Accelerated and Inexact Forward-Backward Algorithms

Abstract

Abstract. We propose a convergence analysis of accelerated forward-backward splitting methods for composite function minimization, when the proximity operator is not available in closed form, and can only be computed up to a certain precision. We prove that the 1/k 2 convergence rate for the function values can be achieved if the admissible errors are of a certain type and satisfy a sufficiently fast decay condition. Our analysis is based on the machinery of estimate sequences first introduced by Nesterov for the study of accelerated gradient descent algorithms. Furthermore, we give a global complexity analysis, taking into account the cost of computing admissible approximations of the proximal point. An experimental analysis is also presented.

Accelerated Alternating Descent Methods for Dykstra-Like Problems

“…A natural development would be to analyze better the behaviour of the inexact scheme, which in practice seems to be quite efficient in our application. The correct framework for this analysis should probably be the framework of inexact accelerated schemes, as studied in [21,1], however for this we would need to better estimate the errors which are introduced by the method (AAMM-inexact) and which seem much smaller than one could naturally expect.…”

Robust Principal Component Analysis Based on Low-Rank and Block-Sparse Matrix Decomposition

“…There has been recent work on this (see, e.g., [44,78,85]) but often under the assumption that the computed point is feasible, i.e., it is inside dom f . There has been recent work on this (see, e.g., [44,78,85]) but often under the assumption that the computed point is feasible, i.e., it is inside dom f .…”

Structured Sparsity Models for Brain Decoding from fMRI Data

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“…The efficacy of accelerated proximal methods when the proximity operator is computed numerically has been studied in [17], [18]. We use the weaker requirements in [17] on the decay of the errors.…”