# Kernel Expansions for High-Dimensional Mean-Field Control with Non-local Interactions

arXiv: https://arxiv.org/abs/2405.10922 · HTML: https://mlsanity.com/paper/2405.10922/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 May 2024.

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

- [End-to-End Training of High-Dimensional Optimal Control with Implicit Hamiltonians via Jacobian-Free Backpropagation](https://mlsanity.com/paper/2510.00359/) (2025)
- [On the Convergence of Jacobian-Free Backpropagation for Optimal Control Problems with Implicit Hamiltonians](https://mlsanity.com/paper/2602.00921/) (2026)
- [End-to-End Learning of Safe Optimal Feedback Control in High Dimensions with Control Barrier Function Layers](https://mlsanity.com/paper/2607.20674/) (2026)

## Abstract (arXiv)

Mean-field control (MFC) problems aim to find the optimal policy to control massive populations of interacting agents. These problems are crucial in areas such as economics, physics, and biology. We consider the non-local setting, where the interactions between agents are governed by a suitable kernel. For N agents, the interaction cost has O(N2) complexity, which can be prohibitively slow to evaluate and differentiate when N is large. To this end, we propose an efficient primal-dual algorithm that utilizes basis expansions of the kernels. The basis expansions reduce the cost of computing the interactions, while the primal-dual methodology decouples the agents at the expense of solving for a moderate number of dual variables. We also demonstrate that our approach can further be structured in a multi-resolution manner, where we estimate optimal dual variables using a moderate N and solve decoupled trajectory optimization problems for large N. We illustrate the effectiveness of our method on an optimal control of 5000 interacting quadrotors.