Tight Bounds for $\gamma$-Regret via the Decision-Estimation Coefficient

• URL: http://arxiv.org/abs/2303.03327v2
• Date: Fri, 21 Jul 2023 17:54:14 GMT
• Title: Tight Bounds for $\gamma$-Regret via the Decision-Estimation Coefficient
• Authors: Margalit Glasgow and Alexander Rakhlin
• Abstract summary: We give a statistical characterization of the $gamma$-regret for arbitrary structured bandit problems. The $gamma$-regret emerges in structured bandit problems over a function class $mathcalF$ where finding an exact optimum of $f in mathcalF$ is intractable.
• Score: 88.86699022151598
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this work, we give a statistical characterization of the $\gamma$-regret for arbitrary structured bandit problems, the regret which arises when comparing against a benchmark that is $\gamma$ times the optimal solution. The $\gamma$-regret emerges in structured bandit problems over a function class $\mathcal{F}$ where finding an exact optimum of $f \in \mathcal{F}$ is intractable. Our characterization is given in terms of the $\gamma$-DEC, a statistical complexity parameter for the class $\mathcal{F}$, which is a modification of the constrained Decision-Estimation Coefficient (DEC) of Foster et al., 2023 (and closely related to the original offset DEC of Foster et al., 2021). Our lower bound shows that the $\gamma$-DEC is a fundamental limit for any model class $\mathcal{F}$: for any algorithm, there exists some $f \in \mathcal{F}$ for which the $\gamma$-regret of that algorithm scales (nearly) with the $\gamma$-DEC of $\mathcal{F}$. We provide an upper bound showing that there exists an algorithm attaining a nearly matching $\gamma$-regret. Due to significant challenges in applying the prior results on the DEC to the $\gamma$-regret case, both our lower and upper bounds require novel techniques and a new algorithm.

Related papers

• p-Mean Regret for Stochastic Bandits [52.828710025519996]
  We introduce a simple, unified UCB-based algorithm that achieves novel $p$-mean regret bounds. Our framework encompasses both average cumulative regret and Nash regret as special cases.
  arXiv  Detail & Related papers  (2024-12-14T08:38:26Z)
• Inverting the Leverage Score Gradient: An Efficient Approximate Newton Method [10.742859956268655]
  This paper aims to recover the intrinsic model parameters given the leverage scores gradient. We specifically scrutinize the inversion of the leverage score gradient, denoted as $g(x)$.
  arXiv  Detail & Related papers  (2024-08-21T01:39:42Z)
• A simple and improved algorithm for noisy, convex, zeroth-order optimisation [59.51990161522328]
  We construct an algorithm that returns a point $hat xin barmathcal X$ such that $f(hat x)$ is as small as possible. We prove that this method is such that the $f(hat x) - min_xin barmathcal X f(x)$ is of smaller order than $d2/sqrtn$ up to poly-logarithmic terms.
  arXiv  Detail & Related papers  (2024-06-26T18:19:10Z)
• Fast $(1+\varepsilon)$-Approximation Algorithms for Binary Matrix Factorization [54.29685789885059]
  We introduce efficient $(1+varepsilon)$-approximation algorithms for the binary matrix factorization (BMF) problem. The goal is to approximate $mathbfA$ as a product of low-rank factors. Our techniques generalize to other common variants of the BMF problem.
  arXiv  Detail & Related papers  (2023-06-02T18:55:27Z)
• Instance-Dependent Bounds for Zeroth-order Lipschitz Optimization with Error Certificates [0.0]
  We study the problem of zeroth-order (black-box) optimization of a Lipschitz function $f$ defined on a compact subset $mathcal X$ of $mathbb Rd$. We characterize the optimal number of evaluations of any Lipschitz function $f$ to find and certify an approximater of $f$ at accuracy $varepsilon$.
  arXiv  Detail & Related papers  (2021-02-03T09:51:03Z)
• Near-Optimal Regret Bounds for Contextual Combinatorial Semi-Bandits with Linear Payoff Functions [53.77572276969548]
  We show that the C$2$UCB algorithm has the optimal regret bound $tildeO(dsqrtkT + dk)$ for the partition matroid constraints. For general constraints, we propose an algorithm that modifies the reward estimates of arms in the C$2$UCB algorithm.
  arXiv  Detail & Related papers  (2021-01-20T04:29:18Z)
• Revisiting Modified Greedy Algorithm for Monotone Submodular Maximization with a Knapsack Constraint [75.85952446237599]
  We show that a modified greedy algorithm can achieve an approximation factor of $0.305$. We derive a data-dependent upper bound on the optimum. It can also be used to significantly improve the efficiency of such algorithms as branch and bound.
  arXiv  Detail & Related papers  (2020-08-12T15:40:21Z)

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.