Abstract
We analyze several generic proximal splitting algorithms well suited for large-scale convex nonsmooth optimization. We derive sublinear and linear convergence results with new rates on the function value suboptimality or distance to the solution, as well as new accelerated versions, using varying stepsizes. In addition, we propose distributed variants of these algorithms, which can be accelerated as well. While most existing results are ergodic, our nonergodic results significantly broaden our understanding of primal–dual optimization algorithms.
Highlights
We propose new algorithms for the generic convex optimization problem:
Where M ≥ 1 is typically the number of parallel computing nodes in a distributed setting; the Km: X → Um are linear operators; X and Um are real Hilbert spaces; R and Hm are proper, closed, convex functions with values in R ∪ {+∞}, the proximity operators of which are easy to compute; and the Fm are convex LFm-smooth functions; that is ∇Fm is LFm-Lipschitz continuous, for some LFm > 0. This template problem covers most convex optimization problems met in signal and image processing, operations research, control, machine learning, and many other fields, and our goal is to propose new generic distributed algorithms able to deal with nonsmooth functions using their proximity operators, with acceleration in presence of strong convexity
Our contributions are the following: (1) New algorithms: We propose the first distributed algorithms to solve (Eq 1) in whole generality, with proved convergence to an exact solution, and having the full splitting, or decoupling, property: ∇Fm, proxHm, Km and K*m are applied at the m-th node, and the proximity operator of R
Summary
We propose new algorithms for the generic convex optimization problem:. :. Where M ≥ 1 is typically the number of parallel computing nodes in a distributed setting; the Km: X → Um are linear operators; X and Um are real Hilbert spaces (all spaces are supposed of finite dimension); R and Hm are proper, closed, convex functions with values in R ∪ {+∞}, the proximity operators of which are easy to compute; and the Fm are convex LFm-smooth functions; that is ∇Fm is LFm-Lipschitz continuous, for some LFm > 0 This template problem covers most convex optimization problems met in signal and image processing, operations research, control, machine learning, and many other fields, and our goal is to propose new generic distributed algorithms able to deal with nonsmooth functions using their proximity operators, with acceleration in presence of strong convexity
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