Bayesian Design Principles for Frequentist Sequential Learning

TL;DR

This paper introduces a Bayesian-inspired framework for sequential learning that optimizes frequentist regret, enabling the development of simple, prior-free algorithms applicable across stochastic, adversarial, and non-stationary environments.

Contribution

It presents a novel optimization approach to generate algorithmic beliefs, unifying Bayesian principles with frequentist regret minimization in a generic, prior-free manner.

Findings

Developed a new algorithm for multi-armed bandits with best-of-all-worlds performance.

Proposed the concept of Algorithmic Information Ratio as a complexity measure.

Demonstrated applicability to linear bandits, bandit convex optimization, and reinforcement learning.

Abstract

We develop a general theory to optimize the frequentist regret for sequential learning problems, where efficient bandit and reinforcement learning algorithms can be derived from unified Bayesian principles. We propose a novel optimization approach to generate "algorithmic beliefs" at each round, and use Bayesian posteriors to make decisions. The optimization objective to create "algorithmic beliefs," which we term "Algorithmic Information Ratio," represents an intrinsic complexity measure that effectively characterizes the frequentist regret of any algorithm. To the best of our knowledge, this is the first systematical approach to make Bayesian-type algorithms prior-free and applicable to adversarial settings, in a generic and optimal manner. Moreover, the algorithms are simple and often efficient to implement. As a major application, we present a novel algorithm for multi-armed bandits that achieves the "best-of-all-worlds" empirical performance in the stochastic, adversarial, and non-stationary environments. And we illustrate how these principles can be used in linear bandits, bandit convex optimization, and reinforcement learning.