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 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 $$\delta $$ δ -subgradients in our $$(\delta ,B)$$ ( 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.