In today’s data-rich environment, context-based dynamic pricing has gained much attention. To model the demand as a function of price and context, the existing literature either adopts a parametric model or a non-parametric model. The former is easier to implement but may suffer from model mis-specification, whereas the latter is more robust but does not leverage many structural properties of the underlying problem. This paper combines these two approaches by studying the context-based dynamic pricing with online learning, where the unknown expected demand admits a semi-parametric partially linear structure. Specifically, we consider two demand models, whose expected demand at price $p$ and context $x \in \mathbb{R}^d$ is given by $bp+g(x)$ and $ f(p)+ a^\top x$ respectively. We assume that $g(x)$ is $\beta$-H{\"o}lder continuous in the first model, and $f(p)$ is $k$th-order smooth with an additional parameter $\delta$ in the second model. For both models, we design an efficient online learning algorithm with provable regret upper bounds, and establish matching lower bounds. This enables us to characterize the statistical complexity for the two learning models, whose optimal regret rates are $\widetilde \Theta(\sqrt T \vee T^{\frac{d}{d+2\beta}})$ and $\widetilde \Theta(\sqrt T \vee (\delta T^{k+1})^{\frac{1}{2k+1}})$ respectively. The numerical results demonstrate that our learning algorithms are more effective than benchmark algorithms, and also reveal the effects of parameters $d$, $\beta$ and $\delta$ on the algorithm's empirical regret, which are consistent with our theoretical findings.

This paper investigates the impact of pre-existing offline data on online learning, in the context of dynamic pricing. We study a single-product dynamic pricing problem over a selling horizon of T periods. The demand in each period is determined by the price of the product according to a linear demand model with unknown parameters. We assume that the seller already has some pre-existing offline data before the start of the selling horizon. The seller wants to utilize both the pre-existing offline data and the sequential online data to minimize the regret of the online learning process. We characterize the joint effect of the size, location and dispersion of the offline data on the optimal regret of the online learning process. Our results reveal surprising transformations of the optimal regret rate with respect to the size of the offline data, which we refer to as phase transitions. In addition, our results demonstrate that the location and dispersion of the offline data also have an intrinsic effect on the optimal regret, and we quantify this effect via the inverse-square law.