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Interaction Matters: A Note on Non-asymptotic Local Convergence of Generative Adversarial Networks

Tengyuan Liang, James Stokes

TL;DR

This work develops a non-asymptotic local convergence theory for smooth two-player zero-sum games, motivated by GAN optimization. It reveals how the off-diagonal interaction term can slow SGA in stable settings while enabling exponential convergence for unstable equilibria via four stabilized dynamics: OMD, CO, IU, and PM. The authors provide explicit learning-rate guidance and demonstrate improved rates for OMD, with experiments on simple Gaussian-density tasks illustrating local convergence behavior and landscape sensitivity. The results bridge several stabilization methods, offering a unified lens on saddle-point dynamics and informing practical GAN training considerations, while acknowledging limitations in extending local results to global distribution learning."

Abstract

Motivated by the pursuit of a systematic computational and algorithmic understanding of Generative Adversarial Networks (GANs), we present a simple yet unified non-asymptotic local convergence theory for smooth two-player games, which subsumes several discrete-time gradient-based saddle point dynamics. The analysis reveals the surprising nature of the off-diagonal interaction term as both a blessing and a curse. On the one hand, this interaction term explains the origin of the slow-down effect in the convergence of Simultaneous Gradient Ascent (SGA) to stable Nash equilibria. On the other hand, for the unstable equilibria, exponential convergence can be proved thanks to the interaction term, for four modified dynamics proposed to stabilize GAN training: Optimistic Mirror Descent (OMD), Consensus Optimization (CO), Implicit Updates (IU) and Predictive Method (PM). The analysis uncovers the intimate connections among these stabilizing techniques, and provides detailed characterization on the choice of learning rate. As a by-product, we present a new analysis for OMD proposed in Daskalakis, Ilyas, Syrgkanis, and Zeng [2017] with improved rates.

Interaction Matters: A Note on Non-asymptotic Local Convergence of Generative Adversarial Networks

TL;DR

This work develops a non-asymptotic local convergence theory for smooth two-player zero-sum games, motivated by GAN optimization. It reveals how the off-diagonal interaction term can slow SGA in stable settings while enabling exponential convergence for unstable equilibria via four stabilized dynamics: OMD, CO, IU, and PM. The authors provide explicit learning-rate guidance and demonstrate improved rates for OMD, with experiments on simple Gaussian-density tasks illustrating local convergence behavior and landscape sensitivity. The results bridge several stabilization methods, offering a unified lens on saddle-point dynamics and informing practical GAN training considerations, while acknowledging limitations in extending local results to global distribution learning."

Abstract

Motivated by the pursuit of a systematic computational and algorithmic understanding of Generative Adversarial Networks (GANs), we present a simple yet unified non-asymptotic local convergence theory for smooth two-player games, which subsumes several discrete-time gradient-based saddle point dynamics. The analysis reveals the surprising nature of the off-diagonal interaction term as both a blessing and a curse. On the one hand, this interaction term explains the origin of the slow-down effect in the convergence of Simultaneous Gradient Ascent (SGA) to stable Nash equilibria. On the other hand, for the unstable equilibria, exponential convergence can be proved thanks to the interaction term, for four modified dynamics proposed to stabilize GAN training: Optimistic Mirror Descent (OMD), Consensus Optimization (CO), Implicit Updates (IU) and Predictive Method (PM). The analysis uncovers the intimate connections among these stabilizing techniques, and provides detailed characterization on the choice of learning rate. As a by-product, we present a new analysis for OMD proposed in Daskalakis, Ilyas, Syrgkanis, and Zeng [2017] with improved rates.

Paper Structure

This paper contains 12 sections, 7 theorems, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Stable Case: Non-asymptotic Local Convergence
  3. Unstable Case: Local Bi-Linear Problem
  4. (Improved) Optimistic Mirror Descent
  5. (Modified) Predictive Methods
  6. Implicit Updates
  7. Consensus Optimization
  8. Experiments
  9. Learning Covariance of Multivariate Gaussian
  10. Mixture of Gaussians
  11. Conclusions and Future Work
  12. Technical Proofs

Key Result

Theorem 1

Consider $U(\theta, \omega): \mathbb{R}^p \times \mathbb{R}^q \rightarrow \mathbb{R}$ that satisfies Assumption asmp:convex-concave for some radius $r>0$ near a stable local Nash equilibrium $(\theta_*, \omega_*)$ as in Definition def:slne. Suppose the initialization satisfies $(\theta_0, \omega_0) ($\alpha, \beta$ defined in Eqn. eq:alpha_beta) obtains an $\epsilon$-minimizer such that $(\theta_

Figures (3)

  • Figure 1: Distance to Nash equilibrium as a function of gradient iteration for the bilinear game assuming $p=q=5$, $\gamma = 1$ and $r=0.5$. The components of the interaction matrix $C$ were chosen i.i.d. uniform on $[0,1]$.
  • Figure 2: Evaluation metrics for covariance learning (top) and mixture of Gaussians learning (bottom) using different dynamical systems after $t=10^5$ and $t=5\cdot 10^4$ training iterations, respectively and 16 random seeds. Note that for covariance learning, we use the log-scale on $y$-axis.
  • Figure 3: Density plots of best and worst generator distribution measured by empirical Wasserstein-1 distance from the target distribution, across all baselines amongst 16 random seeds (excluding non-convergent runs) after training for $5\times 10^4$ iterations. Top: Consensus ($W_1=0.093$). Bottom: OMD ($W_1=0.367$).

Theorems & Definitions (17)

  • Definition 1.1: Local Nash Equilibrium
  • Definition 2.1: Stable Local Nash Equilibrium
  • Theorem 1: Exponential Convergence: SGA
  • Remark 1
  • Corollary 1: Simple Lower Bound for SGA
  • Theorem 2: Informal: Unstable Case
  • Theorem 3: Exponential Convergence: OMD
  • Theorem 4: Exponential Convergence: PM
  • Theorem 5: Exponential Convergence: IU
  • Theorem 6: Exponential Convergence: CO
  • ...and 7 more