A Lyapunov Analysis of Accelerated PDHG Algorithms

The generalized Lasso is a remarkably versatile and extensively utilized model across a broad spectrum of domains, including statistics, machine learning, and image science. Among the optimization techniques employed to address the challenges posed by this model, saddle-point methods stand out for their effectiveness. In particular, the primal-dual hybrid gradient (PDHG) algorithm has emerged as a highly popular choice, celebrated for its robustness and efficiency in finding optimal solutions. Recently, the underlying mechanism of the PDHG algorithm has been elucidated through the high-resolution ordinary differential equation (ODE) and the implicit-Euler scheme as detailed in [Li and Shi, 2024a]. This insight has spurred the development of several accelerated variants of the PDHG algorithm, originally proposed by [Chambolle and Pock, 2011]. By employing discrete Lyapunov analysis, we establish that the PDHG algorithm with iteration-varying step sizes, converges at a rate near $O(1/k^2)$. Furthermore, for the specific setting where $\tau_{k+1}\sigma_k = s^2$ and $\theta_k = \tau_{k+1}/\tau_k \in (0, 1)$ as proposed in [Chambolle and Pock, 2011], an even faster convergence rate of $O(1/k^2)$ can be achieved. To substantiate these findings, we design a novel discrete Lyapunov function. This function is distinguished by its succinctness and straightforwardness, providing a clear and elegant proof of the enhanced convergence properties of the PDHG algorithm under the specified conditions. Finally, we utilize the discrete Lyapunov function to establish the optimal linear convergence rate when both the objective functions are strongly convex.