊讶的结果为一些异质设置-例如,而Hazan和Levy(2014)表明,$\widetilde{O}(d^{3/2}\sqrt{T})$遗憾是可实现的一系列$T$光滑和强凸$d$维函数,我们的算法揭示了即使它们中的$T^{3/4}$不是强凸的,并且有时即使它们中的常数部分不是强凸的,也是可以实现的。我们的方法受到Bartlett等人(2007)的框架的启发,他们研究了类似的异质环境,但具有更强的梯度反馈。扩展他们的框架,强盗反馈设置需要新的想法,如解除可行域,并使用一个几何齐次自和谐障碍正则化。

英文摘要:

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 $\widetilde{O}(d^{3/2}\sqrt{T})$ 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 $T^{3/4}$ 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.

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