This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope (FBE), which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established when the cost function satisfies the Kurdyka-\L ojasiewicz property without imposing convexity requirements on the smooth function. Two prominent special cases of the investigated setting are regularized finite sum minimization and the sharing problem; in particular, an immediate byproduct of our analysis leads to novel convergence results and rates for the popular Finito/MISO algorithm in the nonsmooth and nonconvex setting with very general sampling strategies. This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope (FBE), which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established when the cost function satisfies the Kurdyka-\L ojasiewicz property without imposing convexity requirements on the smooth function. Two prominent special cases of the investigated setting are regularized finite sum minimization and the sharing problem; in particular, an immediate byproduct of our analysis leads to novel convergence results and rates for the popular Finito/MISO algorithm in the nonsmooth and nonconvex setting with very general sampling strategies.

中文翻译：

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非光滑非凸问题的块坐标和增量聚合近端梯度方法

本文分析了用于最小化可分离平滑函数和（不可分离）非光滑函数之和的块坐标近端梯度方法，这两个函数都允许是非凸的。我们分析中的主要工具是前向后向包络 (FBE)，它用作特别合适的连续实值李雅普诺夫函数。当成本函数满足 Kurdyka-\L ojasiewicz 性质而不对平滑函数施加凸性要求时，就建立了全局和线性收敛结果。研究设置的两个突出的特殊情况是正则化有限和最小化和共享问题；特别是，我们分析的一个直接副产品导致了流行的 Finito/MISO 算法在非光滑和非凸设置中采用非常通用的采样策略的新颖收敛结果和速率。本文分析了用于最小化可分离平滑函数和（不可分离）非光滑函数之和的块坐标近端梯度方法，这两个函数都允许是非凸的。我们分析中的主要工具是前向后向包络 (FBE)，它用作特别合适的连续实值李雅普诺夫函数。当成本函数满足 Kurdyka-\L ojasiewicz 性质而不对平滑函数施加凸性要求时，就建立了全局和线性收敛结果。研究设置的两个突出的特殊情况是正则化有限和最小化和共享问题；特别是，我们分析的一个直接副产品导致流行的 Finito/MISO 算法在非光滑和非凸设置中使用非常通用的采样策略产生新的收敛结果和速率。