Stage-wise Conservative Linear Bandits

TL;DR

This paper introduces two algorithms for stage-wise conservative linear bandits that ensure safety constraints while optimizing rewards, providing regret bounds and adaptability to various constraint settings.

Contribution

The paper proposes novel algorithms, SCLTS and SCLUCB, for safe linear bandit optimization with regret guarantees and flexibility for different safety constraint scenarios.

Findings

SCLTS and SCLUCB achieve regret bounds of O(√T log^{3/2}T) and O(√T log T).

Algorithms limit the number of baseline actions to O(log T).

Methods adapt to constraints with bandit feedback and unknown baseline actions.

Abstract

We study stage-wise conservative linear stochastic bandits: an instance of bandit optimization, which accounts for (unknown) safety constraints that appear in applications such as online advertising and medical trials. At each stage, the learner must choose actions that not only maximize cumulative reward across the entire time horizon but further satisfy a linear baseline constraint that takes the form of a lower bound on the instantaneous reward. For this problem, we present two novel algorithms, stage-wise conservative linear Thompson Sampling (SCLTS) and stage-wise conservative linear UCB (SCLUCB), that respect the baseline constraints and enjoy probabilistic regret bounds of order O(\sqrt{T} \log^{3/2}T) and O(\sqrt{T} \log T), respectively. Notably, the proposed algorithms can be adjusted with only minor modifications to tackle different problem variations, such as constraints with bandit-feedback, or an unknown sequence of baseline actions. We discuss these and other improvements over the state-of-the-art. For instance, compared to existing solutions, we show that SCLTS plays the (non-optimal) baseline action at most O(\log{T}) times (compared to O(\sqrt{T})). Finally, we make connections to another studied form of safety constraints that takes the form of an upper bound on the instantaneous reward. While this incurs additional complexity to the learning process as the optimal action is not guaranteed to belong to the safe set at each round, we show that SCLUCB can properly adjust in this setting via a simple modification.