We consider dynamic pricing with covariates under a generalized linear demand model: a seller can dynamically adjust the price of a product over a horizon of $T$ time periods, and at each time period $t$, the demand of the product is jointly determined by the price and an observable covariate vector $x_t\in\mathbb{R}^d$ through a generalized linear model with unknown co-efficients. Most of the existing literature assumes the covariate vectors $x_t$'s are independently and identically distributed (i.i.d.); the few papers that relax this assumption either sacrifice model generality or yield sub-optimal regret bounds. In this paper, we show that UCB and Thompson sampling-based pricing algorithms can achieve an $O(d\sqrt{T}\log T)$ regret upper bound without assuming any statistical structure on the covariates $x_t$. Our upper bound on the regret matches the lower bound up to logarithmic factors. We thus show that (i) the i.i.d. assumption is not necessary for obtaining low regret, and (ii) the regret bound can be independent of the (inverse) minimum eigenvalue of the covariance matrix of the $x_t$'s, a quantity present in previous bounds. Moreover, we consider a constrained setting of the dynamic pricing problem where there is a limited and unreplenishable inventory and we develop theoretical results that relate the best achievable algorithm performance to a variation measure with respect to the temporal distribution shift of the covariates. We also discuss conditions under which a better regret is achievable and demonstrate the proposed algorithms' performance with numerical experiments.

翻译：我们研究了广义线性需求模型下带协变量的动态定价问题：卖方可在$T$个时期的时间跨度内动态调整产品价格，每个时期$t$的产品需求由价格与可观测协变量向量$x_t\in\mathbb{R}^d$通过含有未知系数的广义线性模型共同决定。现有文献大多假设协变量向量$x_t$独立同分布(i.i.d.)；少数放宽该假设的研究要么牺牲模型通用性，要么获得次优的遗憾界。本文证明，在不假设协变量$x_t$具有任何统计结构的情况下，基于UCB和汤普森采样的定价算法可实现$O(d\sqrt{T}\log T)$的遗憾上界。该遗憾上界与下界仅相差对数因子。我们由此表明：(i)独立同分布假设并非获取低遗憾的必要条件；(ii)遗憾界可独立于$x_t$协方差矩阵的（逆）最小特征值（该参数存在于以往研究结论中）。此外，我们考虑了库存有限且不可补给的约束型动态定价问题，建立了最优算法性能与协变量时间分布偏移变异度量之间的理论关系。本文还讨论了实现更优遗憾的条件，并通过数值实验验证了所提出算法的性能。