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GANs as Gradient Flows that Converge

Yu-Jui Huang, Yuchong Zhang

TL;DR

This work casts unsupervised learning as a gradient flow in the space of probability densities, showing that a distribution-dependent ODE drives the current density toward the unknown data distribution $\rho_d$. It reveals a precise equivalence between simulating this gradient flow and training a GAN, offering a cooperative navigator-calibrator interpretation that explains GAN divergence through unintended MSE fitting. The analysis hinges on a nonlinear Fokker–Planck equation, a Trevisan superposition-based construction of a solution to the density-dependent ODE, and a convergence proof establishing $\rho^{Y_t}\to\rho_d$ in $L^1$ as $t\to\infty$. The results provide a mathematically rigorous backbone for gradient-flow approaches to implicit generative modeling and offer principled insights for improving GAN stability and convergence in practice.

Abstract

This paper approaches the unsupervised learning problem by gradient descent in the space of probability density functions. A main result shows that along the gradient flow induced by a distribution-dependent ordinary differential equation (ODE), the unknown data distribution emerges as the long-time limit. That is, one can uncover the data distribution by simulating the distribution-dependent ODE. Intriguingly, the simulation of the ODE is shown equivalent to the training of generative adversarial networks (GANs). This equivalence provides a new "cooperative" view of GANs and, more importantly, sheds new light on the divergence of GANs. In particular, it reveals that the GAN algorithm implicitly minimizes the mean squared error (MSE) between two sets of samples, and this MSE fitting alone can cause GANs to diverge. To construct a solution to the distribution-dependent ODE, we first show that the associated nonlinear Fokker-Planck equation has a unique weak solution, by the Crandall-Liggett theorem for differential equations in Banach spaces. Based on this solution to the Fokker-Planck equation, we construct a unique solution to the ODE, using Trevisan's superposition principle. The convergence of the induced gradient flow to the data distribution is obtained by analyzing the Fokker-Planck equation.

GANs as Gradient Flows that Converge

TL;DR

This work casts unsupervised learning as a gradient flow in the space of probability densities, showing that a distribution-dependent ODE drives the current density toward the unknown data distribution . It reveals a precise equivalence between simulating this gradient flow and training a GAN, offering a cooperative navigator-calibrator interpretation that explains GAN divergence through unintended MSE fitting. The analysis hinges on a nonlinear Fokker–Planck equation, a Trevisan superposition-based construction of a solution to the density-dependent ODE, and a convergence proof establishing in as . The results provide a mathematically rigorous backbone for gradient-flow approaches to implicit generative modeling and offer principled insights for improving GAN stability and convergence in practice.

Abstract

This paper approaches the unsupervised learning problem by gradient descent in the space of probability density functions. A main result shows that along the gradient flow induced by a distribution-dependent ordinary differential equation (ODE), the unknown data distribution emerges as the long-time limit. That is, one can uncover the data distribution by simulating the distribution-dependent ODE. Intriguingly, the simulation of the ODE is shown equivalent to the training of generative adversarial networks (GANs). This equivalence provides a new "cooperative" view of GANs and, more importantly, sheds new light on the divergence of GANs. In particular, it reveals that the GAN algorithm implicitly minimizes the mean squared error (MSE) between two sets of samples, and this MSE fitting alone can cause GANs to diverge. To construct a solution to the distribution-dependent ODE, we first show that the associated nonlinear Fokker-Planck equation has a unique weak solution, by the Crandall-Liggett theorem for differential equations in Banach spaces. Based on this solution to the Fokker-Planck equation, we construct a unique solution to the ODE, using Trevisan's superposition principle. The convergence of the induced gradient flow to the data distribution is obtained by analyzing the Fokker-Planck equation.
Paper Structure (22 sections, 18 theorems, 145 equations, 1 figure, 1 algorithm)

This paper contains 22 sections, 18 theorems, 145 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gradient Descent for Functions on $\mathcal{P}(\mathbb{R}^d)$
  4. Connection to the Lions and Wasserstein Derivatives
  5. Problem Formulation
  6. Main Results and Contributions
  7. Simulating ODE \ref{['Y GAN']}
  8. Connection to GANs
  9. Implication for the Divergence of GANs
  10. The Nonlinear Fokker-Planck Equation
  11. Preparations
  12. Existence and Uniqueness
  13. Solutions to Density-Dependent ODE \ref{['Y GAN']}
  14. Convergence to Data Distribution $\rho_{\operatorname{d}}$
  15. Proofs for Section \ref{['sec:preliminaries']}
  16. ...and 7 more sections

Key Result

Proposition 5

Let $G:\mathcal{P}(\mathbb{R}^d)\to\mathbb{R}$ has a linear functional derivative $\frac{\delta G}{\delta \rho}:\mathcal{P}(\mathbb{R}^d)\times\mathbb{R}^d\to\mathbb{R}$. Consider $\rho\in\mathcal{P}(\mathbb{R}^d)$ and the measure $\mu(\cdot) := \int_\cdot \rho(y) dy$ on $\mathbb{R}^d$. If $\rho$ is

Figures (1)

  • Figure 1: An initial $G:\mathbb{R}\to\mathbb{R}$ maps samples $\{z^{(i)}\}_{i=1}^6$ of $Z$ to $\{x^{(i)}\}_{i=1}^6$. Two distinct $\tilde{G}:\mathbb{R}\to\mathbb{R}$ serve to relocate $\{x^{(i)}\}_{i=1}^6$ to approximate $\{y^{(i)}\}_{i=1}^6$, the new points obtained in \ref{['new points']}, under the criterion of point-wise similarity and set-wise similarity, respectively. The red circles represent $\tilde{x}^{(i)} := \tilde{G}(z^{(i)})$, $i=1,...,6$.

Theorems & Definitions (36)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 4
  • Proposition 5
  • Lemma 6
  • Theorem 7
  • Proposition 8
  • Definition 9
  • Remark 10
  • ...and 26 more