Randomized Block-Coordinate Optimistic Gradient Algorithms for Root-Finding Problems

In this paper, we develop two new randomized block-coordinate optimistic gradient algorithms to approximate a solution of nonlinear equations, which are called root-finding problems. Our first algorithm is non-accelerated with constant stepsizes, and achieves $\mathcal{O}(1/k)$ best-iterate convergence rate on $\mathbb{E}[ \Vert Gx^k\Vert^2]$ when the underlying operator $G$ is Lipschitz continuous and the equation $Gx = 0$ admits a weak Minty solution, where $\mathbb{E}[\cdot]$ is the expectation and $k$ is the iteration counter. Our second method is a new accelerated randomized block-coordinate optimistic gradient algorithm. We establish both $\mathcal{O}(1/k^2)$ and $o(1/k^2)$ last-iterate convergence rates on both $\mathbb{E}[ \Vert Gx^k\Vert^2]$ and $\mathbb{E}[ \Vert x^{k+1} - x^{k}\Vert^2]$ for this algorithm under the co-coerciveness of $G$. Then, we apply our methods to a class of finite-sum nonlinear inclusions which covers various applications in machine learning and statistical learning, especially in federated learning and network optimization. We obtain two new federated learning-type algorithms for this problem class with rigorous convergence rate guarantees.