Policy Optimization Using Semiparametric Models for Dynamic Pricing

In this paper, we study the contextual dynamic pricing problem where the market value of a product is linear in their observed features plus some market noise. Products are sold one at a time, and only a binary response indicating the success or failure of a sale is observed. Our model setting is similar to \cite{JN19} except that we expand the demand curve to a semiparametric model and need to learn dynamically both parametric and nonparametric components. We propose a dynamic statistical learning and decision-making policy that combines semiparametric estimation from a generalized linear model with an unknown link and online decision making to minimize regret (maximize revenue). Under mild conditions, we show that for a market noise c.d.f. $F(\cdot)$ with $m$-th order derivative, our policy achieves a regret upper bound of $\tilde{\cO}_{d}(T^{\frac{2m+1}{4m-1}})$ for $m\geq 2$, where $T$ is time horizon and $\tilde{\cO}_{d}$ is the order that hides logarithmic terms and the dimensionality of feature $d$. The upper bound is further reduced to $\tilde{\cO}_{d}(\sqrt{T})$ if $F$ is super smooth whose Fourier transform decays exponentially. In terms of dependence on the horizon $T$, these upper bounds are close to $\Omega(\sqrt{T})$, the lower bound where the market noise distribution belongs to a parametric class. We further generalize these results to the case when the product features are dynamically dependent, satisfying some strong mixing conditions.