Contextual Bandits

Contextual bandits with stage-wise constraints model decisions that must control risk while learning from their outcomes. We study this problem when only the selected action's reward and cost are observed, with a general reward class and a convex class of nonlinear cost functions. We propose CS-UCCB, which combines offline regression, cost confidence sets, and upper counterfactual confidence bounds. Each regression fit defines a valid cost confidence set, while a geometry derived from covering entropy controls cumulative cost uncertainty along the sampled decisions. Under polylogarithmic covering entropy for both classes and polylogarithmic offline estimation error, the finite-action algorithm guarantees, with high probability, feasibility at every round and $\widetilde{\mathcal O}(\sqrt T)$ regret without an eluder-dimension term. We also extend the analysis to infinite actions under an explicit reward-design condition and compare our bound with existing eluder-dimension bounds.

The contextual bandit models sequential decision-making problems in which rewards depend on both the chosen action and observed context. In many domains such as medicine, business, and engineering, prior knowledge provides structural information about the reward function that can be exploited to improve optimization efficiency. This paper studies settings where, for each observable context, the conditional mean reward is known to be concave with respect to the action variable. To leverage this structure, we develop a contextual bandit algorithm that conditions a Bayesian Gaussian Process posterior on concavity constraints. We propose a novel reward model that combines a concavity-preserving regression spline basis with a constrained GP posterior, yielding a tractable shape-constrained estimator. Building on this model, we construct a UCB algorithm and establish new posterior concentration inequalities for the constrained posterior, leading to regret guarantees that are never worse than applying standard GP-UCB without concavity information and can be strictly tighter when concavity is informative. Experiments on benchmark problems and a Warfarin dosing test application demonstrate substantial reductions in cumulative regret relative to state-of-the-art baselines.