# Learning to Price with Resource Constraints: From Full Information to Machine-Learned Prices

arXiv: https://arxiv.org/abs/2501.14155 · HTML: https://mlsanity.com/paper/2501.14155/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 67 references that are in the map.
- First posted on arXiv 24 Jan 2025.

## Builds on (most cited first, 30 of 32)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Competitive caching with machine learned advice](https://mlsanity.com/paper/1802.05399/) (2018)
- [Algorithms with Predictions](https://mlsanity.com/paper/2006.09123/) (2020)
- [Bypassing the Monster: A Faster and Simpler Optimal Algorithm for Contextual Bandits under Realizability](https://mlsanity.com/paper/2003.12699/) (2020)
- [Linear Contextual Bandits with Knapsacks](https://mlsanity.com/paper/1507.06738/) (2015)
- [Improving Online Algorithms via ML Predictions](https://mlsanity.com/paper/2407.17712/) (2024)
- [Optimal Robustness-Consistency Trade-offs for Learning-Augmented Online Algorithms](https://mlsanity.com/paper/2010.11443/) (2020)
- [Dynamic Pricing in High-dimensions](https://mlsanity.com/paper/1609.07574/) (2016)
- [Dual Mirror Descent for Online Allocation Problems](https://mlsanity.com/paper/2002.10421/) (2020)
- [Resourceful Contextual Bandits](https://mlsanity.com/paper/1402.6779/) (2014)
- [An efficient algorithm for contextual bandits with knapsacks, and an extension to concave objectives](https://mlsanity.com/paper/1506.03374/) (2015)
- [Upper Counterfactual Confidence Bounds: a New Optimism Principle for Contextual Bandits](https://mlsanity.com/paper/2007.07876/) (2020)
- [Online Linear Programming: Dual Convergence, New Algorithms, and Regret Bounds](https://mlsanity.com/paper/1909.05499/) (2019)
- [Non-stationary Bandits with Knapsacks](https://mlsanity.com/paper/2205.12427/) (2022)
- [Logarithmic Regret in Feature-based Dynamic Pricing](https://mlsanity.com/paper/2102.10221/) (2021)
- [Towards Agnostic Feature-based Dynamic Pricing: Linear Policies vs Linear Valuation with Unknown Noise](https://mlsanity.com/paper/2201.11341/) (2022)
- [A Re-solving Heuristic with Uniformly Bounded Loss for Network Revenue Management](https://mlsanity.com/paper/1802.06192/) (2018)
- [Semi-parametric dynamic contextual pricing](https://mlsanity.com/paper/1901.02045/) (2019)
- [Online Linear Optimization with Many Hints](https://mlsanity.com/paper/2010.03082/) (2020)
- [Structured Linear Contextual Bandits: A Sharp and Geometric Smoothed Analysis](https://mlsanity.com/paper/2002.11332/) (2020)
- [The Bayesian Prophet: A Low-Regret Framework for Online Decision Making](https://mlsanity.com/paper/1901.05028/) (2019)
- [Online Pricing with Offline Data: Phase Transition and Inverse Square Law](https://mlsanity.com/paper/1910.08693/) (2019)
- [Degeneracy is OK: Logarithmic Regret for Network Revenue Management with Indiscrete Distributions](https://mlsanity.com/paper/2210.07996/) (2022)
- [Logarithmic Regret from Sublinear Hints](https://mlsanity.com/paper/2111.05257/) (2021)
- [Online Contextual Decision-Making with a Smart Predict-then-Optimize Method](https://mlsanity.com/paper/2206.07316/) (2022)
- [Non-monotonic Resource Utilization in the Bandits with Knapsacks Problem](https://mlsanity.com/paper/2209.12013/) (2022)
- [Bandits with Knapsacks beyond the Worst-Case](https://mlsanity.com/paper/2002.00253/) (2020)
- [Contextual Decision-Making with Knapsacks Beyond the Worst Case](https://mlsanity.com/paper/2211.13952/) (2022)
- [High-dimensional Linear Bandits with Knapsacks](https://mlsanity.com/paper/2311.01327/) (2023)

## Built on by (most cited first, 3 of 3)

- [Constrained Online Decision-Making: A Unified Framework](https://mlsanity.com/paper/2505.07101/) (2025)
- [Adaptive Resolving Methods for Markov Decision Processes with Function Approximations](https://mlsanity.com/paper/2505.12037/) (2025)
- [Prediction-Assisted Pricing and Admission for LLM APIs with Stochastic Token Consumption](https://mlsanity.com/paper/2609.00710/) (2026)

## Abstract (arXiv)

We study the dynamic pricing problem with knapsack, addressing the challenge of balancing exploration and exploitation under resource constraints. We introduce three algorithms tailored to different informational settings: a Boundary Attracted Re-solve Method for full information, an online learning algorithm for scenarios with no prior information, and an estimate-then-select re-solve algorithm that leverages machine-learned informed prices with known upper bound of estimation errors. The Boundary Attracted Re-solve Method achieves logarithmic regret without requiring the non-degeneracy condition, while the online learning algorithm attains an optimal O(√T) regret. Our estimate-then-select approach bridges the gap between these settings, providing improved regret bounds when reliable offline data is available. Numerical experiments validate the effectiveness and robustness of our algorithms across various scenarios. This work advances the understanding of online resource allocation and dynamic pricing, offering practical solutions adaptable to different informational structures.