We study off-policy learning (OPL) in contextual bandits, which plays a key role in a wide range of real-world applications such as recommendation systems and online advertising. Typical OPL in contextual bandits assumes an unconstrained environment where a policy can select the same item infinitely. However, in many practical applications, including coupon allocation and e-commerce, limited supply constrains items through budget limits on distributed coupons or inventory restrictions on products. In these settings, greedily selecting the item with the highest expected reward for the current user may lead to early depletion of that item, making it unavailable for future users who could potentially generate higher expected rewards. As a result, OPL methods that are optimal in unconstrained settings may become suboptimal in limited supply settings. To address the issue, we provide a theoretical analysis showing that conventional greedy OPL approaches may fail to maximize the policy performance, and demonstrate that policies with superior performance must exist in limited supply settings. Based on this insight, we introduce a novel method called Off-Policy learning with Limited Supply (OPLS). Rather than simply selecting the item with the highest expected reward, OPLS focuses on items with relatively higher expected rewards compared to the other users, enabling more efficient allocation of items with limited supply. Our empirical results on both synthetic and real-world datasets show that OPLS outperforms existing OPL methods in contextual bandit problems with limited supply.

**Off-Policy Evaluation (OPE)** aims to assess the effectiveness of counterfactual policies using offline logged data and is frequently utilized to identify the top-$k$ promising policies for deployment in online A/B tests. Existing evaluation metrics for OPE estimators primarily focus on the "accuracy" of OPE or that of downstream policy selection, neglecting risk-return tradeoff and *efficiency* in subsequent online policy deployment. To address this issue, we draw inspiration from portfolio evaluation in finance and develop a new metric, called **SharpeRatio@k**, which measures the risk-return tradeoff and efficiency of policy portfolios formed by an OPE estimator under varying online evaluation budgets ($k$). We first demonstrate, in two example scenarios, that our proposed metric can clearly distinguish between conservative and high-stakes OPE estimators and reliably identify the most *efficient* estimator capable of forming superior portfolios of candidate policies that maximize return with minimal risk during online deployment, while existing evaluation metrics produce only degenerate results. To facilitate a quick, accurate, and consistent evaluation of OPE via SharpeRatio@k, we have also implemented the proposed metric in an open-source software. Using SharpeRatio@k and the software, we conduct a benchmark experiment of various OPE estimators regarding their risk-return tradeoff, presenting several future directions for OPE research.

We explore off-policy evaluation and learning (OPE/L) in contextual combinatorial bandits (CCB), where a policy selects a subset in the action space. For example, it might choose a set of furniture pieces (a bed and a drawer) from available items (bed, drawer, chair, etc.) for interior design sales. This setting is widespread in fields such as recommender systems and healthcare, yet OPE/L of CCB remains unexplored in the relevant literature. Typical OPE/L methods such as regression and importance sampling can be applied to the CCB problem, however, they face significant challenges due to high bias or variance, exacerbated by the exponential growth in the number of available subsets. To address these challenges, we introduce a concept of factored action space, which allows us to decompose each subset into binary indicators. This formulation allows us to distinguish between the “main effect” derived from the main actions, and the “residual effect”, originating from the supplemental actions, facilitating more effective OPE. Specifically, our estimator, called OPCB, leverages an importance sampling-based approach to unbiasedly estimate the main effect, while employing regression-based approach to deal with the residual effect with low variance. OPCB achieves substantial variance reduction compared to conventional importance sampling methods and bias reduction relative to regression methods under certain conditions, as illustrated in our theoretical analysis. Experiments demonstrate OPCB’s superior performance over typical methods in both OPE and OPL.