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.