Optimal Order Simple Regret for Gaussian Process Bandits

• URL: http://arxiv.org/abs/2108.09262v1
• Date: Fri, 20 Aug 2021 16:49:32 GMT
• Title: Optimal Order Simple Regret for Gaussian Process Bandits
• Authors: Sattar Vakili, Nacime Bouziani, Sepehr Jalali, Alberto Bernacchia, Da-shan Shiu
• Abstract summary: We show that a pure exploration algorithm is significantly tighter than the existing bounds. We show that this regret is optimal up to logarithmically for cases where a lower bound on a kernel is known.
• Score: 6.84224661918159
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Consider the sequential optimization of a continuous, possibly non-convex, and expensive to evaluate objective function $f$. The problem can be cast as a Gaussian Process (GP) bandit where $f$ lives in a reproducing kernel Hilbert space (RKHS). The state of the art analysis of several learning algorithms shows a significant gap between the lower and upper bounds on the simple regret performance. When $N$ is the number of exploration trials and $\gamma_N$ is the maximal information gain, we prove an $\tilde{\mathcal{O}}(\sqrt{\gamma_N/N})$ bound on the simple regret performance of a pure exploration algorithm that is significantly tighter than the existing bounds. We show that this bound is order optimal up to logarithmic factors for the cases where a lower bound on regret is known. To establish these results, we prove novel and sharp confidence intervals for GP models applicable to RKHS elements which may be of broader interest.

Related papers

• Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path [80.60592344361073]
  We study the Shortest Path (SSP) problem with a linear mixture transition kernel. An agent repeatedly interacts with a environment and seeks to reach certain goal state while minimizing the cumulative cost. Existing works often assume a strictly positive lower bound of the iteration cost function or an upper bound of the expected length for the optimal policy.
  arXiv  Detail & Related papers  (2024-02-14T07:52:00Z)
• A Robust Phased Elimination Algorithm for Corruption-Tolerant Gaussian Process Bandits [118.22458816174144]
  We propose a novel robust elimination-type algorithm that runs in epochs, combines exploration with infrequent switching to select a small subset of actions, and plays each action for multiple time instants. Our algorithm, GP Robust Phased Elimination (RGP-PE), successfully balances robustness to corruptions with exploration and exploitation. We perform the first empirical study of robustness in the corrupted GP bandit setting, and show that our algorithm is robust against a variety of adversarial attacks.
  arXiv  Detail & Related papers  (2022-02-03T21:19:36Z)
• Misspecified Gaussian Process Bandit Optimization [59.30399661155574]
  Kernelized bandit algorithms have shown strong empirical and theoretical performance for this problem. We introduce a emphmisspecified kernelized bandit setting where the unknown function can be $epsilon$--uniformly approximated by a function with a bounded norm in some Reproducing Kernel Hilbert Space (RKHS) We show that our algorithm achieves optimal dependence on $epsilon$ with no prior knowledge of misspecification.
  arXiv  Detail & Related papers  (2021-11-09T09:00:02Z)
• Gaussian Process Bandit Optimization with Few Batches [49.896920704012395]
  We introduce a batch algorithm inspired by finite-arm bandit algorithms. We show that it achieves the cumulative regret upper bound $Oast(sqrtTgamma_T)$ using $O(loglog T)$ batches within time horizon $T$. In addition, we propose a modified version of our algorithm, and characterize how the regret is impacted by the number of batches.
  arXiv  Detail & Related papers  (2021-10-15T00:54:04Z)
• Randomized Exploration for Reinforcement Learning with General Value Function Approximation [122.70803181751135]
  We propose a model-free reinforcement learning algorithm inspired by the popular randomized least squares value iteration (RLSVI) algorithm. Our algorithm drives exploration by simply perturbing the training data with judiciously chosen i.i.d. scalar noises. We complement the theory with an empirical evaluation across known difficult exploration tasks.
  arXiv  Detail & Related papers  (2021-06-15T02:23:07Z)
• Lenient Regret and Good-Action Identification in Gaussian Process Bandits [43.03669155559218]
  We study the problem of Gaussian process (GP) bandits under relaxed optimization criteria stating that any function value above a certain threshold is "good enough" On the practical side, we consider the problem of finding a single "good action" according to a known pre-specified threshold, and introduce several good-action identification algorithms that exploit knowledge of the threshold.
  arXiv  Detail & Related papers  (2021-02-11T01:16:58Z)
• On Information Gain and Regret Bounds in Gaussian Process Bandits [8.499604625449907]
  Consider the optimization of an expensive to evaluate and possibly non- sequential objective function $f$ from noisy feedback observations. For the Matern family of kernels, lower bounds on $gamma_T$, and regret under the frequentist setting, are known.
  arXiv  Detail & Related papers  (2020-09-15T10:15:29Z)
• Exploiting Higher Order Smoothness in Derivative-free Optimization and Continuous Bandits [99.70167985955352]
  We study the problem of zero-order optimization of a strongly convex function. We consider a randomized approximation of the projected gradient descent algorithm. Our results imply that the zero-order algorithm is nearly optimal in terms of sample complexity and the problem parameters.
  arXiv  Detail & Related papers  (2020-06-14T10:42:23Z)
• Multi-Scale Zero-Order Optimization of Smooth Functions in an RKHS [19.252319300590653]
  A black-box function $f:mathcalX mapsto mathbbR$ is optimized under the assumption that $f$ is H"older smooth and has bounded norm in the RKHS associated with a given kernel $K$. In this paper, we propose a new algorithm (textttLP-GP-UCB) where the usual GP surrogate model is augmented with Local Polynomial (LP) estimators of the H" smooth parameter $f$.
  arXiv  Detail & Related papers  (2020-05-11T01:55:39Z)
• Regret and Belief Complexity Trade-off in Gaussian Process Bandits via Information Thresholding [42.669970064867556]
  We show how to characterize the trade-off between regret bounds of GP bandit algorithms and complexity of the posterior distributions. We observe state of the art accuracy and complexity trade-offs for GP bandit algorithms applied to global optimization.
  arXiv  Detail & Related papers  (2020-03-23T21:05:15Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.