Optimal Nonparametric Dynamic Pricing with Censored Demand and Adversarial Inventory

2026-09-26

AI Summary

This study addresses the nonparametric dynamic pricing problem under censored demand and adversarial inventory constraints, where existing methods are limited by linear assumptions and inefficient data utilization. To overcome these limitations, this work proposes a shared-estimation-based dual-grid UCB algorithm, a threshold UCB algorithm, and a randomized posterior price reduction technique that enhance data efficiency through synergistic optimization. Theoretically, we rigorously establish the minimax optimality of the proposed Threshold-UCB algorithm, achieving an optimal regret bound of O(T^{2/3}) in a general nonparametric setting. Empirical evaluations demonstrate that the proposed approach significantly outperforms existing baseline algorithms.

Abstract

We study online dynamic pricing with censored demand, where an arbitrary inventory level is revealed before pricing and may adapt to past observations, while demand follows an unknown, price-dependent distribution that is stationary over time. For a horizon of $T$ rounds, Xu et al. [2026] achieved $\widetilde{\mathcal{O}}(\sqrt{T})$ regret under restrictive structural assumptions including linear demand, price-independent additive noise, and conditions relating inventory levels to the noise support. Our first contribution is to extend this framework to a substantially more general and statistically harder nonparametric setting, requiring only the natural assumption that expected sales are nonincreasing in price and allowing nonlinear demand curves and price-dependent noise. For this model, we first propose a simple baseline, Double-Grid-UCB, which discretizes both price and inventory and achieves $\widetilde{\mathcal{O}}(T^{3/4})$ expected regret using separate revenue estimates for each price-inventory grid pair. Then, we develop Threshold-UCB, which improves the expected regret to $\widetilde{\mathcal{O}}(T^{2/3})$. Unlike Double-Grid-UCB, Threshold-UCB reuses sales observations across inventory levels through shared estimates of demand-tail probabilities, allowing the same data to support revenue upper bounds for multiple inventories rather than a single inventory bin. We also complement this upper bound with an $\Omega(T^{2/3})$ lower bound via a reduction from stochastic posted pricing, establishing its minimax optimality. Finally, extensive experiments across inventory processes, demand functions, and noise models demonstrate consistently superior performance of Threshold-UCB over benchmark algorithms.

Problem

• dynamic pricing
• censored demand
• nonparametric
• adversarial inventory
• regret minimization

Innovation

• Nonparametric Dynamic Pricing
• Censored Demand
• Adversarial Inventory
• Threshold-UCB
• Minimax Optimality