Variable smoothing for convex optimization problems using stochastic gradients
For researchers in convex optimization, this work offers a new algorithmic approach that extends smoothing techniques to stochastic settings, though it is incremental as it builds on existing methods.
The paper proposes novel algorithms for structured convex optimization by combining Moreau envelope smoothing with stochastic gradient techniques, achieving efficient solutions for large-scale problems like total variation denoising and deblurring.
We aim to solve a structured convex optimization problem, where a nonsmooth function is composed with a linear operator. When opting for full splitting schemes, usually, primal-dual type methods are employed as they are effective and also well studied. However, under the additional assumption of Lipschitz continuity of the nonsmooth function which is composed with the linear operator we can derive novel algorithms through regularization via the Moreau envelope. Furthermore, we tackle large scale problems by means of stochastic oracle calls, very similar to stochastic gradient techniques. Applications to total variational denoising and deblurring are provided.