AI 中文总结
本文针对审查需求下的动态定价问题，提出非参数方法Threshold-UCB，实现极小极大最优遗憾界，并通过实验验证其优越性。
AI 中文摘要
我们研究带有审查需求的在线动态定价问题，其中任意库存水平在定价前被揭示，并且可能适应过去的观测，而需求遵循一个未知的、依赖于价格的分布，该分布随时间平稳。对于$T$轮的时间范围，Xu等人[2026]在线性需求、价格无关加性噪声以及库存水平与噪声支撑相关的条件下实现了$\tilde{\mathcal{O}}(\sqrt{T})$的遗憾界。我们的第一个贡献是将该框架扩展到更一般且统计上更困难的非参数设置，仅要求期望销售量随价格非增的自然假设，并允许非线性需求曲线和价格相关噪声。对于该模型，我们首先提出一个简单的基线算法Double-Grid-UCB，该算法对价格和库存进行离散化，并使用每个价格-库存网格对的独立收入估计实现$\tilde{\mathcal{O}}(T^{3/4})$的期望遗憾。然后，我们开发了Threshold-UCB，将期望遗憾改进至$\tilde{\mathcal{O}}(T^{2/3})$。与Double-Grid-UCB不同，Threshold-UCB通过共享需求尾部概率的估计来跨库存水平重用销售观测，使得相同数据能够支持多个库存的收入上界，而非单一库存区间。我们还通过从随机标价问题的归约补充了$\Omega(T^{2/3})$的下界，确立了其极小极大最优性。最后，跨库存过程、需求函数和噪声模型的广泛实验表明，Threshold-UCB在基准算法中始终表现出优越的性能。
英文摘要
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 $Ω(T^{2/3})$ 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.
