Nonparametric Contextual Pricing and Inventory Learning under Censored Demand

In online retailing, when a product sells out, a retailer often sees only the units sold, not how many customers would have bought it had inventory been available. However, the inventory level determines how much demand is revealed, and this information can influence subsequent decisions and future profits. We study an online selling problem in which, in each round, the seller observes a market context and then makes pricing and stocking decisions based on censored sales data from previous rounds. The challenge is to learn a context-dependent pricing and stocking policy without a parametric demand model or fully observed profit, since lost-sales costs depend on unobserved unmet demand. To overcome this difficulty, we propose a Surrogate-Profit UCB (SP-UCB) algorithm that turns each incomplete sales record into a reliable guide for both inventory and price decisions, using data from past rounds with similar market conditions. This design allows us to learn while serving customers, without a separate exploration phase or the need to recover all demand hidden by stockouts. We prove the minimax optimality of the proposed algorithm, with strictly faster rates when expected profit varies more smoothly with price. Comprehensive numerical experiments have been conducted to confirm the effectiveness of the proposed algorithm.