Policy Optimization Using Semi-parametric Models for Dynamic Pricing

In this paper, we study the contextual dynamic pricing problem where the market value of a product is linear in its observed features plus some market noise. Products are sold one at a time, and only a binary response indicating success or failure of a sale is observed. Our model setting is similar to Javanmard and Nazerzadeh [2019] 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(·) with m-th order derivative (m≥ 2), our policy achieves a regret upper bound of tildeOd(T^frac2m+14m-1), where T is time horizon and tildeOd is the order that hides logarithmic terms and the dimensionality of feature d. The upper bound is further reduced to tildeOd(√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(√T), the lower bound where F belongs to a parametric class. We further generalize these results to the case with dynamically dependent product features under the strong mixing condition.