Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methods (1902.08297v3)

Abstract: Recent applications that arise in machine learning have surged significant interest in solving min-max saddle point games. This problem has been extensively studied in the convex-concave regime for which a global equilibrium solution can be computed efficiently. In this paper, we study the problem in the non-convex regime and show that an \varepsilon--first order stationary point of the game can be computed when one of the player's objective can be optimized to global optimality efficiently. In particular, we first consider the case where the objective of one of the players satisfies the Polyak-{\L}ojasiewicz (PL) condition. For such a game, we show that a simple multi-step gradient descent-ascent algorithm finds an \varepsilon--first order stationary point of the problem in \widetilde{\mathcal{O}}(\varepsilon^{-2}) iterations. Then we show that our framework can also be applied to the case where the objective of the "max-player" is concave. In this case, we propose a multi-step gradient descent-ascent algorithm that finds an \varepsilon--first order stationary point of the game in \widetilde{\cal O}(\varepsilon^{-3.5}) iterations, which is the best known rate in the literature. We applied our algorithm to a fair classification problem of Fashion-MNIST dataset and observed that the proposed algorithm results in smoother training and better generalization.

• The paper introduces iterative gradient descent-ascent algorithms to compute ε-first order Nash equilibria under the PL condition with O(ε⁻²) complexity.
• It extends to non-convex concave settings using accelerated projected gradient ascent, achieving a best-known convergence rate of O(ε⁻³.5).
• The study validates its approach on a fair classification task with Fashion-MNIST, underscoring its practical and theoretical impact on machine learning optimization.

Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methods

The paper investigates iterative first-order methods to solve non-convex min-max saddle-point problems in the emerging discipline of machine learning. While the convex-concave regime of this problem has been thoroughly examined, solving non-convex regimes poses significant challenges due to the lack of guaranteed existence of local Nash equilibria and computational complexity. The authors extend the analysis to non-convex settings where one player’s objective can be optimized to global optimality efficiently, particularly leveraging the Polyak-Łojasiewicz (PL) condition.

Problem Setting and Challenges

The primary focus is on the typical min-max problem expressed as:

$$\min_{\theta \in \Theta} \max_{\alpha \in A} \, f(\theta, \alpha)$$

where $\Theta$ and $A$ are convex sets and $f$ is non-convex in $\theta$. The challenges in dealing with such problems include the absence of guarantees for local Nash equilibria in non-convex settings and the inefficiency of gradient descent-ascent algorithms in reaching equilibrium solutions. Thus, the paper aims for the computation of first-order Nash equilibria – a more reachable goal for non-convex games.

Methodological Contributions

The authors propose two noteworthy iterative algorithms to find $\varepsilon$-first order stationary points (or Nash equilibria):

1. Multi-step Gradient Descent-Ascent for PL Games:
   • The paper first addresses scenarios where one player’s objective function adheres to the PL condition, reinforcing the stability and smoothness of the optimization landscape.
   • It introduces a multi-step gradient descent-ascent algorithm, demonstrating convergence to an $\varepsilon$-approximate solution with a complexity of $\widetilde{\mathcal{O}}(\varepsilon^{-2})$ iterations.
   • The optimality of this complexity is highlighted in relation to similar non-convex optimization problems.
2. Non-Convex Concave Setting:
   • For situations where the max-player’s objective is concave, the authors propose an alternative framework that effectively addresses the convergence issues identified in previous studies.
   • Utilizing accelerated projected gradient ascent, the paper’s algorithm achieves the best-known rate complexity of $\widetilde{\mathcal{O}}(\varepsilon^{-3.5})$.

Practical Implications

In terms of applications, the paper substantiates the algorithm's practicality using a fair classification problem on the Fashion-MNIST dataset. This case paper demonstrates improved generalization and training stability, emphasizing the relevance of the proposed solutions in real-world scenarios involving fairness concerns in machine learning models.

Theoretical Implications and Future Directions

The theoretical groundwork laid by the paper informs future advancements in AI and optimization. The paper offers a foundation for addressing non-convex non-concave challenges, which could expand the versatility of adversarial learning models, generative adversarial networks (GANs), and other fields reliant on robust optimization.

Future developments could explore adaptive methods to dynamically adjust algorithmic parameters in response to problem-specific characteristics, potentially extending these approaches to broader classes of machine learning problems. Exploring the integration of stochastic elements to account for real-world data variability would also be a valuable avenue for research.

Conclusion

This paper contributes a significant advancement to understanding and solving non-convex min-max optimization problems by harnessing the properties of the PL condition and innovative iterative methods. The empirical and theoretical findings emphasize the potential of these algorithms to meet the demands of complex machine learning applications where traditional methods fall short. The work extends the frontiers of optimization in AI, presenting a pathway for future explorations in this domain.