# Dual Mirror Descent for Online Allocation Problems

arXiv: https://arxiv.org/abs/2002.10421 · HTML: https://mlsanity.com/paper/2002.10421/

## Key facts

- Cited by 51 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 24 Feb 2020.

## Built on by (most cited first, 30 of 51)

- [Online Bidding Algorithms for Return-on-Spend Constrained Advertisers](https://mlsanity.com/paper/2208.13713/) (2022)
- [No-Regret Algorithms in non-Truthful Auctions with Budget and ROI Constraints](https://mlsanity.com/paper/2404.09832/) (2024)
- [Optimal Regularized Online Allocation by Adaptive Re-Solving](https://mlsanity.com/paper/2209.00399/) (2022)
- [Joint Online Learning and Decision-making via Dual Mirror Descent](https://mlsanity.com/paper/2104.09750/) (2021)
- [Infrequent Resolving Algorithm for Online Linear Programming](https://mlsanity.com/paper/2408.00465/) (2024)
- [Approximately Stationary Bandits with Knapsacks](https://mlsanity.com/paper/2302.14686/) (2023)
- [No-Regret Learning Under Adversarial Resource Constraints: A Spending Plan Is All You Need!](https://mlsanity.com/paper/2506.13244/) (2025)
- [Optimal Spend Rate Estimation and Pacing for Ad Campaigns with Budgets](https://mlsanity.com/paper/2202.05881/) (2022)
- [Best of Many Worlds Guarantees for Online Learning with Knapsacks](https://mlsanity.com/paper/2202.13710/) (2022)
- [Learning to Control Unknown Strongly Monotone Games](https://mlsanity.com/paper/2407.00575/) (2024)
- [Computing Optimal Regularizers for Online Linear Optimization](https://mlsanity.com/paper/2410.17336/) (2024)
- [Safe Online Bid Optimization with Return on Investment and Budget Constraints](https://mlsanity.com/paper/2201.07139/) (2022)
- [Analysis of Dual-Based PID Controllers through Convolutional Mirror Descent](https://mlsanity.com/paper/2202.06152/) (2022)
- [Learning to Price with Resource Constraints: From Full Information to Machine-Learned Prices](https://mlsanity.com/paper/2501.14155/) (2025)
- [Primal-Dual Guided Decoding for Constrained Discrete Diffusion](https://mlsanity.com/paper/2605.09749/) (2026)
- [MNL-Bandit with Knapsacks: a near-optimal algorithm](https://mlsanity.com/paper/2106.01135/) (2021)
- [A Deep Reinforcement Learning Approach for Online Parcel Assignment](https://mlsanity.com/paper/2109.03467/) (2021)
- [Stochastic Online Fisher Markets: Static Pricing Limits and Adaptive Enhancements](https://mlsanity.com/paper/2205.00825/) (2022)
- [Trading-off price for data quality to achieve fair online allocation](https://mlsanity.com/paper/2306.13440/) (2023)
- [Online Bidding under RoS Constraints without Knowing the Value](https://mlsanity.com/paper/2503.03195/) (2025)
- [A New Benchmark for Online Learning with Budget-Balancing Constraints](https://mlsanity.com/paper/2503.14796/) (2025)
- [Multi-task Offline Reinforcement Learning for Online Advertising in Recommender Systems](https://mlsanity.com/paper/2506.23090/) (2025)
- [Learning Potentials for Dynamic Matching and Application to Heart Transplantation](https://mlsanity.com/paper/2602.08878/) (2026)
- [Cross-Epoch Adaptive Rollout Optimization for RL Post-Training](https://mlsanity.com/paper/2606.05606/) (2026)
- [Learning-Assisted Algorithm Unrolling for Online Optimization with Budget Constraints](https://mlsanity.com/paper/2212.01689/) (2022)
- [An Online Algorithm for Chance Constrained Resource Allocation](https://mlsanity.com/paper/2303.03254/) (2023)
- [Online Resource Allocation in Episodic Markov Decision Processes](https://mlsanity.com/paper/2305.10744/) (2023)
- [Speeding up Policy Simulation in Supply Chain RL](https://mlsanity.com/paper/2406.01939/) (2024)
- [Binary-Integer-Programming Based Algorithm for Expert Load Balancing in Mixture-of-Experts Models](https://mlsanity.com/paper/2502.15451/) (2025)
- [A Theoretical Framework for Auxiliary-Loss-Free Load Balancing of Sparse Mixture-of-Experts in Large-Scale AI Models](https://mlsanity.com/paper/2512.03915/) (2025)

## Abstract (arXiv)

We consider online allocation problems with concave revenue functions and resource constraints, which are central problems in revenue management and online advertising. In these settings, requests arrive sequentially during a finite horizon and, for each request, a decision maker needs to choose an action that consumes a certain amount of resources and generates revenue. The revenue function and resource consumption of each request are drawn independently and at random from a probability distribution that is unknown to the decision maker. The objective is to maximize cumulative revenues subject to a constraint on the total consumption of resources. We design a general class of algorithms that achieve sub-linear expected regret compared to the hindsight optimal allocation. Our algorithms operate in the Lagrangian dual space: they maintain a dual multiplier for each resource that is updated using online mirror descent. By choosing the reference function accordingly, we recover dual sub-gradient descent and dual exponential weights algorithm. The resulting algorithms are simple, efficient, and shown to attain the optimal order of regret when the length of the horizon and the initial number of resources are scaled proportionally. We discuss applications to online bidding in repeated auctions with budget constraints and online proportional matching with high entropy.