The Practicality of Stochastic Optimization in Imaging Inverse Problems
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In this work we investigate the practicality of stochastic gradient descent and recently introduced variants with variance-reduction techniques in imaging inverse problems. Such algorithms have been shown in the machine learning literature to have optimal complexities in theory, and provide great improvement empirically over the deterministic gradient methods. Surprisingly, in some tasks such as image deblurring, many of such methods fail to converge faster than the accelerated deterministic gradient methods, even in terms of epoch counts. We investigate this phenomenon and propose a theory-inspired mechanism to characterize whether an inverse problem should be preferred to be solved by stochastic optimization techniques. We derive conditions on the structure of the inverse problem for being a suitable application of stochastic gradient methods, using standard tools in numerical linear algebra. Based on our analysis, we provide the practitioners convenient ways to examine whether they should use stochastic gradient methods or the classical deterministic gradient methods to solve a given inverse problem. Our results also provide guidance on choosing appropriately the partition minibatch schemes. Finally, we propose an accelerated primal-dual SGD algorithm in order to tackle another key bottleneck of stochastic optimization which is the heavy computation of proximal operators. The proposed method has fast convergence rate in practice, and is able to efficiently handle non-smooth regularization terms which are coupled with linear operators.
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