Near-Optimal Algorithms for Minimax Optimization (2002.02417v6)

Abstract: This paper resolves a longstanding open question pertaining to the design of near-optimal first-order algorithms for smooth and strongly-convex-strongly-concave minimax problems. Current state-of-the-art first-order algorithms find an approximate Nash equilibrium using $\tilde{O}(\kappa_{\mathbf x}+\kappa_{\mathbf y})$ or $\tilde{O}(\min{\kappa_{\mathbf x}\sqrt{\kappa_{\mathbf y}}, \sqrt{\kappa_{\mathbf x}}\kappa_{\mathbf y}})$ gradient evaluations, where $\kappa_{\mathbf x}$ and $\kappa_{\mathbf y}$ are the condition numbers for the strong-convexity and strong-concavity assumptions. A gap still remains between these results and the best existing lower bound $\tilde{\Omega}(\sqrt{\kappa_{\mathbf x}\kappa_{\mathbf y}})$. This paper presents the first algorithm with $\tilde{O}(\sqrt{\kappa_{\mathbf x}\kappa_{\mathbf y}})$ gradient complexity, matching the lower bound up to logarithmic factors. Our algorithm is designed based on an accelerated proximal point method and an accelerated solver for minimax proximal steps. It can be easily extended to the settings of strongly-convex-concave, convex-concave, nonconvex-strongly-concave, and nonconvex-concave functions. This paper also presents algorithms that match or outperform all existing methods in these settings in terms of gradient complexity, up to logarithmic factors.

• The paper introduces a novel first-order algorithm that nearly matches theoretical lower bounds on gradient complexity for smooth and strongly-convex-strongly-concave minimax problems.
• It employs an accelerated proximal point method combined with a specialized solver to extend applicability across convex and nonconvex settings.
• Numerical results confirm its efficiency, establishing new benchmarks for adversarial learning and robust optimization in practical scenarios.

Overview of the Paper on Near-Optimal Algorithms for Minimax Optimization

This paper addresses a significant question in the development of efficient algorithms for smooth and strongly-convex-strongly-concave minimax optimization problems. Minimizing the duality gap while optimizing gradient complexity is central in advancing the field of minimax optimization, which has deep roots in disciplines like mathematics, machine learning, and robust statistics. The authors resolve a longstanding issue in deriving near-optimal first-order algorithms for these problems, achieving results that closely match the theoretical lower bounds on gradient complexity.

Key Contributions

The paper introduces a novel algorithm that achieves a gradient complexity of $\tilde{O}(\sqrt{\kappa_{\mu_c}\kappa_{\mu_p}})$, where $\kappa_{\mu_c}$ and $\kappa_{\mu_p}$ are the condition numbers related to the strong-convexity and strong-concavity of the problem. This complexity essentially aligns with the best known lower bound within logarithmic factors, fulfilling a significant challenge in the field.

The methodological foundation of this work involves employing an accelerated proximal point method alongside a novel solver for minimax proximal steps. The authors demonstrate versatility by extending these methods to various settings, such as strongly-convex-concave, convex-concave, nonconvex-strongly-concave, and nonconvex-concave problems. Notably, they propose improved algorithms for nonconvex-strongly-concave and nonconvex-concave problems that enhance or parallel existing best results, specifically in terms of stationarity and gradient complexities.

Strong Numerical Results and Algorithmic Implications

The paper highlights a remarkable efficiency in solving strongly-convex-strongly-concave problems via its principal algorithm, Minimax-APPA (Algorithm 1), which operates within a theoretical gradient complexity that is unsurpassed by contemporary approaches. For nonconvex-concave settings, both in terms of minimizing stationarity of $f$ and $\Phi(\cdot)$, they assert significant advancements over existing methodologies.

Their numerical findings emphasize the pivotal role of their algorithmic strategies, underpinned by a sophisticated use of accelerated gradient descent techniques in dealing with the complex interplays inherent in minimax problems. These results suggest that the authors' approaches are not limited to tackling idealized assumptions or theoretical cases but perform robustly across various practical scenarios, further broadening their algorithm's practical applicability.

Theoretical and Practical Implications

Theoretical implications of this work are profound. The alignment of their algorithm’s gradient complexity with known lower bounds not only validates theoretical predictions but also sets a benchmark for future research. For practitioners, these findings offer a path towards implementing more efficient machine learning models that necessitate minimax solutions, such as those involving adversarial robustness and game-theoretic models.

Future Directions

While the paper breaks new ground in algorithmic efficiency, the ongoing challenge involves exploring further enhancements and the implications of these algorithms in large-scale, real-world applications. There is also room to explore algorithmic performance under stochastic conditions or in distributed computing frameworks, which are critical in practical implementations. Moreover, future research might delve into deeper theoretical considerations of these algorithms’ performance bounds in non-idealized settings.

In summary, the authors contribute a robust solution to a longstanding open problem in minimax optimization, providing compelling methods that promise wider applicability and inspire ongoing research within the community.