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.

翻译：广义Lasso模型在统计学、机器学习和图像科学等众多领域中展现出卓越的通用性并被广泛应用。在求解该模型所引发优化问题的诸多方法中，鞍点方法因其高效性而备受瞩目。其中，原始-对偶混合梯度（PDHG）算法以其鲁棒性和求解效率成为广受欢迎的选择。近期，[Li and Shi, 2024a] 通过高分辨率常微分方程（ODE）和隐式欧拉格式阐明了PDHG算法的内在机制。这一洞见推动了由 [Chambolle and Pock, 2011] 最初提出的多种PDHG加速变体的发展。通过离散李雅普诺夫分析，我们证明了具有时变步长的PDHG算法以接近 $O(1/k^2)$ 的速率收敛。此外，对于 [Chambolle and Pock, 2011] 提出的特定参数设置，即 $\tau_{k+1}\sigma_k = s^2$ 且 $\theta_k = \tau_{k+1}/\tau_k \in (0, 1)$，算法可实现更快的 $O(1/k^2)$ 收敛速率。为验证这些结论，我们构建了一个新颖的离散李雅普诺夫函数。该函数以其简洁性和直观性著称，为PDHG算法在特定条件下增强的收敛特性提供了清晰而优雅的证明。最后，我们利用该离散李雅普诺夫函数，在目标函数均为强凸的情况下，确立了最优的线性收敛速率。