A Unified Analysis of Extra-gradient and Optimistic Gradient Methods for Saddle Point Problems: Proximal Point Approach (1901.08511v4)

Abstract: In this paper we consider solving saddle point problems using two variants of Gradient Descent-Ascent algorithms, Extra-gradient (EG) and Optimistic Gradient Descent Ascent (OGDA) methods. We show that both of these algorithms admit a unified analysis as approximations of the classical proximal point method for solving saddle point problems. This viewpoint enables us to develop a new framework for analyzing EG and OGDA for bilinear and strongly convex-strongly concave settings. Moreover, we use the proximal point approximation interpretation to generalize the results for OGDA for a wide range of parameters.

• The paper's main contribution is demonstrating that EG and OGDA approximate the Proximal Point method, ensuring linear convergence in various saddle point settings.
• It achieves linear convergence for bilinear problems with complexity O(κ log(1/ε)) and generalizes OGDA to allow differing step sizes while maintaining efficiency.
• The study extends the analysis to strongly convex-strongly concave cases, offering insights that could impact parameter selection in GANs and robust optimization.

Unified Analysis of Extra-gradient and Optimistic Gradient Methods for Saddle Point Problems

The paper presents a comprehensive theoretical examination of Extra-gradient (EG) and Optimistic Gradient Descent Ascent (OGDA) methods for saddle point problems, proposing that these methods can be viewed as approximations of the Proximal Point (PP) method. This approach provides a unified framework to analyze these two gradient-based methods in both bilinear and strongly convex-strongly concave settings.

Key Contributions

The primary contribution lies in demonstrating that both EG and OGDA methods are approximations of the Proximal Point method, a classical approach in optimization that enjoys linear convergence properties. This perspective facilitates a deeper understanding of why these methods exhibit convergence in bilinear cases and enables the generalization of OGDA to a broader range of parameter settings. Significant results include:

• Linear Convergence for Bilinear Problems: Both EG and OGDA converge linearly for bilinear saddle point problems with a complexity of $O(\kappa \log(1/\epsilon))$, where $\kappa$ is the condition number of the problem.
• Generalization of OGDA: A new generalized form of OGDA is introduced, allowing different step sizes for gradient descent and the "negative-momentum" term. It is shown that linear convergence can still be achieved with these modifications under certain conditions.
• Strongly Convex-Strongly Concave Case: The paper extends the analysis to show that even for functions that are strongly convex in one variable and strongly concave in another, both EG and generalized OGDA maintain linear convergence with a similar complexity bound.

Comparative Analysis

The analysis provides a comparative table showing convergence complexities from previous works, highlighting the improvements achieved in this paper. For bilinear settings, the paper matches linear convergence rates previously established but through a more cohesive framework, thereby broadening the applicability and understanding of these iterative methods. For convex-concave cases, the paper establishes new results for OGDA, contributing to the theoretical understanding of its convergence behavior.

Implications and Future Directions

The implications of this work are twofold: practical and theoretical. Practically, the unification of gradient methods under the proximal point framework may influence how algorithmic parameters are chosen for efficient convergence in applications such as Generative Adversarial Networks (GANs) and robust optimization tasks. Theoretically, this work posits a new direction for analyzing other gradient methods—potentially including stochastic variants—using the proximal point lens, likely stimulating further research in optimization theory. Additionally, the framework could be expanded to encompass stochastic approximation scenarios, considering the increasing complexity and stochasticity in modern machine learning applications.

In conclusion, this paper lays a foundation for understanding and expanding upon common gradient-based approaches to saddle point problems by situating them within the proximal point method's theoretical confines. As iterative methods continue to develop, this unified approach offers a promising avenue for the analysis and generalization of diverse optimization algorithms.