An Optimal Algorithm for Bandit Convex Optimization with Strongly-Convex and Smooth Loss
Abstract
We consider non-stochastic bandit convex optimization with strongly-convex and smooth loss functions. For this problem, Hazan and Levy have proposed an algorithm with a regret bound of $\tilde{O}(d^{3/2} \sqrt{T})$ given access to an $O(d)$-self-concordant barrier over the feasible region, where $d$ and $T$ stand for the dimensionality of the feasible region and the number of rounds, respectively. However, there are no known efficient ways for constructing self-concordant barriers for general convex sets, and a $\tilde{O}(\sqrt{d})$ gap has remained between the upper and lower bounds, as the known regret lower bound is $\Omega(d\sqrt{T})$. Our study resolves these two issues by introducing an algorithm that achieves an optimal regret bound of $\tilde{O}(d \sqrt{T})$ under a mild assumption, without self-concordant barriers. More precisely, the algorithm requires only a membership oracle for the feasible region, and it achieves an optimal regret bound of $\tilde{O}(d\sqrt{T})$ under the assumption that the optimal solution is an interior of the feasible region. Even without this assumption, our algorithm achieves $\tilde{O}(d^{3/2}\sqrt{T})$-regret.
BibTeX
@InProceedings{pmlr-v108-ito20a,
title = {An Optimal Algorithm for Bandit Convex Optimization with Strongly-Convex and Smooth Loss},
author = {Ito, Shinji},
booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics},
pages = {2229--2239},
year = {2020},
editor = {Chiappa, Silvia and Calandra, Roberto},
volume = {108},
series = {Proceedings of Machine Learning Research},
month = {26--28 Aug},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v108/ito20a/ito20a.pdf},
url = {https://proceedings.mlr.press/v108/ito20a.html},
abstract = {We consider non-stochastic bandit convex optimization with strongly-convex and smooth loss functions. For this problem, Hazan and Levy have proposed an algorithm with a regret bound of $\tilde{O}(d^{3/2} \sqrt{T})$ given access to an $O(d)$-self-concordant barrier over the feasible region, where $d$ and $T$ stand for the dimensionality of the feasible region and the number of rounds, respectively. However, there are no known efficient ways for constructing self-concordant barriers for general convex sets, and a $\tilde{O}(\sqrt{d})$ gap has remained between the upper and lower bounds, as the known regret lower bound is $\Omega(d\sqrt{T})$. Our study resolves these two issues by introducing an algorithm that achieves an optimal regret bound of $\tilde{O}(d \sqrt{T})$ under a mild assumption, without self-concordant barriers. More precisely, the algorithm requires only a membership oracle for the feasible region, and it achieves an optimal regret bound of $\tilde{O}(d\sqrt{T})$ under the assumption that the optimal solution is an interior of the feasible region. Even without this assumption, our algorithm achieves $\tilde{O}(d^{3/2}\sqrt{T})$-regret.}
}