Numerical approximation of McKean-Vlasov SDEs via stochastic gradient descent

TL;DR

This work introduces an SGD-based framework to numerically approximate McKean-Vlasov SDEs without relying on interacting particle systems, by casting the solution as a fixed-point problem for the curve $\bar{\gamma}(t)=\mathbb{E}[\varphi(X_t)]$ and projecting onto a finite-dimensional basis for SGD optimization. The method hinges on a fixed-point map $\Psi$ defined via an auxiliary SDE and on a variational loss $F(\gamma)$, with a rigorous finite-dimensional surrogate $G(a)$ that can be minimized by SGD; convergence to stationary points is established under suitable smoothness and growth conditions, leveraging tangent-process estimates. Numerical studies on Kuramoto-type, polynomial-drift, and convolution-type MV-SDEs show the SGD approach achieves competitive accuracy with significantly reduced simulation costs compared to IPS benchmarks, especially when using mini-batches and low-degree polynomial bases. The results provide a theoretical and computational foundation for solving MV-SDEs via SGD, including stability and convergence analysis, and extendable to broader coefficient regimes and projection schemes.

Abstract

We propose a novel approach to numerically approximate McKean-Vlasov stochastic differential equations (MV-SDE) using stochastic gradient descent (SGD) while avoiding the use of interacting particle systems (IPS) {and the associated simulation costs required to achieve the ``propagation of chaos'' limit}. The SGD technique is deployed to solve a Euclidean minimization problem, obtained by first representing the MV-SDE as a minimization problem over the set of continuous functions of time, and then approximating the domain with a finite-dimensional subspace. Convergence is established by proving certain intermediate stability and moment estimates of the relevant stochastic processes, including the tangent processes. Numerical experiments illustrate the competitive performance of our SGD based method compared to the IPS benchmarks. This work offers a theoretical foundation for using the SGD method in the context of numerical approximation of MV-SDEs, and provides analytical tools to study its stability and convergence.

Figures (6)

• Figure 1: Kuramoto-Shinomoto-Sakaguchi MV-SDE. Comparison of the output curves $\mathscr{L} {\bf a}_m=(\mathscr{L} {\bf a}_m^1, \mathscr{L} {\bf a}_m^2)$ of the SGD algorithm versus the benchmark curves ${\bar{\gamma}}^{\text{MC}}=({\bar{\gamma}}^{\text{MC}}_1 , {\bar{\gamma}}^{\text{MC}}_2)$ for all values of $n$, for timestep size $h=10^{-2}$, $T=0.5$, $M=1000$, $x_0=0.5$ and $\sigma=0.5$.
• Figure 2: Kuramoto-Shinomoto-Sakaguchi MV-SDE. Comparison of the output curves $\mathscr{L} {\bf a}_m=(\mathscr{L} {\bf a}_m^1, \mathscr{L} {\bf a}_m^2)$ of the SGD algorithm versus the benchmark curves ${\bar{\gamma}}^{\text{MC}}=({\bar{\gamma}}^{\text{MC}}_1 , {\bar{\gamma}}^{\text{MC}}_2)$ for all values of $n$, for timestep size $h=10^{-2}$, $T=1$, $M=1000$, $x_0=0.5$ and $\sigma=0.5$.
• Figure 3: Kuramoto-Shinomoto-Sakaguchi MV-SDE. Comparison of the output curves $\mathscr{L} {\bf a}_m=(\mathscr{L} {\bf a}_m^1, \mathscr{L} {\bf a}_m^2)$ of the SGD algorithm versus the benchmark curves ${\bar{\gamma}}^{\text{MC}}=({\bar{\gamma}}^{\text{MC}}_1 , {\bar{\gamma}}^{\text{MC}}_2)$ for all values of $n$, for timestep size $h=10^{-2}$, $T=2$, $M=1000$, $x_0=0.5$ and $\sigma=0.5$.
• Figure 4: Polynomial drift MV-SDE. Comparison of the output curves $\mathscr{L} {\bf a}_m=(\mathscr{L} {\bf a}_m^1, \mathscr{L} {\bf a}_m^2)$ of the SGD algorithm versus the benchmark curves ${\bar{\gamma}}^{\text{MC}}=({\bar{\gamma}}^{\text{MC}}_1 , {\bar{\gamma}}^{\text{MC}}_2)$ for all values of $T$, for timestep size $h=10^{-2}$, $n=3$, $M=1000$, $x_0=1$ and $\delta=0.8$.
• Figure 5: Monte Carlo method densities $\tilde{w}^{(K),\text{MC}}_T(x)$ over the interval $[-3,4]$, for $T=1$ and $K = 3, 5, 10, 20$. The benchmark vector ${\bar{\gamma}}^{\text{MC}}(T)$ was computed with $N_{0}=10^7$ particles. Here $X_0 \sim \mathcal{N}_{(0,1)}$ and $\sigma = 0.1$. In the model \ref{['eq:MKV_con']} we had $X_0 \sim \mathcal{N}_{(0,1)}$ and $\sigma = 0.1$.

Theorems & Definitions (37)

• Remark 2.1
• Proposition 2.4
• Remark 2.5: Growth assumptions
• Remark 2.6: Regularity assumptions
• Example 2.7
• Proposition 2.9: Higher-order smoothness of $\bar{\gamma}$
• Remark 2.10
• Example 2.11
• Proposition 2.12
• Lemma 2.13