We consider a high-dimensional dynamic pricing problem under non-stationarity, where a firm sells products to $T$ sequentially arriving consumers that behave according to an unknown demand model with potential changes at unknown times. The demand model is assumed to be a high-dimensional generalized linear model (GLM), allowing for a feature vector in $\mathbb R^d$ that encodes products and consumer information. To achieve optimal revenue (i.e., least regret), the firm needs to learn and exploit the unknown GLMs while monitoring for potential change-points. To tackle such a problem, we first design a novel penalized likelihood-based online change-point detection algorithm for high-dimensional GLMs, which is the first algorithm in the change-point literature that achieves optimal minimax localization error rate for high-dimensional GLMs. A change-point detection assisted dynamic pricing (CPDP) policy is further proposed and achieves a near-optimal regret of order $O(s\sqrt{\Upsilon_T T}\log(Td))$, where $s$ is the sparsity level and $\Upsilon_T$ is the number of change-points. This regret is accompanied with a minimax lower bound, demonstrating the optimality of CPDP (up to logarithmic factors). In particular, the optimality with respect to $\Upsilon_T$ is seen for the first time in the dynamic pricing literature, and is achieved via a novel accelerated exploration mechanism. Extensive simulation experiments and a real data application on online lending illustrate the efficiency of the proposed policy and the importance and practical value of handling non-stationarity in dynamic pricing.

翻译：本文研究了非平稳条件下的高维动态定价问题，其中企业向$T$个顺序到达的消费者销售产品，这些消费者的行为遵循未知的需求模型，且模型可能在未知时间点发生变动。需求模型采用高维广义线性模型（GLM），允许使用$\mathbb R^d$维特征向量编码产品与消费者信息。为实现最优收益（即最小遗憾值），企业需在学习与利用未知GLM的同时，监测潜在的变点。为应对该问题，我们首先针对高维GLM设计了一种基于惩罚似然的在线变点检测算法，这是变点检测文献中首个实现高维GLM最优极小化定位误差率的算法。进一步提出一种变点检测辅助的动态定价（CPDP）策略，其遗憾值达到接近最优的$O(s\sqrt{\Upsilon_T T}\log(Td))$阶，其中$s$为稀疏度，$\Upsilon_T$为变点个数。该遗憾值伴随一个极小化下界，证明了CPDP（对数因子意义下）的最优性。特别地，关于$\Upsilon_T$的最优性在动态定价文献中首次出现，并通过一种新颖的加速探索机制实现。大量仿真实验及在线借贷真实数据应用验证了所提策略的有效性，以及处理非平稳性在动态定价中的重要性与实用价值。