We consider the following interacting particle system in $\mathbb{R}^d$:

$$
\begin{aligned}
\dot{x}_i(t) &= -\frac{1}{N}\sum _{j=1}^{N}\nabla K(x_i(t)-x_j(t)) -\frac{1}{N} \sum _{j=1} ^{N}K\big(x_i(t)-x_j(t)\bigr)\nabla V (x_j(t)), \\\
x_i(0) &= x_i ^0 \in \mathbb R^d, \qquad i=1,\cdots,N.
\end{aligned} \tag{1}
$$

We refer to each of the $N$ functions $x_i(\cdot) \in \mathbb{R}^d$ as a particle. The function $K : \mathbb{R}^d \mapsto \mathbb{R}$ is a smooth, symmetric, and positive definite kernel. The function $V : \mathbb{R}^d \to \mathbb{R}$ is a smooth potential such that $e^{-V(x)}$ is integrable. More specific assumptions about $K$ and $V$ are given below.

We are interested in the macroscopic behavior of the particle system $(1)$ as $N \to \infty$ in the framework of mean field limit. Formally this mean field limit is described by the following non-local, nonlinear partial differential equation (PDE):
$$
\begin{aligned}
&\partial_t \rho = \nabla \cdot \left(\rho\left(K \ast (\nabla \rho+\nabla V\rho)\right)\right), \\\
&\rho(0,\cdot) = \rho_0(\cdot).
\end{aligned} \tag{2}
$$

In this article, we will prove the well-posedness of (1) and (2).

The Stein Variational Gradient Descent (SVGD) algorithm was first introduced by Liu and Wang [LW16], whose idea is to transport a set of $N$ particles $\{x_i\}_{i=1}^N$ in $\mathbb R^d$ so that their empirical measure

$$
\mu^N:=\frac{1}{N}\sum_{i=1}^N\delta_{x_i}
$$

approximates the target probability measure

$$
\rho_\infty (x)\ dx=Z^{-1} e^{-V(x)}\ dx
$$

with an unknown normalization factor $Z$.

Optimization over the space of probability measures is not only widely applicable, but also offers a useful perspective for analyzing certain complicated finite-dimensional nonconvex optimization problems. In particular, lifting such problems to optimization problems over probability measures can lead to better structural properties, such as convexity. Mean-field Langevin dynamics provides a representative example of this idea. Its central motivation is that some highly nonconvex optimization problems arising in neural network training become better behaved when reformulated as the optimization of a functional on the space of probability measures. This viewpoint also makes it possible to build a theoretical foundation for understanding the convergence of SGD. In what follows, we briefly introduce this perspective, mainly based on the paper by [Hu, Kaitong, et al]. The main analytical framework of this theory can be illustrated by Figure 2.

This is the talk I presented in the Optimization Group Seminar, titled: An Introduction to Optimization over the Space of Probability Measures: From Sampling to Wasserstein Gradient Flow.