Second Order Methods for Bandit Optimization and Control

• URL: http://arxiv.org/abs/2402.08929v2
• Date: Thu, 03 Oct 2024 04:08:17 GMT
• Title: Second Order Methods for Bandit Optimization and Control
• Authors: Arun Suggala, Y. Jennifer Sun, Praneeth Netrapalli, Elad Hazan,
• Abstract summary: We show that our algorithm achieves optimal (in terms of terms of convex functions that we call $kappa$-2020) regret bounds for a large class of convex functions. We also investigate the adaptation of our second-order bandit algorithm to online convex optimization with memory.
• Score: 34.51425758864638
• License:
• Abstract: Bandit convex optimization (BCO) is a general framework for online decision making under uncertainty. While tight regret bounds for general convex losses have been established, existing algorithms achieving these bounds have prohibitive computational costs for high dimensional data. In this paper, we propose a simple and practical BCO algorithm inspired by the online Newton step algorithm. We show that our algorithm achieves optimal (in terms of horizon) regret bounds for a large class of convex functions that we call $\kappa$-convex. This class contains a wide range of practically relevant loss functions including linear, quadratic, and generalized linear models. In addition to optimal regret, this method is the most efficient known algorithm for several well-studied applications including bandit logistic regression. Furthermore, we investigate the adaptation of our second-order bandit algorithm to online convex optimization with memory. We show that for loss functions with a certain affine structure, the extended algorithm attains optimal regret. This leads to an algorithm with optimal regret for bandit LQR/LQG problems under a fully adversarial noise model, thereby resolving an open question posed in \citep{gradu2020non} and \citep{sun2023optimal}. Finally, we show that the more general problem of BCO with (non-affine) memory is harder. We derive a $\tilde{\Omega}(T^{2/3})$ regret lower bound, even under the assumption of smooth and quadratic losses.

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