Neural Architectures for Modeling Distributional Dependence in Diffusion Processes

Core Concepts

Explicitly including distributional dependence in the parameterization of stochastic differential equations can improve modeling capabilities for temporal data with interaction under an exchangeability assumption, while maintaining strong performance for standard Itô-SDEs.

Abstract

The paper proposes neural architectures for representing McKean-Vlasov stochastic differential equations (MV-SDEs), which model the behavior of an infinite number of interacting particles by imposing a dependence on the particle density.
The key highlights are:

The authors present three neural architectures for representing MV-SDEs: the empirical measure (EM) architecture, the implicit measure (IM) architecture, and the marginal law (ML) architecture. These architectures differ in how they model the expectation in the mean-field term of the MV-SDE.

The authors develop maximum likelihood-based estimators to infer the parameters of the MV-SDE from data, without requiring prior knowledge of the drift structure.

The authors analyze the implicit regularization and richer probability flows associated with the proposed MV-SDE architectures, and show how they can outperform standard neural Itô-SDE models in time series and generative modeling tasks.

The experiments on synthetic and real-world datasets demonstrate the effectiveness of the proposed architectures in modeling temporal data with interaction, while maintaining strong performance on standard Itô-SDEs.

Stats

The paper does not provide specific numerical values or statistics to support the key logics. The results are presented in the form of sample paths, mean squared error plots, and energy distance comparisons.

Quotes

"Explicitly including distributional dependence in the parameterization of the SDE is effective in modeling temporal data with interaction under an exchangeability assumption while maintaining strong performance for standard Itô-SDEs due to the richer class of probability flows associated with MV-SDEs."
"The fact that the interaction of many particles can cause blowups leads to a remarkable property of MV-SDEs that allows discontinuous paths. The major benefit of this property is that we do not need to consider an additional jump noise process – we only need to specify a particular interaction between the particles to induce the jump behavior."

Key Insights Distilled From

Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processes

by Haoming Yang... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09402.pdf

Deeper Inquiries

How can the proposed architectures be extended to handle heterogeneous agents or non-exchangeable particle systems?

The proposed architectures can be extended to handle heterogeneous agents or non-exchangeable particle systems by introducing depth into the architecture. This can be achieved by having multiple measures represented by weight matrices to take the expectation with respect to. Each measure can correspond to a different type of agent or particle, allowing for a more diverse representation of the system. By incorporating this heterogeneity, the architecture can better capture the interactions and dynamics of non-exchangeable particle systems, where each particle may have unique characteristics or behaviors.

What are the theoretical convergence rates of the proposed estimators, and how can they be improved?

The theoretical convergence rates of the proposed estimators can be analyzed by studying the rate at which the estimators converge to the true parameters of the MV-SDEs. To improve the convergence rates, one approach could be to apply a multilevel scheme, as discussed in previous research. This scheme can help improve accuracy while reducing computational cost by iteratively refining the estimates at different levels of resolution. Additionally, optimizing the learning rate and regularization parameters in the optimization process can also enhance the convergence rates of the estimators.

Can the proposed architectures be used to model point processes by leveraging the discontinuous sample paths of MV-SDEs?

Yes, the proposed architectures can be used to model point processes by leveraging the discontinuous sample paths of MV-SDEs. The discontinuous paths in MV-SDEs allow for the representation of point processes through the interactions between particles that induce jumps. By incorporating the explicit distributional dependence in the architectures, the models can capture the dynamics of point processes where events occur at specific points in time. The richer class of densities modeled by MV-SDEs, including the ability to induce jumps in the sample paths, makes them well-suited for modeling point processes with discontinuities. By leveraging the discontinuous paths, the architectures can effectively capture the complex dynamics of point processes in various applications.