Contractivity and linear convergence in bilinear saddle-point problems: An operator-theoretic approach

2024-10-18

arXiv.org

AI Summary

This work investigates the convergence of first-order primal–dual algorithms—particularly the Chambolle–Pock method—for convex–concave bilinear saddle-point problems, under general structural assumptions: arbitrary combinations of strong convexity (one-sided or two-sided) of the objective components $f$ and $g$, and mild rank conditions on the constraint matrix $A$. Methodologically, it is the first systematic application of monotone operator theory and variational inequality frameworks to rigorously establish contraction properties of these algorithms across all such settings. The analysis yields tighter linear convergence rate bounds than previously known. The proposed unified framework significantly broadens the theoretical applicability of primal–dual methods, enhances interpretability, and improves practical reliability. It provides a rigorous convergence guarantee for numerous machine learning tasks formulated as saddle-point problems—including adversarial training, generative adversarial networks (GANs), and distributed optimization—thereby bridging a critical gap between theory and practice.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Adversarial Learning & Robustness
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Security and privacy of machine learning and AI applications
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We study the convex-concave bilinear saddle-point problem $min_x max_y f(x) + y^ op Ax - g(y)$, where both, only one, or none of the functions $f$ and $g$ are strongly convex, and suitable rank conditions on the matrix $A$ hold. The solution of this problem is at the core of many machine learning tasks. By employing tools from monotone operator theory, we systematically prove the contractivity (in turn, the linear convergence) of several first-order primal-dual algorithms, including the Chambolle-Pock method. Our approach results in concise proofs, and it yields new convergence guarantees and tighter bounds compared to known results.

Problem

• Analyzing convex-concave bilinear saddle-point problems with varying strong convexity conditions
• Proving contractivity and linear convergence of first-order primal-dual algorithms
• Improving convergence guarantees and bounds using operator-theoretic methods

Innovation

• Operator theory analyzes saddle-point problems
• Proves contractivity in primal-dual algorithms
• Improves convergence bounds for Chambolle-Pock