用于不满足Lipschitz梯度连续性的非凸正则化有限和最小化的Bregman Finito/MISO
Bregman Finito/MISO for nonconvex regularized finite sum minimization without Lipschitz gradient continuity
AI总结：
该研究提出两种基于相对光滑性的非凸正则化有限和最小化算法，以Bregman Moreau包络为Lyapunov函数简化分析，在不同采样和凸性条件下得到收敛结果。
摘要：
我们提出两种用于非凸正则化有限和最小化的算法，将典型的Lipschitz可微性假设放宽为相对光滑性概念。第一种是Finito/MISO的Bregman扩展，在随机采样下适用于完全非凸问题，在非光滑项凸的情况下适用于本质循环采样场景。第二种是低内存变体，借鉴SVRG和SARAH的思路，同样支持完全非凸形式。我们采用Bregman Moreau包络作为Lyapunov函数，使分析过程极为简洁。在随机采样情形下，当代价函数强凸时可实现线性收敛，且对和式中的单个函数无凸性要求。对于本质循环变体和低内存变体，当代价函数满足Kurdyka-Łojasiewicz性质时，可得到全局收敛与线性收敛结果。
英文摘要：
We introduce two algorithms for nonconvex regularized finite sum minimization, where typical Lipschitz differentiability assumptions are relaxed to the notion of relative smoothness. The first one is a Bregman extension of Finito/MISO, studied for fully nonconvex problems when the sampling is randomized, or under convexity of the nonsmooth term when it is essentially cyclic. The second algorithm is a low-memory variant, in the spirit of SVRG and SARAH, that also allows for fully nonconvex formulations. Our analysis is made remarkably simple by employing a Bregman Moreau envelope as Lyapunov function. In the randomized case, linear convergence is established when the cost function is strongly convex, yet with no convexity requirements on the individual functions in the sum. For the essentially cyclic and low-memory variants, global and linear convergence results are established when the cost function satisfies the Kurdyka-Łojasiewicz property.
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