Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gradient Flows as Optimal Controlled Evolutions: From Rn to Wasserstein product spaces

Yongxin Chen, Tryphon Georgiou, Michele Pavon

TL;DR

The paper develops a unified variational framework showing that gradient descent in Euclidean space, the gradient flow of relative entropy in Wasserstein space, and gradient flows on Wasserstein product spaces can be interpreted as optimal controlled evolutions. By connecting classical calculus of variations to optimal transport and Schrödinger-bridge theory, it derives, in the Euclidean setting, a control problem whose solution reproduces gradient descent; in Wasserstein space, it recasts the Fokker–Planck equation as a gradient flow of $D(\rho\|\bar{\rho})$ and extends this to a two-component product space with opposite fluxes. The results yield a natural variational interpretation of dissipation and open avenues for coordinated evolution of interacting populations, with potential applications to microrobotic swarms and density-control problems. The work highlights deep links between entropy, Fisher information, and control costs, and suggests extensions to multi-population settings and efficient numerical solvers for coupled flux systems.

Abstract

We show that the continuous-time gradient descent in Rn can be viewed as an optimal controlled evolution for a suitable action functional; a similar result holds for stochastic gradient descent. We then provide an analogous characterization for the Wasserstein gradient flow of the (relative) entropy, with an action that mirrors the classical case where the Euclidean gradient is replaced by the Wasserstein gradient of the relative entropy. In the small-step limit, these continuous-time actions align with the Jordan Kinderlehrer Otto scheme. Next, we consider gradient flows for the relative entropy over a Wasserstein product space-a study motivated by the stochastic-control formulation of Schrodinger bridges. We characterize the product-space steepest descent as the solution to a variational problem with two control velocities and a product-space Wasserstein gradient, and we show that the induced fluxes in the two components are equal and opposite. This framework suggests applications to the optimal control evolution of microrobotic swarms that can communicate their present distribution to the other swarm.

Gradient Flows as Optimal Controlled Evolutions: From Rn to Wasserstein product spaces

TL;DR

The paper develops a unified variational framework showing that gradient descent in Euclidean space, the gradient flow of relative entropy in Wasserstein space, and gradient flows on Wasserstein product spaces can be interpreted as optimal controlled evolutions. By connecting classical calculus of variations to optimal transport and Schrödinger-bridge theory, it derives, in the Euclidean setting, a control problem whose solution reproduces gradient descent; in Wasserstein space, it recasts the Fokker–Planck equation as a gradient flow of and extends this to a two-component product space with opposite fluxes. The results yield a natural variational interpretation of dissipation and open avenues for coordinated evolution of interacting populations, with potential applications to microrobotic swarms and density-control problems. The work highlights deep links between entropy, Fisher information, and control costs, and suggests extensions to multi-population settings and efficient numerical solvers for coupled flux systems.

Abstract

We show that the continuous-time gradient descent in Rn can be viewed as an optimal controlled evolution for a suitable action functional; a similar result holds for stochastic gradient descent. We then provide an analogous characterization for the Wasserstein gradient flow of the (relative) entropy, with an action that mirrors the classical case where the Euclidean gradient is replaced by the Wasserstein gradient of the relative entropy. In the small-step limit, these continuous-time actions align with the Jordan Kinderlehrer Otto scheme. Next, we consider gradient flows for the relative entropy over a Wasserstein product space-a study motivated by the stochastic-control formulation of Schrodinger bridges. We characterize the product-space steepest descent as the solution to a variational problem with two control velocities and a product-space Wasserstein gradient, and we show that the induced fluxes in the two components are equal and opposite. This framework suggests applications to the optimal control evolution of microrobotic swarms that can communicate their present distribution to the other swarm.

Paper Structure

This paper contains 13 sections, 3 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Gradient descent as optimal controlled evolution
  3. Elements of optimal mass transport theory
  4. The static problem
  5. The dynamic problem
  6. The Fokker-Planck equation as a gradient flow in Wasserstein space
  7. The Fokker-Planck as an optimal controlled evolution on Wasserstein space
  8. Relative entropy as a functional on Wasserstein product spaces
  9. Gradient descent on Wasserstein Product Spaces as optimal controlled evolution
  10. Recap and outlook
  11. Weighted gradient descents
  12. Newton's method
  13. Stochastic gradient descent

Key Result

Theorem 1

Let $f$ be a $C^1$ function on ${\mathbb R}^n$. Assume that $\nabla f$ is Lipschitz continuous. Consider the corresponding Gradient Flow on $[0,T]$ Let ${\cal X}$ denote the family of $C^1[0,T]$, ${\mathbb R}^n$-valued paths such that $x(0)=x_0$. Let $\cal U$ be the family of continuous, ${\mathbb R}^n$-valued control functions on $[0,T]$. Consider the following calculus of variations/optimal con

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 3
  • Remark 4
  • Theorem 5
  • proof