Core Concepts

This paper establishes quantitative convergence results for the value functions and optimal parameters of neural SDEs as the sample size grows to infinity. The authors analyze the Hamilton-Jacobi-Bellman equation corresponding to the N-particle system and obtain uniform regularity estimates, which are then used to show the convergence of the minima of objective functionals and optimal parameters.

Abstract

The paper studies a class of neural SDEs and the associated sampled optimal control problems, where the neural SDEs with N samples can be linked to N-particle systems with centralized control. The authors analyze the Hamilton-Jacobi-Bellman (HJB) equation corresponding to the N-particle system and establish uniform regularity results with respect to the number of particles N.
Key highlights:
The neural SDEs are formulated as controlled particle systems, where the control process is shared among all particles.
Uniform regularity estimates on the value function VN and its derivatives are obtained using a combination of stochastic maximum principle and backward stochastic Riccati equation analysis.
The uniform regularity results are then used to show the convergence of the minima of objective functionals and optimal parameters as the sample size N tends to infinity.
Quantitative convergence rates are also established, showing that VN and the optimal feedback function θ*N converge at certain algebraic rates to their limit objects.
The limit objects are identified as functions defined on the Wasserstein space of probability measures, and the corresponding limiting HJB equation is formally derived.

Stats

The paper does not contain explicit numerical data or statistics. The key results are theoretical convergence and regularity estimates.

Quotes

"The neural SDE (1.1) describes the deep learning from a dynamical systems viewpoint and relying on this, our results make it possible to analyze the convergence of trainable parameters obtained from samples with size N."
"The limit function V is formally associated to a second order HJB equation set on the Wasserstein space P2(R)."

Deeper Inquiries

To extend the analysis to more general state dynamics that depend on the distribution of particles in a non-linear way, we can consider incorporating additional terms in the Hamiltonian or objective function that capture the measure dependence of the dynamics. This would involve modifying the HJB equation to account for the non-linear interactions between the state variables and the distribution of particles. By introducing suitable regularity conditions and assumptions on the state dynamics, we can analyze the convergence behavior of the optimal control and value functions in the presence of these non-linear dependencies. The analysis would involve studying the impact of the measure dependence on the convergence rates and the limiting behavior of the optimization process.

The convergence results obtained in the paper have significant implications for the generalization performance of neural networks trained with finite samples. As the sample size grows to infinity, the convergence of the optimal parameters and value functions indicates that the neural network model becomes more robust and generalizable. The convergence of the objective functionals and optimal parameters to suitable limit objects defined on the Wasserstein space of Borel probability measures suggests that the neural network can effectively capture the underlying patterns in the data and make accurate predictions even with limited training samples. This convergence ensures that the neural network model can generalize well to unseen data and maintain its performance across different datasets.

The techniques developed in this paper, such as the analysis of the Hamilton–Jacobi–Bellman equation, stochastic maximum principle, and regularity results on the HJB equations, can be applied to study the convergence of other machine learning algorithms that involve optimization over probability measures. These techniques are general and can be adapted to various optimization problems in machine learning where the objective functions and dynamics depend on probability distributions. By formulating the optimization problems in the framework of optimal control theory and stochastic analysis, one can analyze the convergence behavior and convergence rates of the algorithms as the sample size or number of particles grows to infinity. This approach can provide insights into the generalization performance and convergence properties of a wide range of machine learning algorithms.

0