Minimax Optimality in Contextual Dynamic Pricing with General Valuation Models

2024-06-24

arXiv.org

AI Summary

This paper studies contextual dynamic pricing: a decision-maker sets personalized prices in real time based on product and user features (context), observes only binary purchase feedback (i.e., whether the customer’s private valuation exceeds the price), and faces valuations generated by an unknown function of context corrupted by arbitrary i.i.d. noise. We propose the first minimax-optimal algorithm for general valuation function classes—requiring neither linearity nor smoothness assumptions—and relying solely on a generic offline regression oracle. Our approach integrates noise distribution discretization, hierarchical data partitioning, and an upper-confidence-bound pricing strategy. We establish a regret upper bound of $ ilde{mathcal{O}}( ho_{mathcal{V}}^{1/3}(delta) T^{2/3})$, which is tight up to logarithmic factors with the information-theoretic lower bound. Empirical results demonstrate substantial improvements over existing methods under nonlinear valuation structures and non-Gaussian noise.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Dynamic pricing, the practice of adjusting prices based on contextual factors, has gained significant attention due to its impact on revenue maximization. In this paper, we address the contextual dynamic pricing problem, which involves pricing decisions based on observable product features and customer characteristics. We propose a novel algorithm that achieves improved regret bounds while minimizing assumptions about the problem. Our algorithm discretizes the unknown noise distribution and combines the upper confidence bounds with a layered data partitioning technique to effectively regulate regret in each episode. These techniques effectively control the regret associated with pricing decisions, leading to the minimax optimality. Specifically, our algorithm achieves a regret upper bound of $ ilde{mathcal{O}}( ho_{mathcal{V}}^{frac{1}{3}}(delta) T^{frac{2}{3}})$, where $ ho_{mathcal{V}}(delta)$ represents the estimation error of the valuation function. Importantly, this bound matches the lower bound up to logarithmic terms, demonstrating the minimax optimality of our approach. Furthermore, our method extends beyond linear valuation models commonly used in dynamic pricing by considering general function spaces. We simplify the estimation process by reducing it to general offline regression oracles, making implementation more straightforward.

Problem

• Develops minimax-optimal algorithm for contextual dynamic pricing
• Handles unknown noise distribution and general valuation models
• Achieves tight regret bounds without knowing Lipschitz constant

Innovation

• Discretizes noise range for candidate prices
• Applies layered data partitioning for tighter bounds
• Uses offline regression oracles for general functions