[Paper Review] A Single-Loop Smoothed Gradient Descent-Ascent Algorithm for Nonconvex-Concave Min-Max Problems

Jiawei Zhang, Peijun Xiao+2 more|arXiv (Cornell University)|Oct 29, 2020

Sparse and Compressive Sensing TechniquesEngineering37 references17 citations

TL;DR

This paper proposes a single-loop smoothed gradient descent-ascent (GDA) algorithm for nonconvex-concave min-max problems, stabilizing oscillations through a smoothing scheme. It achieves an optimal $O(1/ heta^2)$ iteration complexity for minimizing the pointwise maximum of nonconvex functions and $O(1/ heta^4)$ for general nonconvex-concave problems, outperforming prior multi-loop methods.

ABSTRACT

Nonconvex-concave min-max problem arises in many machine learning applications including minimizing a pointwise maximum of a set of nonconvex functions and robust adversarial training of neural networks. A popular approach to solve this problem is the gradient descent-ascent (GDA) algorithm which unfortunately can exhibit oscillation in case of nonconvexity. In this paper, we introduce a "smoothing" scheme which can be combined with GDA to stabilize the oscillation and ensure convergence to a stationary solution. We prove that the stabilized GDA algorithm can achieve an $O(1/ε^2)$ iteration complexity for minimizing the pointwise maximum of a finite collection of nonconvex functions. Moreover, the smoothed GDA algorithm achieves an $O(1/ε^4)$ iteration complexity for general nonconvex-concave problems. Extensions of this stabilized GDA algorithm to multi-block cases are presented. To the best of our knowledge, this is the first algorithm to achieve $O(1/ε^2)$ for a class of nonconvex-concave problem. We illustrate the practical efficiency of the stabilized GDA algorithm on robust training.

Motivation & Objective

• To address oscillation issues in gradient descent-ascent (GDA) for nonconvex-concave min-max problems.
• To achieve optimal iteration complexity $O(1/\epsilon^2)$ for minimizing the pointwise maximum of nonconvex functions.
• To develop a single-loop, multi-block algorithm suitable for large-scale and distributed learning.
• To improve convergence and stability in robust adversarial training of neural networks.

Proposed method

• Introduces a smoothing scheme to stabilize GDA by regularizing the update direction.
• Applies a single-loop structure where each iteration uses a closed-form update, avoiding inner loops.
• Uses a probabilistic simplex formulation to convert finite max problems into concave min-max problems.
• Employs gradient ascent over the probability simplex to approximate the inner maximization in adversarial training.
• Applies projection to the simplex using an $O(d\log d)$ algorithm for efficient computation.
• Extends the method to multi-block settings for distributed optimization.

Experimental results

Research questions

• RQ1 Can a single-loop GDA algorithm achieve $O(1/\epsilon^2)$ iteration complexity for nonconvex-concave min-max problems?
• RQ2 Can smoothing stabilize GDA and prevent oscillations in nonconvex settings?
• RQ3 Can the algorithm be extended to multi-block and distributed optimization scenarios?
• RQ4 Does the smoothed GDA outperform existing multi-loop algorithms in convergence speed and robustness?

Key findings

• The smoothed GDA achieves $O(1/\epsilon^2)$ iteration complexity for minimizing the pointwise maximum of nonconvex functions, matching the theoretical lower bound.
• For general nonconvex-concave problems, the algorithm achieves $O(1/\epsilon^4)$ iteration complexity, improving upon prior $O(1/\epsilon^{2.5})$ bounds.
• The algorithm is the first single-loop method to achieve $O(1/\epsilon^2)$ complexity for this class of problems.
• Experiments on MNIST and CIFAR10 show faster convergence and competitive robust accuracy (38.5%) compared to prior work.
• The method outperforms the algorithm in [20] in convergence speed on CIFAR10 after 30 epochs.
• The smoothed GDA is effective for robust adversarial training, achieving state-of-the-art performance with minimal hyperparameter tuning.