# Contextual Online Pricing with (Biased) Offline Data

arXiv: https://arxiv.org/abs/2507.02762 · HTML: https://mlsanity.com/paper/2507.02762/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 35 references that are in the map.
- First posted on arXiv 3 Jul 2025.

## Builds on (most cited first, 15 of 15)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Off-Dynamics Reinforcement Learning: Training for Transfer with Domain Classifiers](https://mlsanity.com/paper/2006.13916/) (2020)
- [Meta-Thompson Sampling](https://mlsanity.com/paper/2102.06129/) (2021)
- [Bayesian decision-making under misspecified priors with applications to meta-learning](https://mlsanity.com/paper/2107.01509/) (2021)
- [Leveraging Offline Data in Online Reinforcement Learning](https://mlsanity.com/paper/2211.04974/) (2022)
- [Dynamic Pricing with Demand Covariates](https://mlsanity.com/paper/1604.07463/) (2016)
- [Warm-starting Contextual Bandits: Robustly Combining Supervised and Bandit Feedback](https://mlsanity.com/paper/1901.00301/) (2019)
- [Taking a hint: How to leverage loss predictors in contextual bandits?](https://mlsanity.com/paper/2003.01922/) (2020)
- [On the Prior Sensitivity of Thompson Sampling](https://mlsanity.com/paper/1506.03378/) (2015)
- [Multi-Armed Bandits With Machine Learning-Generated Surrogate Rewards](https://mlsanity.com/paper/2506.16658/) (2025)
- [Online Pricing with Offline Data: Phase Transition and Inverse Square Law](https://mlsanity.com/paper/1910.08693/) (2019)
- [Hybrid Transfer Reinforcement Learning: Provable Sample Efficiency from Shifted-Dynamics Data](https://mlsanity.com/paper/2411.03810/) (2024)
- [Data-pooling Reinforcement Learning for Personalized Healthcare Intervention](https://mlsanity.com/paper/2211.08998/) (2022)
- [Combining Offline Causal Inference and Online Bandit Learning for Data Driven Decision](https://mlsanity.com/paper/2001.05699/) (2020)
- [Provable Domain Adaptation for Offline Reinforcement Learning with Limited Samples](https://mlsanity.com/paper/2408.12136/) (2024)

## Built on by (most cited first, 4 of 4)

- [Direction-Aware Offline-to-Online Learning in Linear Contextual Bandits](https://mlsanity.com/paper/2604.24016/) (2026)
- [Best-Arm Identification with Generative Proxy](https://mlsanity.com/paper/2607.06879/) (2026)
- [Offline-Online Reinforcement Learning for Linear Mixture MDPs](https://mlsanity.com/paper/2604.11994/) (2026)
- [Sample-Mean Anchored Thompson Sampling for Offline-to-Online Learning with Distribution Shift](https://mlsanity.com/paper/2605.10289/) (2026)

## Abstract (arXiv)

We study contextual online pricing with biased offline data. For the scalar price elasticity case, we identify the instance-dependent quantity δ2 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 λmin(hatSigma) of the offline data, and δ2 jointly determine the statistical complexity. An Optimism-in-the-Face-of-Uncertainty (OFU) policy achieves a minimax-optimal, instance-dependent regret bound tildeObig(d√T wedge (V2T + fracdTλmin(hatSigma) + (N wedge T) δ2)big). For general price elasticity, we establish a worst-case, minimax-optimal rate tildeObig(d√T wedge (V2T + fracdT λmin(hatSigma))big) 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.