A General Framework for Bandit Problems Beyond Cumulative Objectives

Asaf Cassel*, Shie Mannor, Assaf Zeevi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The stochastic multiarmed 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, which 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.

Original languageEnglish
Pages (from-to)2196-2232
Number of pages37
JournalMathematics of Operations Research
Volume48
Issue number4
DOIs
StatePublished - 2023

Keywords

  • multiarmed bandit
  • optimism principle
  • planning
  • reinforcement learning
  • risk
  • upper confidence bound

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