The Inexact Cyclic Block Proximal Gradient Method and Properties of Inexact Proximal Maps

This paper expands the cyclic block proximal gradient method for block separable composite minimization by allowing for inexactly computed gradients and pre-conditioned proximal maps. The resultant algorithm, the inexact cyclic block proximal gradient (I-CBPG) method, shares the same convergence rat...

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Veröffentlicht in:Journal of optimization theory and applications 2024-05, Vol.201 (2), p.668-698
Hauptverfasser: Farias Maia, Leandro, Gutman, David Huckleberry, Hughes, Ryan Christopher
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Hughes, Ryan Christopher
description This paper expands the cyclic block proximal gradient method for block separable composite minimization by allowing for inexactly computed gradients and pre-conditioned proximal maps. The resultant algorithm, the inexact cyclic block proximal gradient (I-CBPG) method, shares the same convergence rate as its exactly computed analogue provided the allowable errors decrease sufficiently quickly or are pre-selected to be sufficiently small. We provide numerical experiments that showcase the practical computational advantage of I-CBPG for certain fixed tolerances of approximation error and for a dynamically decreasing error tolerance regime in particular. Our experimental results indicate that cyclic methods with dynamically decreasing error tolerance regimes can actually outpace their randomized siblings with fixed error tolerance regimes. We establish a tight relationship between inexact pre-conditioned proximal map evaluations and δ -subgradients in our ( δ , B ) -Second Prox theorem. This theorem forms the foundation of our convergence analysis and enables us to show that inexact gradient computations can be subsumed within a single unifying framework.
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subjects Algorithms
Applications of Mathematics
Approximation
Calculus of Variations and Optimal Control
Optimization
Computation
Convergence
Descent
Engineering
Experiments
Gradients
Mathematics
Mathematics and Statistics
Methods
Operations Research/Decision Theory
Optimization
Theorems
Theory of Computation
title The Inexact Cyclic Block Proximal Gradient Method and Properties of Inexact Proximal Maps
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