A General Framework for Bandit Problems Beyond Cumulative Objectives

TL;DR

This paper introduces a unified framework for multi-armed bandit problems with complex, non-cumulative objectives, providing conditions for tractable oracle policies and designing optimism-based algorithms for various risk-sensitive criteria.

Contribution

It develops a systematic approach for bandit problems with non-cumulative objectives, identifying conditions for oracle policy tractability and guiding the design of UCB algorithms.

Findings

Conditions for oracle policy tractability are established.

Framework applied to objectives like CVaR, mean-variance, and Sharpe ratio.

Guidelines for designing optimism-based policies for complex objectives.

Abstract

The stochastic multi-armed bandit (MAB) problem is a common model for sequential decision problems. In the standard setup, a decision maker has to choose at every instant between several competing arms, each of them provides a scalar random variable, referred to as a "reward." Nearly all research on this topic considers the total cumulative reward as the criterion of interest. This work focuses on other natural objectives that cannot be cast as a sum over rewards, but rather more involved functions of the reward stream. Unlike the case of cumulative criteria, in the problems we study here the oracle policy, that knows the problem parameters a priori and is used to "center" the regret, is not trivial. We provide a systematic approach to such problems, and derive general conditions under which the oracle policy is sufficiently tractable to facilitate the design of optimism-based (upper confidence bound) learning policies. These conditions elucidate an interesting interplay between the arm reward distributions and the performance metric. Our main findings are illustrated for several commonly used objectives such as conditional value-at-risk, mean-variance trade-offs, Sharpe-ratio, and more.