# Adversarial Bandits with Knapsacks

arXiv: https://arxiv.org/abs/1811.11881 · HTML: https://mlsanity.com/paper/1811.11881/

## Key facts

- Cited by 73 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Nov 2018.

## Built on by (most cited first, 30 of 73)

- [A Unifying Framework for Online Optimization with Long-Term Constraints](https://mlsanity.com/paper/2209.07454/) (2022)
- [No-Regret Learning in Time-Varying Zero-Sum Games](https://mlsanity.com/paper/2201.12736/) (2022)
- [Learning Equilibria in Matching Markets from Bandit Feedback](https://mlsanity.com/paper/2108.08843/) (2021)
- [Contextual Bandits with Packing and Covering Constraints: A Modular Lagrangian Approach via Regression](https://mlsanity.com/paper/2211.07484/) (2022)
- [Stochastic Bandits for Multi-platform Budget Optimization in Online Advertising](https://mlsanity.com/paper/2103.10246/) (2021)
- [Non-stationary Bandits with Knapsacks](https://mlsanity.com/paper/2205.12427/) (2022)
- [Optimistic Whittle Index Policy: Online Learning for Restless Bandits](https://mlsanity.com/paper/2205.15372/) (2022)
- [Autobidders with Budget and ROI Constraints: Efficiency, Regret, and Pacing Dynamics](https://mlsanity.com/paper/2301.13306/) (2023)
- [Online Bidding Algorithms for Return-on-Spend Constrained Advertisers](https://mlsanity.com/paper/2208.13713/) (2022)
- [No-Regret Learning in Bilateral Trade via Global Budget Balance](https://mlsanity.com/paper/2310.12370/) (2023)
- [The Symmetry between Arms and Knapsacks: A Primal-Dual Approach for Bandits with Knapsacks](https://mlsanity.com/paper/2102.06385/) (2021)
- [An α-regret analysis of Adversarial Bilateral Trade](https://mlsanity.com/paper/2210.06846/) (2022)
- [Bandit Algorithms for Precision Medicine](https://mlsanity.com/paper/2108.04782/) (2021)
- [Non-monotonic Resource Utilization in the Bandits with Knapsacks Problem](https://mlsanity.com/paper/2209.12013/) (2022)
- [Beyond Primal-Dual Methods in Bandits with Stochastic and Adversarial Constraints](https://mlsanity.com/paper/2405.16118/) (2024)
- [No-Regret Algorithms in non-Truthful Auctions with Budget and ROI Constraints](https://mlsanity.com/paper/2404.09832/) (2024)
- [Joint Online Learning and Decision-making via Dual Mirror Descent](https://mlsanity.com/paper/2104.09750/) (2021)
- [Mitigating Disparity while Maximizing Reward: Tight Anytime Guarantee for Improving Bandits](https://mlsanity.com/paper/2208.09254/) (2022)
- [Approximately Stationary Bandits with Knapsacks](https://mlsanity.com/paper/2302.14686/) (2023)
- [Online Learning under Adversarial Nonlinear Constraints](https://mlsanity.com/paper/2306.03655/) (2023)
- [Bandits with Replenishable Knapsacks: the Best of both Worlds](https://mlsanity.com/paper/2306.08470/) (2023)
- [No-Regret is not enough! Bandits with General Constraints through Adaptive Regret Minimization](https://mlsanity.com/paper/2405.06575/) (2024)
- [Versatile Dueling Bandits: Best-of-both-World Analyses for Online Learning from Preferences](https://mlsanity.com/paper/2202.06694/) (2022)
- [Bandit Sequential Posted Pricing via Half-Concavity](https://mlsanity.com/paper/2312.12794/) (2023)
- [Contractual Reinforcement Learning: Pulling Arms with Invisible Hands](https://mlsanity.com/paper/2407.01458/) (2024)
- [Online Decision Deferral under Budget Constraints](https://mlsanity.com/paper/2409.20489/) (2024)
- [No-Regret Learning Under Adversarial Resource Constraints: A Spending Plan Is All You Need!](https://mlsanity.com/paper/2506.13244/) (2025)
- [Lazy Lagrangians with Predictions for Online Learning](https://mlsanity.com/paper/2201.02890/) (2022)
- [Best of Many Worlds Guarantees for Online Learning with Knapsacks](https://mlsanity.com/paper/2202.13710/) (2022)
- [The Survival Bandit Problem](https://mlsanity.com/paper/2206.03019/) (2022)

## Abstract (arXiv)

We consider Bandits with Knapsacks (henceforth, BwK), a general model for multi-armed bandits under supply/budget constraints. In particular, a bandit algorithm needs to solve a well-known knapsack problem: find an optimal packing of items into a limited-size knapsack. The BwK problem is a common generalization of numerous motivating examples, which range from dynamic pricing to repeated auctions to dynamic ad allocation to network routing and scheduling. While the prior work on BwK focused on the stochastic version, we pioneer the other extreme in which the outcomes can be chosen adversarially. This is a considerably harder problem, compared to both the stochastic version and the "classic" adversarial bandits, in that regret minimization is no longer feasible. Instead, the objective is to minimize the competitive ratio: the ratio of the benchmark reward to the algorithm's reward. We design an algorithm with competitive ratio O(log T) relative to the best fixed distribution over actions, where T is the time horizon; we also prove a matching lower bound. The key conceptual contribution is a new perspective on the stochastic version of the problem. We suggest a new algorithm for the stochastic version, which builds on the framework of regret minimization in repeated games and admits a substantially simpler analysis compared to prior work. We then analyze this algorithm for the adversarial version and use it as a subroutine to solve the latter.