Adversarial Rewards in Universal Learning for Contextual Bandits

TL;DR

This paper investigates the limits of universal learning in contextual bandits with evolving, potentially adversarial rewards, revealing fundamental impossibilities and conditions for learnability beyond traditional models.

Contribution

It demonstrates the impossibility of optimistic universal learning in adversarial reward settings and provides necessary and sufficient conditions for universal learning under various adversarial models.

Findings

Universal learning is impossible with adversarial rewards in general.

Conditions for universal learning depend on the reward model.

Learnable processes are larger than i.i.d. but smaller than supervised learning.

Abstract

We study the fundamental limits of learning in contextual bandits, where a learner's rewards depend on their actions and a known context, which extends the canonical multi-armed bandit to the case where side-information is available. We are interested in universally consistent algorithms, which achieve sublinear regret compared to any measurable fixed policy, without any function class restriction. For stationary contextual bandits, when the underlying reward mechanism is time-invariant, Blanchard et. al (2022) characterized learnable context processes for which universal consistency is achievable; and further gave algorithms ensuring universal consistency whenever this is achievable, a property known as optimistic universal consistency. It is well understood, however, that reward mechanisms can evolve over time, possibly adversarially, and depending on the learner's actions. We show that optimistic universal learning for contextual bandits with adversarial rewards is impossible in general, contrary to all previously studied settings in online learning -- including standard supervised learning. We also give necessary and sufficient conditions for universal learning under various adversarial reward models, and an exact characterization for online rewards. In particular, the set of learnable processes for these reward models is still extremely general -- larger than i.i.d., stationary or ergodic -- but in general strictly smaller than that for supervised learning or stationary contextual bandits, shedding light on new adversarial phenomena.