The paper considers the problem of approximating solutions of large-scale consistent linear systems Ax = b, where the full matrix A is not accessible simultaneously. The authors aim to find the unique solution characterized by a general strongly convex function f(x), subject to the linear constraints Ax = b.
The key highlights and insights are:
The authors propose a block (accelerated) randomized Bregman-Kaczmarz method that only uses a block of constraints in each iteration to tackle this problem.
They consider a dual formulation of the problem to deal efficiently with the linear constraints. Using convex tools, they show that the dual function satisfies the Polyak-Lojasiewicz (PL) property, provided that the primal objective function is strongly convex and satisfies some mild assumptions.
The authors transfer the algorithm to the primal space, which combined with the PL property, allows them to obtain linear convergence rates for their proposed method.
The authors provide a theoretical analysis of the convergence of their proposed method under different assumptions on the objective function and demonstrate its superior efficiency and speed compared to existing methods for the same problem.
The authors also propose a restart scheme for their accelerated method that exhibits faster convergence than the standard counterpart.
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Önemli Bilgiler Şuradan Elde Edildi
by Lionel Tondj... : arxiv.org 04-04-2024
https://arxiv.org/pdf/2310.17338.pdfDaha Derin Sorular