Contextual Online Pricing with (Biased) Offline Data

2025-07-03

AI Summary

This paper studies contextual online pricing under biased offline data. Addressing challenges including unknown bias and covariate distribution shift in offline data, we propose a robust algorithm that synergistically integrates offline statistical structure with online exploration. Built upon the Optimism-in-the-Face-of-Uncertainty (OFU) framework, it dynamically couples offline covariance estimation, sample size, and bias magnitude. We establish, for the first time, an instance-dependent and minimax-optimal regret bound: $ ilde{mathcal{O}}ig(dsqrt{T} wedge (V^2T + frac{dT}{λ_{min}(hatΣ) + (N wedge T) δ^2})ig)$, which strictly improves upon pure online baselines. The bound explicitly characterizes how bias strength $δ$, offline sample size $N$, feature dimension $d$, and minimum eigenvalue $λ_{min}(hatΣ)$ govern statistical complexity. Our approach naturally extends to stochastic linear bandits.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Economics and fairness of platforms and recommendation systems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study contextual online pricing with biased offline data. For the scalar price elasticity case, we identify the instance-dependent quantity $δ^2$ that measures how far the offline data lies from the (unknown) online optimum. We show that the time length $T$, bias bound $V$, size $N$ and dispersion $λ_{min}(hatΣ)$ of the offline data, and $δ^2$ jointly determine the statistical complexity. An Optimism-in-the-Face-of-Uncertainty (OFU) policy achieves a minimax-optimal, instance-dependent regret bound $ ilde{mathcal{O}}ig(dsqrt{T} wedge (V^2T + frac{dT}{λ_{min}(hatΣ) + (N wedge T) δ^2})ig)$. For general price elasticity, we establish a worst-case, minimax-optimal rate $ ilde{mathcal{O}}ig(dsqrt{T} wedge (V^2T + frac{dT }{λ_{min}(hatΣ)})ig)$ and provide a generalized OFU algorithm that attains it. When the bias bound $V$ is unknown, we design a robust variant that always guarantees sub-linear regret and strictly improves on purely online methods whenever the exact bias is small. These results deliver the first tight regret guarantees for contextual pricing in the presence of biased offline data. Our techniques also transfer verbatim to stochastic linear bandits with biased offline data, yielding analogous bounds.

Problem

• Study contextual online pricing with biased offline data.
• Determine statistical complexity using offline data parameters.
• Achieve minimax-optimal regret bounds with OFU policy.

Innovation

• Optimism-in-the-Face-of-Uncertainty policy for pricing
• Instance-dependent regret bound with offline data
• Generalized OFU algorithm for price elasticity