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This paper addresses the limitations of traditional safety-constrained contextual bandit models that rely solely on expected cost thresholds, which can be inadequate in heteroscedastic environments. The authors introduce the High-Probability Constrained UCB algorithm, which optimally balances exploration for reward with a conservative estimation of the safe action set, ensuring that the realized costs remain within safe limits. Their results demonstrate a significant reduction in safety violations compared to existing expected-cost constrained approaches, achieving a tight regret bound of $\tilde{\mathcal{O}}(d\sqrt{T})$ for linear models.
Enforcing realized-cost safety can dramatically reduce violations in contextual bandits, outperforming traditional expected-cost methods.
Contextual bandits are a standard framework for sequential decision-making under uncertainty, with applications in clinical trials, dosage selection, recommendation systems, and autonomous systems. Safety is central in many of these applications, since a single unsafe decision in settings such as dosage selection or autonomous driving can have catastrophic consequences. A common way to model safety in bandit problems is to associate each action with both a reward signal and a cost signal, and to optimize reward subject to constraints on cost. Most existing safety-constrained bandit models enforce safety by requiring the expected cost of each action to remain below a prescribed threshold. However, this may be insufficient in heteroscedastic settings, where the chosen action affects not only the expected reward and cost, but also the variability of the observed outcomes. We study contextual bandits with one-dimensional continuous actions and stage-wise high-probability constraints on the realized cost. We propose High-Probability Constrained UCB, an optimistic-pessimistic algorithm that explores for reward while conservatively estimating the safe action set. For linear reward and cost models, we prove a tight $\tilde{\mathcal{O}}(d\sqrt{T})$ regret bound, and we extend the analysis to general function classes using the eluder dimension. Experiments show that enforcing realized-cost safety substantially reduces violations compared with expected-cost constrained baselines.
