Optimal Contextual Pricing under Agnostic Non-Lipschitz Demand

TL;DR

This paper introduces a new polynomial-time pricing algorithm that achieves near-optimal regret bounds in complex demand scenarios with non-Lipschitz noise, closing a long-standing gap in contextual pricing theory.

Contribution

It presents the Conservative-Markdown Redirect-UCB Pricing algorithm, which handles non-Lipschitz demand and achieves optimal regret bounds matching known lower limits.

Findings

Achieves  O(T^{2/3}) regret, improving over previous bounds.

Matches the lower bounds of Kleinberg and Leighton (2003) up to logarithmic factors.

Closes the open regret gap in linear-valuation contextual pricing under agnostic noise.

Abstract

We study contextual dynamic pricing with linear valuations and bounded-support agnostic noise, whose induced demand curve may be non-Lipschitz with arbitrary jumps and atoms. Such discontinuities break the cross-context interpolation arguments used by smooth-demand pricing algorithms, while the best previous method achieved only $\tilde O(T^{3/4})$ regret. We propose Conservative-Markdown Redirect-UCB Pricing, a polynomial-time algorithm that combines randomized parameter estimation, conservative residual-grid probing, and confidence-based one-step redirection. Our algorithm achieves $\tilde O(T^{2/3})$ optimal regret, matching the known lower bounds of Kleinberg and Leighton (2003) up to logarithmic factors and improving over the previous upper bound of Xu and Wang (2022). Under stochastic well-conditioned contexts, this closes the long-existing open regret gap in linear-valuation contextual pricing under agnostic non-Lipschitz noise distribution.