[Paper Review] Analysis of Nonautonomous Adversarial Systems

Arash Mehrjou|arXiv (Cornell University)|Mar 13, 2018

Generative Adversarial Networks and Image Synthesis2 references3 citations

TL;DR

This paper analyzes the dynamics of nonautonomous generative adversarial networks (GANs) using a minimalistic 'tiny-GAN' model, demonstrating that annealing the data distribution over time stabilizes training by suppressing persistent oscillations in generator and discriminator parameters. The key contribution is a theoretical and simulation-based proof that longer annealing times reduce oscillation amplitudes, leading to convergence toward the desired data distribution.

ABSTRACT

Generative adversarial networks are used to generate images but still their convergence properties are not well understood. There have been a few studies who intended to investigate the stability properties of GANs as a dynamical system. This short writing can be seen in that direction. Among the proposed methods for stabilizing training of GANs, ß-GAN was the first who proposed a complete annealing strategy to change high-level conditions of the GAN objective. In this note, we show by a simple example how annealing strategy works in GANs. The theoretical analysis is supported by simple simulations.

Motivation & Objective

• To understand the convergence and stability properties of GANs as a dynamical system under time-varying data distributions.
• To investigate how annealing strategies—specifically in β-GAN—improve training stability in nonautonomous GANs.
• To model the training process as a continuous dynamical system with time-varying parameters to analyze oscillatory behavior.
• To provide theoretical and simulation-based evidence that longer annealing times reduce persistent oscillations in the generator and discriminator.

Proposed method

• Formulates a minimal nonautonomous GAN model ('tiny-GAN') with a single-point data distribution parameterized by α, a single-point generator at θ, and a linear discriminator at ψ.
• Introduces a time-varying data distribution α(t) that evolves via an annealing process α(t) = α₀ + (αᵣ − α₀)(1 − e⁻ᵗᐟᵀ), where αᵣ is the target real data distribution.
• Models the GAN training as a continuous dynamical system with three states: θ (generator), ψ (discriminator), and α (data distribution), governed by a system of ODEs.
• Applies Laplace transforms to the ODE system to analytically derive the time-domain behavior of ψ(t) and θ(t), enabling closed-form analysis of oscillations.
• Derives the amplitude of persistent oscillations in ψ(t) as 𝒜 = K / (1 + a²) × √(1 + a²), where K ∝ 1/T and a = 1/T, showing amplitude decreases as T increases.
• Uses inverse Laplace transforms to express ψ(t) as a sum of sinusoidal and decaying exponential terms, with the sinusoidal component responsible for sustained oscillations.

Experimental results

Research questions

• RQ1 How does annealing the data distribution in GANs affect the stability of the training dynamics in a continuous-time framework?
• RQ2 What is the analytical behavior of the generator and discriminator parameters under a time-varying data distribution?
• RQ3 Why does β-GAN’s annealing strategy reduce oscillations compared to standard GANs?
• RQ4 How does the annealing time T influence the amplitude of persistent oscillations in the discriminator's parameters?
• RQ5 Can the dynamics of nonautonomous GANs be modeled and analyzed using linearized continuous dynamical systems with tractable solutions?

Key findings

• The amplitude of persistent oscillations in the discriminator's parameters ψ(t) is proportional to K / (1 + a²) × √(1 + a²), where K ∝ (αᵣ − α₀)/T and a = 1/T.
• As the annealing time T increases, the oscillation amplitude decreases, approaching zero in the limit T → ∞, indicating improved stability.
• The generator parameter θ(t) converges to a steady-state oscillation around the true data distribution αᵣ, with the oscillation amplitude governed by the integral of ψ₁(τ) over time.
• The analytical solution shows that the oscillatory component in ψ(t) is a sinusoid with frequency 1 rad/s, and its amplitude scales inversely with T.
• The steady-state behavior of θ(t) is θ(t) → αᵣ + Ψ₁(t), where Ψ₁(t) is a sinusoidal integral of the oscillatory ψ₁(τ), confirming persistent oscillation around the target distribution.
• The theoretical analysis confirms that longer annealing times suppress oscillations, providing a dynamical justification for the empirical success of β-GAN.