Semi-Parametric Contextual Pricing with General Smoothness

Abstract

We study the contextual pricing problem, where in each round a seller observes a context, sets a price, and receives a binary purchase signal. We adopt a semi-parametric model in which the demand follows a linear parametric form composed with an unknown link function from a $\beta$-Hölder class. Prior work established regret rates of $\tilde{\mathcal{O}}(T^{2/3})$ for $\beta=1$ and $\tilde{\mathcal{O}}(T^{3/5})$ for $\beta=2$. Under a uni-modality condition, we propose a unified algorithm that combines the stationary subroutine of Wang & Chen (2025) with local polynomial regression, achieving the general rate $\tilde{\mathcal{O}}(T^{\frac{\beta+1}{2\beta+1}})$ for all $\beta \ge 1$. This recovers and strengthens existing results, while also addressing a gap in the prior analysis for $\beta=2$. Our analysis develops tighter semi-parametric confidence regions, removes derivative lower bound assumptions from earlier work, and offers a sharper exploration–exploitation trade-off. These insights not only extend theoretical guarantees to general $\beta$ but also improve practical performance by reducing the need for long forced-exploration phases.