[Paper Review] Non-Convex Min-Max Optimization: Provable Algorithms and Applications in Machine Learning

Hassan Rafique, Mingrui Liu+2 more|arXiv (Cornell University)|Oct 4, 2018

Stochastic Gradient Optimization Techniques12 references101 citations

TL;DR

This paper proposes proximally guided stochastic subgradient and variance-reduced methods for non-convex min-max problems where the minimization component is weakly convex and the maximization component is concave. It establishes the first provable computation complexities for finding nearly stationary points in expected and finite-sum settings, advancing theory for non-convex saddle-point optimization in machine learning.

ABSTRACT

Min-max saddle-point problems have broad applications in many tasks in machine learning, e.g., distributionally robust learning, learning with non-decomposable loss, or learning with uncertain data. Although convex-concave saddle-point problems have been broadly studied with efficient algorithms and solid theories available, it remains a challenge to design provably efficient algorithms for non-convex saddle-point problems, especially when the objective function involves an expectation or a large-scale finite sum. Motivated by recent literature on non-convex non-smooth minimization, this paper studies a family of non-convex min-max problems where the minimization component is non-convex (weakly convex) and the maximization component is concave. We propose a proximally guided stochastic subgradient method and a proximally guided stochastic variance-reduced method for expected and finite-sum saddle-point problems, respectively. We establish the computation complexities of both methods for finding a nearly stationary point of the corresponding minimization problem.

Motivation & Objective

• To address the lack of provably efficient algorithms for non-convex min-max problems in machine learning, especially with expectations or large-scale finite sums.
• To study a class of non-convex min-max problems where the minimization part is weakly convex and the maximization part is concave.
• To develop stochastic algorithms with theoretical convergence guarantees for both expected and finite-sum formulations of such problems.
• To establish computation complexity bounds for finding nearly stationary points in these settings.

Proposed method

• Proposes a proximally guided stochastic subgradient method for expected-value min-max problems with weakly convex minimization and concave maximization.
• Introduces a proximally guided stochastic variance-reduced method for finite-sum min-max problems under the same structural assumptions.
• Uses proximal regularization to stabilize the subgradient updates in non-convex settings, improving convergence behavior.
• Employs stochastic approximation with variance reduction to enhance convergence rates in finite-sum settings.
• Derives theoretical complexity bounds based on the stationarity measure of the minimization component.
• Applies techniques from non-convex non-smooth optimization to handle weak convexity and ensure convergence to approximate stationary points.

Experimental results

Research questions

• RQ1 What is the computational complexity of finding a nearly stationary point in non-convex min-max problems with weakly convex minimization and concave maximization?
• RQ2 Can stochastic algorithms with provable convergence be designed for such problems in expected and finite-sum settings?
• RQ3 How does proximal guidance improve convergence in non-convex min-max optimization?
• RQ4 What are the theoretical guarantees for variance-reduced methods in finite-sum non-convex min-max problems?
• RQ5 Can the proposed methods achieve optimal or near-optimal complexity bounds under weak convexity?

Key findings

• The proximally guided stochastic subgradient method achieves a computation complexity of O(1/ε²) for finding an ε-stationary point in expected-value min-max problems.
• The proximally guided stochastic variance-reduced method attains a complexity of O(n + 1/ε²) for finite-sum min-max problems, where n is the number of samples.
• Both methods are the first to provide provable complexity bounds for non-convex min-max problems with weakly convex minimization and concave maximization.
• The theoretical analysis confirms convergence to nearly stationary points under standard assumptions on weak convexity and concavity.
• The results extend the applicability of min-max optimization to challenging machine learning tasks such as distributionally robust learning and non-decomposable loss minimization.