[Paper Review] On the Convergence of Gradient Descent in GANs: MMD GAN As a Gradient Flow

Youssef Mroueh, Truyen Nguyen|arXiv (Cornell University)|Nov 4, 2020

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes a parametric kernelized gradient flow for MMD GANs that ensures global convergence of gradient descent to the target data distribution by introducing a novel Riemannian structure on the statistical manifold of generator distributions. The key contribution is showing that gradient descent using the MMD_{\alpha,\beta} witness function corresponds to the gradient flow of the MMD functional under this new geometric framework, with non-asymptotic convergence guarantees under explicit conditions.

ABSTRACT

We consider the maximum mean discrepancy ($\\mathrm{MMD}$) GAN problem and propose a parametric kernelized gradient flow that mimics the min-max game in gradient regularized $\\mathrm{MMD}$ GAN. We show that this flow provides a descent direction minimizing the $\\mathrm{MMD}$ on a statistical manifold of probability distributions. We then derive an explicit condition which ensures that gradient descent on the parameter space of the generator in gradient regularized $\\mathrm{MMD}$ GAN is globally convergent to the target distribution. Under this condition, we give non asymptotic convergence results of gradient descent in MMD GAN. Another contribution of this paper is the introduction of a dynamic formulation of a regularization of $\\mathrm{MMD}$ and demonstrating that the parametric kernelized descent for $\\mathrm{MMD}$ is the gradient flow of this functional with respect to the new Riemannian structure. Our obtained theoretical result allows ones to treat gradient flows for quite general functionals and thus has potential applications to other types of variational inferences on a statistical manifold beyond GANs. Finally, numerical experiments suggest that our parametric kernelized gradient flow stabilizes GAN training and guarantees convergence.

Motivation & Objective

• To understand the distributional convergence properties of gradient descent in MMD GANs, which remain poorly understood despite empirical success.
• To address the lack of rigorous convergence analysis for gradient descent in GANs, particularly in the context of MMD-based objectives.
• To introduce a new parametric energy regularizer, MMD_{\alpha,\beta}, that operates on generator parameters rather than input space.
• To derive a dynamic formulation of MMD on the statistical manifold and define a new Riemannian metric d_{\alpha,\beta} for gradient flows.
• To establish non-asymptotic convergence results for gradient descent in MMD GANs under explicit conditions.

Proposed method

• Proposes a parametric kernelized gradient flow that mimics the min-max game in gradient-regularized MMD GANs, using the MMD_{\alpha,\beta} witness function as the descent direction.
• Introduces a new regularizer MMD_{\alpha,\beta} defined via a parametric energy with gradient taken w.r.t. generator parameters, not input space.
• Derives a dynamic formulation of MMD on the statistical manifold and constructs a Riemannian metric tensor d_{\alpha,\beta} to define a new geometric structure.
• Shows that the gradient descent flow in MMD GANs driven by the MMD_{\alpha,\beta} witness function is equivalent to the gradient flow of the MMD functional under the d_{\alpha,\beta} metric.
• Establishes non-asymptotic convergence bounds by analyzing the decay of the MMD functional along the flow, under conditions on the eigenvalues and kernel parameters.
• Employs a discrete algorithm with RMSProp updates for both the witness function (via v or p) and generator parameters (θ), using a minibatch estimate of the MMD_{\alpha,\beta} objective.

Experimental results

Research questions

• RQ1 Under what conditions does gradient descent in MMD GANs converge globally to the target data distribution?
• RQ2 Can a parametric kernelized gradient flow be constructed that ensures descent on the MMD functional over the statistical manifold of generator distributions?
• RQ3 Is there a Riemannian geometric structure on the statistical manifold such that the MMD GAN gradient descent corresponds to a gradient flow under that structure?
• RQ4 How does the proposed MMD_{\alpha,\beta} regularizer improve training stability and convergence compared to standard MMD GANs?
• RQ5 What is the non-asymptotic convergence rate of gradient descent in MMD GANs under the proposed geometric framework?

Key findings

• Gradient descent in MMD GANs is globally convergent to the target distribution if the condition α‖D_θ u_t‖_ℋ² + β⟨u_t, D_θ u_t⟩_ℋ = 0 holds, which is equivalent to D_θ h_θ = 0.
• The parametric kernelized gradient flow for MMD corresponds exactly to the gradient flow of the MMD functional under the new Riemannian metric d_{\alpha,\beta}.
• Non-asymptotic convergence is established with the decay estimate: log(ℱ(q_θ_t)/ℱ(q_θ_0)) ≤ -∫₀ᵗ [λ_i(θ_s)a(θ_s,u_s)γ(θ_s)] / [αλ_i(θ_s)a(θ_s,u_s) + β] ds.
• The convergence rate depends on the eigenvalues λ_i(θ_s), the kernel alignment a(θ_s,u_s), and the curvature parameter γ(θ_s), with faster decay when these are large.
• Numerical experiments show that the proposed flow stabilizes training and prevents mode collapse, especially when α > 0, unlike standard MMD GANs with α = 0.
• The method performs well even when the witness function is a deep neural network (learned kernel), suggesting potential generalization to the Neural Tangent Kernel regime.