# Stochastic Control Methods for Optimization

arXiv: https://arxiv.org/abs/2601.01248 · HTML: https://mlsanity.com/paper/2601.01248/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 69 references that are in the map.
- First posted on arXiv 3 Jan 2026.

## Builds on (most cited first, 30 of 30)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [A Unified Convergence Analysis for Shuffling-Type Gradient Methods](https://mlsanity.com/paper/2002.08246/) (2020)
- [Stochastic Cubic Regularization for Fast Nonconvex Optimization](https://mlsanity.com/paper/1711.02838/) (2017)
- [Stochastic Particle Gradient Descent for Infinite Ensembles](https://mlsanity.com/paper/1712.05438/) (2017)
- [Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks](https://mlsanity.com/paper/1905.07769/) (2019)
- [Recent Theoretical Advances in Non-Convex Optimization](https://mlsanity.com/paper/2012.06188/) (2020)
- [Mean Field Games and Applications: Numerical Aspects](https://mlsanity.com/paper/2003.04444/) (2020)
- [A consensus-based model for global optimization and its mean-field limit](https://mlsanity.com/paper/1604.05648/) (2016)
- [Limit theory for controlled McKean-Vlasov dynamics](https://mlsanity.com/paper/1609.08064/) (2016)
- [Entropic Fictitious Play for Mean Field Optimization Problem](https://mlsanity.com/paper/2202.05841/) (2022)
- [Hopf-Cole transformation via generalized Schr"odinger bridge problem](https://mlsanity.com/paper/1901.09051/) (2019)
- [Soft-constrained Schrodinger Bridge: a Stochastic Control Approach](https://mlsanity.com/paper/2403.01717/) (2024)
- [Mean-field particle swarm optimization](https://mlsanity.com/paper/2108.00393/) (2021)
- [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025)
- [A Probabilistic approach to classical solutions of the master equation for large population equilibria](https://mlsanity.com/paper/1411.3009/) (2014)
- [Numerical Methods for Mean-Field-Type Optimal Control Problems](https://mlsanity.com/paper/1703.10001/) (2017)
- [Rate of convergence for particle approximation of PDEs in Wasserstein space](https://mlsanity.com/paper/2103.00837/) (2021)
- [Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions](https://mlsanity.com/paper/2107.10535/) (2021)
- [On the Optimal Rate for the Convergence Problem in Mean Field Control](https://mlsanity.com/paper/2305.08423/) (2023)
- [Viscosity Solutions of Fully second-order HJB Equations in the Wasserstein Space](https://mlsanity.com/paper/2501.01612/) (2025)
- [Viscosity Solutions of a class of Second Order Hamilton-Jacobi-Bellman Equations in the Wasserstein Space](https://mlsanity.com/paper/2312.10322/) (2023)

## Built on by (most cited first, 1 of 1)

- [Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control](https://mlsanity.com/paper/2603.19463/) (2026)

## Abstract (arXiv)

In this work, we investigate a stochastic control framework for global optimization over both Euclidean spaces and the Wasserstein space of probability measures, where the objective function may be non-convex and/or non-differentiable. In the Euclidean setting, the original minimization problem is approximated by a family of regularized stochastic control problems; using dynamic programming, we analyze the associated Hamilton-Jacobi-Bellman equations and obtain tractable representations via the Cole-Hopf transformation and the Feynman-Kac formula. For optimization over probability measures, we formulate a regularized mean-field control problem characterized by a master equation, and further approximate it by controlled N-particle systems. We establish that, as the regularization parameter tends to zero (and as the particle number tends to infinity for the optimization over probability measures), the value of the control problem converges to the global minimum of the original objective. Building on the resulting probabilistic representations, we propose the Monte Carlo-based numerical schemes that are derivative-free due to the utilization of the Bismut-Elworthy-Li formula and numerical experiments are reported to illustrate the effectiveness of the methods and to support the theoretical convergence rates.