# Online Pricing with Offline Data: Phase Transition and Inverse Square Law

arXiv: https://arxiv.org/abs/1910.08693 · HTML: https://mlsanity.com/paper/1910.08693/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Oct 2019.

## Built on by (most cited first, 13 of 13)

- [Optimal Best-Arm Identification in Bandits with Access to Offline Data](https://mlsanity.com/paper/2306.09048/) (2023)
- [Balancing optimism and pessimism in offline-to-online learning](https://mlsanity.com/paper/2502.08259/) (2025)
- [Best Arm Identification with Possibly Biased Offline Data](https://mlsanity.com/paper/2505.23165/) (2025)
- [Contextual Online Pricing with (Biased) Offline Data](https://mlsanity.com/paper/2507.02762/) (2025)
- [Network Revenue Management with Demand Learning and Fair Resource-Consumption Balancing](https://mlsanity.com/paper/2207.11159/) (2022)
- [Learning to Price with Resource Constraints: From Full Information to Machine-Learned Prices](https://mlsanity.com/paper/2501.14155/) (2025)
- [Hybrid Combinatorial Multi-armed Bandits with Probabilistically Triggered Arms](https://mlsanity.com/paper/2512.21925/) (2025)
- [Optimal Contextual Pricing under Agnostic Non-Lipschitz Demand](https://mlsanity.com/paper/2605.05609/) (2026)
- [Non-Stationary Online Resource Allocation: Learning from a Single Sample](https://mlsanity.com/paper/2602.18114/) (2026)
- [Direction-Aware Offline-to-Online Learning in Linear Contextual Bandits](https://mlsanity.com/paper/2604.24016/) (2026)
- [Phase Transitions in Learning and Earning under Price Protection Guarantee](https://mlsanity.com/paper/2211.01798/) (2022)
- [Continuum-armed Bandit Optimization with Batch Pairwise Comparison Oracles](https://mlsanity.com/paper/2505.22361/) (2025)
- [Offline Clustering of Preference Learning with Active-data Augmentation](https://mlsanity.com/paper/2510.26301/) (2025)

## Abstract (arXiv)

This paper investigates the impact of pre-existing offline data on online learning, in the context of dynamic pricing. We study a single-product dynamic pricing problem over a selling horizon of T periods. The demand in each period is determined by the price of the product according to a linear demand model with unknown parameters. We assume that before the start of the selling horizon, the seller already has some pre-existing offline data. The offline data set contains n samples, each of which is an input-output pair consisting of a historical price and an associated demand observation. The seller wants to utilize both the pre-existing offline data and the sequential online data to minimize the regret of the online learning process. We characterize the joint effect of the size, location and dispersion of the offline data on the optimal regret of the online learning process. Specifically, the size, location and dispersion of the offline data are measured by the number of historical samples n, the distance between the average historical price and the optimal price δ, and the standard deviation of the historical prices σ, respectively. We show that the optimal regret is widetilde Thetaleft(√Twedge fracT(nwedge T)δ2+nσ2right), and design a learning algorithm based on the "optimism in the face of uncertainty" principle, whose regret is optimal up to a logarithmic factor. Our results reveal surprising transformations of the optimal regret rate with respect to the size of the offline data, which we refer to as phase transitions. In addition, our results demonstrate that the location and dispersion of the offline data also have an intrinsic effect on the optimal regret, and we quantify this effect via the inverse-square law.