Algorithms with Logarithmic or Sublinear Regret for Constrained Contextual Bandits

TL;DR

This paper introduces algorithms for constrained contextual bandits that achieve logarithmic or sublinear regret, addressing complex coupling effects caused by budget and time constraints, and extends to systems with unknown distributions.

Contribution

It develops the first algorithms achieving logarithmic regret in constrained contextual bandits, combining an approximation of the oracle with UCB methods for unknown rewards.

Findings

UCB-ALP algorithm achieves near-optimal regret.

Logarithmic regret is possible even with constraints.

Algorithms extend to unknown context distributions.

Abstract

We study contextual bandits with budget and time constraints, referred to as constrained contextual bandits.The time and budget constraints significantly complicate the exploration and exploitation tradeoff because they introduce complex coupling among contexts over time.Such coupling effects make it difficult to obtain oracle solutions that assume known statistics of bandits. To gain insight, we first study unit-cost systems with known context distribution. When the expected rewards are known, we develop an approximation of the oracle, referred to Adaptive-Linear-Programming (ALP), which achieves near-optimality and only requires the ordering of expected rewards. With these highly desirable features, we then combine ALP with the upper-confidence-bound (UCB) method in the general case where the expected rewards are unknown {\it a priori}. We show that the proposed UCB-ALP algorithm achieves logarithmic regret except for certain boundary cases. Further, we design algorithms and obtain similar regret analysis results for more general systems with unknown context distribution and heterogeneous costs. To the best of our knowledge, this is the first work that shows how to achieve logarithmic regret in constrained contextual bandits. Moreover, this work also sheds light on the study of computationally efficient algorithms for general constrained contextual bandits.