Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile (1807.02629v2)

Abstract: Owing to their connection with generative adversarial networks (GANs), saddle-point problems have recently attracted considerable interest in machine learning and beyond. By necessity, most theoretical guarantees revolve around convex-concave (or even linear) problems; however, making theoretical inroads towards efficient GAN training depends crucially on moving beyond this classic framework. To make piecemeal progress along these lines, we analyze the behavior of mirror descent (MD) in a class of non-monotone problems whose solutions coincide with those of a naturally associated variational inequality - a property which we call coherence. We first show that ordinary, "vanilla" MD converges under a strict version of this condition, but not otherwise; in particular, it may fail to converge even in bilinear models with a unique solution. We then show that this deficiency is mitigated by optimism: by taking an "extra-gradient" step, optimistic mirror descent (OMD) converges in all coherent problems. Our analysis generalizes and extends the results of Daskalakis et al. (2018) for optimistic gradient descent (OGD) in bilinear problems, and makes concrete headway for establishing convergence beyond convex-concave games. We also provide stochastic analogues of these results, and we validate our analysis by numerical experiments in a wide array of GAN models (including Gaussian mixture models, as well as the CelebA and CIFAR-10 datasets).

• The paper introduces an extra-gradient step in the Optimistic Mirror Descent algorithm to robustly tackle non-monotone saddle-point problems.
• It establishes coherence properties in saddle-point problems, detailing how strict and null coherence affect convergence.
• Empirical results using GAN training on datasets like CelebA and CIFAR-10 demonstrate OMD's superior stability and performance.

An Examination of Optimistic Mirror Descent in Saddle-Point Problems

The paper "Optimistic Mirror Descent in Saddle-Point Problems: Going the Extra (Gradient) Mile" presents a rigorous exploration of the Optimistic Mirror Descent (OMD) algorithm applied to the challenging landscape of saddle-point problems, providing both theoretical insights and practical validations. The authors Panayotis Mertikopoulos et al. tackle the complexities associated with non-monotone saddle-point problems, particularly in the context of generative adversarial networks (GANs).

Theoretical Framework and Key Contributions

The paper begins by contextualizing saddle-point (SP) problems within the broader domain of optimization and game theory, highlighting their critical relevance in machine learning. Specifically, the authors focus on the difficulties posed by non-monotone SP problems, which do not align with the classical convex-concave framework often assumed in theoretical analyses. The discourse on coherence, a property of SP problems where solutions coincide with those of an associated variational inequality (VI), provides a foundational basis for understanding the nuances of these problems. The authors introduce categories such as strict and null coherence, which play pivotal roles in delineating convergence characteristics of algorithms in SP problems.

Mirror Descent and Its Limitations

The manuscript explicates the mechanisms of the Mirror Descent (MD) algorithm while delineating its inherent limitations when confronted with non-monotone SP problems. Despite its success in online and convex optimization tasks, MD is shown to falter, particularly in problems characterized by null coherence, where it fails to reach convergence and may exhibit cyclical behavior.

Optimistic Mirror Descent: The Extra-Gradient Solution

The primary contribution of the paper is the development and analysis of the Optimistic Mirror Descent algorithm. Through the integration of an extra-gradient step, the authors propose a method to overcome the convergence limitations faced by traditional gradient-based approaches in non-monotone SP domains. This extra-gradient step acts as a foresighted adjustment that improves the algorithm's stability and convergence, even extending to cases where the problem is strictly coherent.

Numerical Results and Experimental Verification

The experimental verification of OMD is thorough, encompassing synthetic Gaussian Mixture Models (GMMs) and real-world image datasets such as CelebA and CIFAR-10. The results demonstrate the enhanced stability and performance of OMD over existing optimization techniques like RMSprop and Adam, specifically emphasizing its role in mitigating issues like mode collapse during GAN training processes. The empirical results, including inception scores and Fréchet distance metrics, underscore OMD's ability to consistently outperform its non-optimistic counterparts in practical applications.

Conclusion and Impact

The exploration of OMD within this paper contributes significantly to the theoretical understanding and practical application of first-order methods in non-convex and non-monotone optimization landscapes. By providing a concrete analytical framework and empirical evidence, the authors effectively address the stability challenges faced in adversarial machine learning frameworks, particularly GANs. This work not only reinforces the valuable role of optimism in gradient-based methods but also opens new avenues for further refinement and adaptation of the extra-gradient approach to complex, high-dimensional optimization problems. Future research could extend this framework to explore broader applications within artificial intelligence and operations research, potentially leading to new algorithmic paradigms in addressing the intricacies of modern machine learning problems.