Harnessing Unimodality in Semiparametric Contextual Pricing via Oracle Price Map Learning

We study contextual dynamic pricing in a semiparametric scalar-index valuation model where the latent value is vt=μ_ast(mathsf ct)+xit, with an unknown utility map μ_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=μ_ast(mathsf c) and the noise tail. Under the β-H"older smoothness of the tail function for β≥ 2 and a revenue-geometry condition that gives a unique, stable, interior maximizer, this oracle map is itself (β-1)-smooth. We exploit such structure through mathsfORBIT, 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 μ_ast(mathsf c)=mathsf c^topθ_ast, an adaptive elliptical exploration scheme constructs the required scalar pilot online without distributional assumptions on the contexts. The resulting policy achieves regret widetildeObig(T^frac2β-14β-3+√dTbig). 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.