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. \(F(\cdot)\) with \(m\)-th order derivative (\(m\geq 2\)), our policy achieves a regret upper bound of \(\tilde{O}_{d}(T^{\frac{2m+1}{4m-1}})\), where \(T\) is time horizon and \(\tilde{O}_{d}\) is the order that hides logarithmic terms and the dimensionality of feature \(d\). The upper bound is further reduced to \(\tilde{O}_{d}(\sqrt{T})\) 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 \(F\) belongs to a parametric class. We further generalize these results to the case with dynamically dependent product features under the strong mixing condition.