Policy Optimization Using Semi-parametric Models for Dynamic Pricing

Abstract

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 O d (T 2m+1 4m−1 ), where T is time horizon and O d is the order that hides logarithmic terms and the dimensionality of feature d. The upper bound is further reduced to O d ( √ 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 Ω( √ 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.