Harnessing Unimodality in Semiparametric Contextual Pricing via Oracle Price Map Learning

arXiv:2605.15411v1 Announce Type: new
Abstract: We study contextual dynamic pricing in a semiparametric scalar-index valuation model where the latent value is $v_t=mu_ast(mathsf c_t)+xi_t$, with an unknown utility map $mu_ast$ and an unknown additive noise distribution. The key decision object is the one-dimensional oracle price map $umapsto p^ast(u)$ induced by the scalar index $u=mu_ast(mathsf c)$ and the noise tail. Under the $beta$-H”older smoothness of the tail function for $betageq 2$ and a revenue-geometry condition that gives a unique, stable, interior maximizer, this oracle map is itself $(beta-1)$-smooth. We exploit such structure through $mathsf{ORBIT}$, a modular coarse-to-fine policy that takes a scalar pilot index as input, localizes a benchmark price in each active bin, and learns a local polynomial approximation of the oracle map inside a trust region via bandit convex optimization. For the baseline linear utility model $mu_ast(mathsf c)=mathsf c^toptheta_ast$, an adaptive elliptical exploration scheme constructs the required scalar pilot online without distributional assumptions on the contexts. The resulting policy achieves regret $widetilde{O}big(T^{frac{2beta-1}{4beta-3}}+sqrt{dT}big)$. For fixed $d$, we establish a matching lower bound in the horizon dependence, unveiling that the nonparametric oracle-map learning term is minimax sharp. The same scalar-pilot interface also yields extensions to sparse high-dimensional linear utility and nonparametric H”older utility.