Smooth Bandit Optimization: Generalization to Holder Space
Yusha Liu, Yining Wang, Aarti Singh
Abstract
We consider bandit optimization of a smooth reward function, where the goal is cumulative regret minimization. This problem has been studied for $\alpha$-Holder continuous (including Lipschitz) functions with $0<\alpha\leq 1$. Our main result is in generalization of the reward function to Holder space with exponent $\alpha>1$ to bridge the gap between Lipschitz bandits and infinitely-differentiable models such as linear bandits. For Holder continuous functions, approaches based on random sampling in bins of a discretized domain suffices as optimal. In contrast, we propose a class of two-layer algorithms that deploy misspecified linear/polynomial bandit algorithms in bins. We demonstrate that the proposed algorithm can exploit higher-order smoothness of the function by deriving a regret upper bound of $\tilde{O}(T^\frac{d+\alpha}{d+2\alpha})$ for when $\alpha>1$, which matches existing lower bound. We also study adaptation to unknown function smoothness over a continuous scale of Holder spaces indexed by $\alpha$, with a bandit model selection approach applied with our proposed two-layer algorithms. We show that it achieves regret rate that matches the existing lower bound for adaptation within the $\alpha\leq 1$ subset.
BibTeX
@InProceedings{pmlr-v130-liu21f,
title = { Smooth Bandit Optimization: Generalization to Holder Space },
author = {Liu, Yusha and Wang, Yining and Singh, Aarti},
booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics},
pages = {2206--2214},
year = {2021},
editor = {Banerjee, Arindam and Fukumizu, Kenji},
volume = {130},
series = {Proceedings of Machine Learning Research},
month = {13--15 Apr},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v130/liu21f/liu21f.pdf},
url = {https://proceedings.mlr.press/v130/liu21f.html},
abstract = { We consider bandit optimization of a smooth reward function, where the goal is cumulative regret minimization. This problem has been studied for $\alpha$-Holder continuous (including Lipschitz) functions with $0<\alpha\leq 1$. Our main result is in generalization of the reward function to Holder space with exponent $\alpha>1$ to bridge the gap between Lipschitz bandits and infinitely-differentiable models such as linear bandits. For Holder continuous functions, approaches based on random sampling in bins of a discretized domain suffices as optimal. In contrast, we propose a class of two-layer algorithms that deploy misspecified linear/polynomial bandit algorithms in bins. We demonstrate that the proposed algorithm can exploit higher-order smoothness of the function by deriving a regret upper bound of $\tilde{O}(T^\frac{d+\alpha}{d+2\alpha})$ for when $\alpha>1$, which matches existing lower bound. We also study adaptation to unknown function smoothness over a continuous scale of Holder spaces indexed by $\alpha$, with a bandit model selection approach applied with our proposed two-layer algorithms. We show that it achieves regret rate that matches the existing lower bound for adaptation within the $\alpha\leq 1$ subset. }
}