Contractivity and linear convergence in bilinear saddle-point problems: An operator-theoretic approach
This provides improved convergence analysis for optimization problems in machine learning, though it is incremental as it builds on existing methods with refined proofs.
The authors tackled the convex-concave bilinear saddle-point problem, common in machine learning, by proving contractivity and linear convergence for first-order primal-dual algorithms like Chambolle-Pock, yielding new guarantees and tighter bounds.
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.