First-order methods with favorable convergence rates have recently become a focal point of much research in the field of convex optimization. These methods have low per-iteration complexity and hence are applicable to very large scale model, such as the ones arising in signal processing, statistics and machine learning. We will first show how these convergence properties extend to a certain class of alternating direction methods - also recently popular for large scale convex problems. All the methods in question employ prox term parameter which is often assumed to be fixed. We will discuss theoretical and practical implications of various strategies for choosing the prox parameter in prox gradient methods and related alternating direction methods. We will show extension of existing convergence rates for both accelerated and classical first-order methods. Practical comparison based on a testing environment for L1 optimization will be presented.
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