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

Colin Dirren, Mattia Bianchi, Panagiotis D. Grontas, John Lygeros, Florian Dorfler

TL;DR: We prove the linear convergence of several algorithms for solving (not strongly) convex bilinear saddle point problems, based on a systematic operator splitting approach.

Abstract: We study the convex-concave bilinear saddle-point problem $\min_x \max_y f(x) + y^\top Ax - g(y)$, 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.

Submission Number: 657