Despite the established convergence theory of Optimistic Gradient Descent Ascent (OGDA) and Extragradient (EG) methods for the convex-concave minimax problems, little is known about the theoretical guarantees of these methods in nonconvex settings. To bridge this gap, for the first time, this paper establishes the convergence of OGDA and EG methods under the nonconvex-strongly-concave (NC-SC) and nonconvex-concave (NC-C) settings by providing a unified analysis through the lens of single-call extra-gradient methods. We further establish lower bounds on the convergence of GDA/OGDA/EG, shedding light on the tightness of our analysis. We also conduct experiments supporting our theoretical results. We believe our results will advance the theoretical understanding of OGDA and EG methods for solving complicated nonconvex minimax real-world problems, e.g., Generative Adversarial Networks (GANs) or robust neural networks training.
UNDERSTANDING OVERPARAMETERIZATION IN GENERATIVE ADVERSARIAL NETWORKS
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. Indeed, most successful GANs used in practice are trained using overparameterized generator and
discriminator networks, both in terms of depth and width. 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 more »
- Publication Date:
- NSF-PAR ID:
- 10272364
- Journal Name:
- International Conference on Learning Representations
- Sponsoring Org:
- National Science Foundation
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