# On kernel-based statistical learning in the mean field limit

arXiv: https://arxiv.org/abs/2310.18074 · HTML: https://mlsanity.com/paper/2310.18074/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 33 references that are in the map.
- First posted on arXiv 27 Oct 2023.

## Builds on (most cited first, 9 of 9)

- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [Mean-Field Optimal Control](https://mlsanity.com/paper/1306.5913/) (2013)
- [Invisible control of self-organizing agents leaving unknown environments](https://mlsanity.com/paper/1504.04064/) (2015)
- [Mean-field optimal control as Gamma-limit of finite agent controls](https://mlsanity.com/paper/1803.04689/) (2018)
- [A collisionless singular Cucker-Smale model with decentralized formation control](https://mlsanity.com/paper/1807.05177/) (2018)
- [Reproducing kernel Hilbert spaces in the mean field limit](https://mlsanity.com/paper/2302.14446/) (2023)
- [Performance bounds for the mean-field limit of constrained dynamics](https://mlsanity.com/paper/1511.08364/) (2015)
- [Kinetic-controlled hydrodynamics for traffic models with driver-assist vehicles](https://mlsanity.com/paper/1807.11476/) (2018)

## Built on by (most cited first, 1 of 1)

- [On Safety in Safe Bayesian Optimization](https://mlsanity.com/paper/2403.12948/) (2024)

## Abstract (arXiv)

In many applications of machine learning, a large number of variables are considered. Motivated by machine learning of interacting particle systems, we consider the situation when the number of input variables goes to infinity. First, we continue the recent investigation of the mean field limit of kernels and their reproducing kernel Hilbert spaces, completing the existing theory. Next, we provide results relevant for approximation with such kernels in the mean field limit, including a representer theorem. Finally, we use these kernels in the context of statistical learning in the mean field limit, focusing on Support Vector Machines. In particular, we show mean field convergence of empirical and infinite-sample solutions as well as the convergence of the corresponding risks. On the one hand, our results establish rigorous mean field limits in the context of kernel methods, providing new theoretical tools and insights for large-scale problems. On the other hand, our setting corresponds to a new form of limit of learning problems, which seems to have not been investigated yet in the statistical learning theory literature.