[Paper Review] Solving Weakly-Convex-Weakly-Concave Saddle-Point Problems as Weakly-Monotone Variational Inequality

Qihang Lin, Mingrui Liu+2 more|arXiv (Cornell University)|Oct 24, 2018

Sparse and Compressive Sensing Techniques28 citations

TL;DR

This paper proposes a novel algorithmic framework for solving weakly convex-weakly concave min-max problems—common in GAN training—by reformulating them as weakly monotone variational inequalities. By iteratively solving strongly monotone subproblems via an inexact proximal point approach, the method achieves first-order convergence to a nearly stationary solution with provable convergence rates.

ABSTRACT

In this paper, we consider first-order convergence theory and algorithms for solving a class of non-convex non-concave min-max saddle-point problems, whose objective function is weakly convex in the variables of minimization and weakly concave in the variables of maximization. It has many important applications in machine learning including training Generative Adversarial Nets (GANs). We propose an algorithmic framework motivated by the inexact proximal point method, where the weakly monotone variational inequality (VI) corresponding to the original min-max problem is solved through approximately solving a sequence of strongly monotone VIs constructed by adding a strongly monotone mapping to the original gradient mapping. We prove first-order convergence to a nearly stationary solution of the original min-max problem of the generic algorithmic framework and establish different rates by employing different algorithms for solving each strongly monotone VI. Experiments verify the convergence theory and also demonstrate the effectiveness of the proposed methods on training GANs.

Motivation & Objective

• To address the challenge of solving non-convex non-concave min-max problems arising in machine learning, particularly in GAN training.
• To develop a first-order convergence theory for algorithms targeting weakly convex-weakly concave saddle-point problems.
• To establish convergence guarantees to nearly stationary solutions under a variational inequality framework.
• To design a generic algorithmic framework that can incorporate various solvers for subproblems while preserving convergence.

Proposed method

• Reformulate the original min-max problem as a weakly monotone variational inequality (VI) using the gradient mapping of the objective function.
• Construct a sequence of strongly monotone VIs by adding a strongly monotone mapping to the original gradient mapping.
• Apply an inexact proximal point method to solve the sequence of strongly monotone VIs approximately.
• Use different first-order solvers for each strongly monotone VI subproblem to achieve varying convergence rates.
• Ensure global convergence to a nearly stationary solution of the original problem through iterative refinement.
• Leverage the weak monotonicity of the original VI to derive convergence rates under appropriate assumptions.

Experimental results

Research questions

• RQ1 Can we establish first-order convergence guarantees for min-max problems with weakly convex-weakly concave structure?
• RQ2 How can we effectively handle non-convex non-concave objectives in min-max optimization using variational inequality theory?
• RQ3 What convergence rates can be achieved by different solvers when applied to the subproblems in the inexact proximal point framework?
• RQ4 Can the proposed framework be practically effective in training GANs despite the non-convexity and non-concavity?

Key findings

• The proposed algorithmic framework achieves first-order convergence to a nearly stationary solution of the original weakly convex-weakly concave min-max problem.
• Different convergence rates are established depending on the choice of solver used for the strongly monotone subproblems.
• The framework is applicable to a broad class of non-convex non-concave problems, including those arising in GAN training.
• Empirical results confirm the theoretical convergence behavior and demonstrate practical effectiveness in training GANs.
• The method provides a unified approach to solving weakly convex-weakly concave problems by transforming them into a sequence of well-conditioned subproblems.
• The inexact proximal point approach ensures robustness and convergence even when subproblems are solved approximately.