  A broad class of unsupervised deep learning methods such as Generative
Adversarial Networks (GANs) involve training of overparameterized models where
the number of parameters of the model exceeds a certain threshold. A large body
of work in supervised learning have shown the importance of model
overparameterization in the convergence of the gradient descent (GD) to
globally optimal solutions. In contrast, the unsupervised setting and GANs in
particular involve non-convex concave mini-max optimization problems that are
often trained using Gradient Descent/Ascent (GDA). The role and benefits of
model overparameterization in the convergence of GDA to a global saddle point
in non-convex concave problems is far less understood. In this work, we present
a comprehensive analysis of the importance of model overparameterization in
GANs both theoretically and empirically. We theoretically show that in an
overparameterized GAN model with a $1$-layer neural network generator and a
linear discriminator, GDA converges to a global saddle point of the underlying
non-convex concave min-max problem. To the best of our knowledge, this is the
first result for global convergence of GDA in such settings. Our theory is
based on a more general result that holds for a broader class of nonlinear
generators and discriminators that obey certain assumptions (including deeper
generators and random feature discriminators). We also empirically study the
role of model overparameterization in GANs using several large-scale
experiments on CIFAR-10 and Celeb-A datasets. Our experiments show that
overparameterization improves the quality of generated samples across various
model architectures and datasets. Remarkably, we observe that
overparameterization leads to faster and more stable convergence behavior of
GDA across the board.
