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This paper expands the Cyclic Block Proximal Gradient method for blockseparable composite minimization by allowing for inexactly computed gradientsand proximal maps. The resultant algorithm, the Inexact Cyclic Block ProximalGradient (I-CBPG) method, shares the same convergence rate as its exactlycomputed analogue provided the allowable errors decrease sufficiently quicklyor are pre-selected to be sufficiently small. We provide numerical experimentsthat showcase the practical computational advantage of I-CBPG for certain fixedtolerances of approximation error and for a dynamically decreasing errortolerance regime in particular. We establish a tight relationship betweeninexact proximal map evaluations and $\delta$-subgradients in our$\delta$-Second Prox Theorem. This theorem forms the foundation of ourconvergence analysis and enables us to show that inexact gradient computationsand other notions of inexact proximal map computation can be subsumed within asingle unifying framework.

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