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Proximal gradient methods have been found to be highly effective for solvingminimization problems with non-negative constraints or L1-regularization. Undersuitable nondegeneracy conditions, it is known that these algorithms identifythe optimal sparsity pattern for these types of problems in a finite number ofiterations. However, it is not known how many iterations this may take. Weintroduce the notion of the "active-set complexity", which in these cases isthe number of iterations before an algorithm is guaranteed to have identifiedthe final sparsity pattern. We further give a bound on the active-setcomplexity of proximal gradient methods in the common case of minimizing thesum of a strongly-convex smooth function and a separable convex non-smoothfunction.

"Active-set complexity" of proximal gradient: How long does it take to find the sparsity pattern?