# Nonparametric inference for density-dependent McKean--Vlasov diffusions

arXiv: https://arxiv.org/abs/2609.01166 · HTML: https://mlsanity.com/paper/2609.01166/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 19 references that are in the map.
- First posted on arXiv 1 Sep 2026.

## Builds on (most cited first, 2 of 2)

- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations](https://mlsanity.com/paper/2206.09527/) (2022)

## Abstract (arXiv)

The present research is devoted to the nonparametric estimation of a density-dependent drift coefficient in a multivariate McKean--Vlasov diffusion from independent observations at a common time, as well as the stationary density. Under certain assumptions on the (known) potential, we reduce the problem to the one-dimensional one and construct a sieve maximum-likelihood estimator based on sparse ReQU neural networks subject to structural and H"older constraints. Using the endpoint-adapted graded approximation, we achieve the rate of left(bnlog n/nright)2(β+1)/(2β+3) for the Kullback-Leibler divergence between the true and estimated stationary densities, with bn being at most a logarithmic factor. Similarly, it is shown that the constructed estimator for the drift coefficient converges to the true one at the rate of left(bnlog n/nright)β/(2β+3) in the L2-metric. A matching Assouad lower bound proves minimax optimality of this bound up to logarithmic factors.