Second Order Methods for Bandit Optimization and Control (2402.08929v2)

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.

Bandit Convex Optimization with Memory: An Efficient Algorithm and Its Analysis

Introduction to BCO-M and Algorithm Development

Bandit Convex Optimization (BCO) with memory addresses decision-making in a setting where the impact of actions persists over time, a scenario common in many real-world applications. The complexity of BCO problems increases when incorporating memory into the optimization process since past actions can affect future outcomes. To address this challenge, we propose an efficient algorithm for Bandit Quadratic Optimization with Memory (BQO-M), focusing on a framework where loss functions are quadratic and decisions have a delayed impact. Leveraging second-order methods and self-concordant barriers, our algorithm delivers practical computation time and a promising regret bound.

Technical Contributions and Regret Analysis

Our primary contributions in the field of bandit convex optimization with memory include a new algorithm leveraged on nuanced second-order methods and the strategic use of self-concordant barriers. This section explores the algorithm's core concepts, focusing on its construction, the unbiased gradient estimation it employs, and its performance in the face of delayed action effects.

Unbiased Gradient Estimator

We design an unbiased gradient estimator as a cornerstone of our algorithm. This estimator accurately captures the direction for optimizing the decision variable despite the delayed feedback mechanism inherent in BCO problems with memory. Our estimator shows that, given a specific set of past decisions, we can formulate an unbiased estimation of the gradient, which is crucial for effective optimization in such settings.

Regret Analysis

Our algorithm's regret analysis illustrates its efficiency and effectiveness. By decomposing the total regret into perturbation loss, movement loss, and underlying regret components, we offer a comprehensive view of different factors contributing to the overall performance. The analysis particularly highlights how the algorithm minimizes the regret bound effectively over time, even with the complexity introduced by the memory aspect of the BCO problem.

Practical Implications and Theoretical Significance

The algorithm's framework for addressing BCO-M showcases significant theoretical contribution by efficiently handling the delayed impact of decisions, a common challenge in online optimization problems. Moreover, the practical implications of our method are profound, offering a viable solution to managing complex decision-making scenarios in real-time systems where past actions influence future outcomes.

Future Directions in BCO-M Research

The research presents a solid foundation for future explorations into more complex forms of BCO problems with memory. Notably, extending the algorithm to handle non-quadratic losses or developing more sophisticated forms of the self-concordant barrier for diverse decision spaces could unlock new capabilities in online optimization. Moreover, investigating the algorithm's applicability in broader contexts, such as dynamic systems control and real-time resource allocation, could further demonstrate its versatility and impact.

Conclusion

This paper marks a significant step forward in Bandit Convex Optimization with memory by introducing an efficient algorithm anchored on second-order methods. Through detailed technical analysis and demonstrated performance efficiency, the research opens up new avenues for tackling complex BCO problems, paving the way for innovative applications in various real-time decision-making scenarios.