Stochastic approximation techniques play a prominent role in solving many large scale problems encountered in machine learning or image/signal processing. In these contexts, the statistics of the data are often unknown a priori or their direct computation is too intensive, and they have thus to be estimated online from the observations. For batch optimization of an objective function being the sum of a data fidelity term and a penalization (e.g. a sparsity promoting function), Majorize-Minimize (MM) methods have recently attracted much interest since they are fast, highly flexible, and effective in ensuring convergence. The goal of this talk is to show how these methods can be successfully extended to the case when the data fidelity term corresponds to a least squares criterion and the cost function is replaced by a sequence of stochastic approximations of it. In this context, we propose an online version of an MM subspace algorithm and we establish its convergence by using suitable probabilistic tools. We also provide new results on the convergence rate of such kind of algorithm. Numerical results illustrate the good practical performance of the proposed algorithm associated with a memory gradient subspace, when applied to both non-adaptive and adaptive linear system identification scenarios.
E. Chouzenoux and J.-C. Pesquet. A Stochastic Majorize-Minimize Subspace Algorithm for Online Penalized Least Squares Estimation. IEEE Transactions on Signal Processing, Vol. 65, No. 18, pages 4770-4783, 2017.
E. Chouzenoux and J.-C. Pesquet. Convergence Rate Analysis of the Majorize-Minimize Subspace Algorithm. IEEE Signal Processing Letters, Vol. 23, No. 9, pages 1284-1288, Septembre 2016.