# MEMOA: Massive Mixtures of Online Agents via Mean-Field Decentralized Nash Equilibria

arXiv: https://arxiv.org/abs/2605.05492 · HTML: https://mlsanity.com/paper/2605.05492/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 47 references that are in the map.
- First posted on arXiv 6 May 2026.

## Builds on (most cited first, 23 of 23)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Federated Optimization in Heterogeneous Networks](https://mlsanity.com/paper/1812.06127/) (2018)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [Ditto: Fair and Robust Federated Learning Through Personalization](https://mlsanity.com/paper/2012.04221/) (2020)
- [Estimating the Carbon Footprint of BLOOM, a 176B Parameter Language Model](https://mlsanity.com/paper/2211.02001/) (2022)
- [Power Hungry Processing: Watts Driving the Cost of AI Deployment?](https://mlsanity.com/paper/2311.16863/) (2023)
- [From Words to Watts: Benchmarking the Energy Costs of Large Language Model Inference](https://mlsanity.com/paper/2310.03003/) (2023)
- [Echo state networks are universal](https://mlsanity.com/paper/1806.00797/) (2018)
- [LLMCarbon: Modeling the end-to-end Carbon Footprint of Large Language Models](https://mlsanity.com/paper/2309.14393/) (2023)
- [Model-sharing Games: Analyzing Federated Learning Under Voluntary Participation](https://mlsanity.com/paper/2010.00753/) (2020)
- [Reducing Cost of LLM Agents with Trajectory Reduction](https://mlsanity.com/paper/2509.23586/) (2025)
- [Budget-Aware Tool Use Enables Effective Agent Scaling](https://mlsanity.com/paper/2511.17006/) (2025)
- [Optimality and Stability in Federated Learning: A Game-theoretic Approach](https://mlsanity.com/paper/2106.09580/) (2021)
- [On Privacy and Personalization in Cross-Silo Federated Learning](https://mlsanity.com/paper/2206.07902/) (2022)
- [How Do AI Agents Spend Your Money? Analyzing and Predicting Token Consumption in Agentic Coding Tasks](https://mlsanity.com/paper/2604.22750/) (2026)
- [Communication-Efficient Massive UAV Online Path Control: Federated Learning Meets Mean-Field Game Theory](https://mlsanity.com/paper/2003.04451/) (2020)
- [Federated Learning as a Mean-Field Game](https://mlsanity.com/paper/2107.03770/) (2021)
- [Linear quadratic mean field games: Asymptotic solvability and relation to the fixed point approach](https://mlsanity.com/paper/1903.08776/) (2019)
- [Joint Local Relational Augmentation and Global Nash Equilibrium for Federated Learning with Non-IID Data](https://mlsanity.com/paper/2308.11646/) (2023)
- [Linear quadratic mean field games: Decentralized O(1/N)-Nash equilibria](https://mlsanity.com/paper/2107.09168/) (2021)
- [Online Federation For Mixtures of Proprietary Agents with Black-Box Encoders](https://mlsanity.com/paper/2505.00216/) (2025)
- [Privacy as Commodity: MFG-RegretNet for Large-Scale Privacy Trading in Federated Learning](https://mlsanity.com/paper/2603.28329/) (2026)

## Abstract (arXiv)

In the modern age of large-scale AI, federated learning has become an increasingly important tool for training large populations of AI agents; however, its computational and communication costs can rapidly fail to scale with the number of agents. This is precisely where decentralized agentic strategies shine: each agent acts autonomously, using only its own state together with a minimal summary of the ensemble, namely the mean-field. We derive the unique optimal decentralized policy in closed form. Optimality is characterized through a worst-client/minimax criterion: minimizing the under-performer regret, namely the maximal online cost incurred by the weakest agent in the ensemble. We further prove that the resulting decentralized policy asymptotically converges, in the large-population limit, to the Nash-optimal centralized policy, whose direct computation is not scalable. We use an online weighting mechanism to optimize the server-computed mixture of client predictions, thereby improving the mean prediction in addition to the previously optimized weakest-client prediction. Numerical experiments verify our theoretical guarantees and demonstrate that our decentralized policy typically outperforms natural greedy decentralized baselines.