Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization

Jonathan Scarlett, Ilija Bogunovic, Volkan Cevher
Proceedings of the 2017 Conference on Learning Theory, PMLR 65:1723-1742, 2017.

Abstract

In this paper, we consider the problem of sequentially optimizing a black-box function $f$ based on noisy samples and bandit feedback. We assume that $f$ is smooth in the sense of having a bounded norm in some reproducing kernel Hilbert space (RKHS), yielding a commonly-considered non-Bayesian form of Gaussian process bandit optimization. We provide algorithm-independent lower bounds on the simple regret, measuring the suboptimality of a single point reported after $T$ rounds, and on the cumulative regret, measuring the sum of regrets over the $T$ chosen points. For the isotropic squared-exponential kernel in $d$ dimensions, we find that an average simple regret of $ε$ requires $T = Ω\big(\frac1ε^2 (\log\frac1ε)^d/2\big)$, and the average cumulative regret is at least $Ω\big( \sqrt{T}(\log T)^d \big)$, thus matching existing upper bounds up to the replacement of $d/2$ by $d+O(1)$ in both cases. For the Matérn-$ν$ kernel, we give analogous bounds of the form $Ω\big( (\frac1ε)^2+d/ν\big)$ and $Ω\big( T^\fracν+ d2ν+ d \big)$, and discuss the resulting gaps to the existing upper bounds.

Cite this Paper

BibTeX
@InProceedings{pmlr-v65-scarlett17a, title = {Lower Bounds on Regret for Noisy {G}aussian Process Bandit Optimization}, author = {Scarlett, Jonathan and Bogunovic, Ilija and Cevher, Volkan}, booktitle = {Proceedings of the 2017 Conference on Learning Theory}, pages = {1723--1742}, year = {2017}, editor = {Kale, Satyen and Shamir, Ohad}, volume = {65}, series = {Proceedings of Machine Learning Research}, month = {07--10 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v65/scarlett17a/scarlett17a.pdf}, url = {https://proceedings.mlr.press/v65/scarlett17a.html}, abstract = {In this paper, we consider the problem of sequentially optimizing a black-box function $f$ based on noisy samples and bandit feedback. We assume that $f$ is smooth in the sense of having a bounded norm in some reproducing kernel Hilbert space (RKHS), yielding a commonly-considered non-Bayesian form of Gaussian process bandit optimization. We provide algorithm-independent lower bounds on the simple regret, measuring the suboptimality of a single point reported after $T$ rounds, and on the cumulative regret, measuring the sum of regrets over the $T$ chosen points. For the isotropic squared-exponential kernel in $d$ dimensions, we find that an average simple regret of $ε$ requires $T = Ω\big(\frac1ε^2 (\log\frac1ε)^d/2\big)$, and the average cumulative regret is at least $Ω\big( \sqrt{T}(\log T)^d \big)$, thus matching existing upper bounds up to the replacement of $d/2$ by $d+O(1)$ in both cases. For the Matérn-$ν$ kernel, we give analogous bounds of the form $Ω\big( (\frac1ε)^2+d/ν\big)$ and $Ω\big( T^\fracν+ d2ν+ d \big)$, and discuss the resulting gaps to the existing upper bounds.} }
Endnote
%0 Conference Paper %T Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization %A Jonathan Scarlett %A Ilija Bogunovic %A Volkan Cevher %B Proceedings of the 2017 Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2017 %E Satyen Kale %E Ohad Shamir %F pmlr-v65-scarlett17a %I PMLR %P 1723--1742 %U https://proceedings.mlr.press/v65/scarlett17a.html %V 65 %X In this paper, we consider the problem of sequentially optimizing a black-box function $f$ based on noisy samples and bandit feedback. We assume that $f$ is smooth in the sense of having a bounded norm in some reproducing kernel Hilbert space (RKHS), yielding a commonly-considered non-Bayesian form of Gaussian process bandit optimization. We provide algorithm-independent lower bounds on the simple regret, measuring the suboptimality of a single point reported after $T$ rounds, and on the cumulative regret, measuring the sum of regrets over the $T$ chosen points. For the isotropic squared-exponential kernel in $d$ dimensions, we find that an average simple regret of $ε$ requires $T = Ω\big(\frac1ε^2 (\log\frac1ε)^d/2\big)$, and the average cumulative regret is at least $Ω\big( \sqrt{T}(\log T)^d \big)$, thus matching existing upper bounds up to the replacement of $d/2$ by $d+O(1)$ in both cases. For the Matérn-$ν$ kernel, we give analogous bounds of the form $Ω\big( (\frac1ε)^2+d/ν\big)$ and $Ω\big( T^\fracν+ d2ν+ d \big)$, and discuss the resulting gaps to the existing upper bounds.
APA
Scarlett, J., Bogunovic, I. & Cevher, V.. (2017). Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization. Proceedings of the 2017 Conference on Learning Theory, in Proceedings of Machine Learning Research 65:1723-1742 Available from https://proceedings.mlr.press/v65/scarlett17a.html.

Related Material