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This paper introduces the stochastic Fej\'{e}r-monotone hybrid steepestdescent method (S-FM-HSDM) to solve affinely constrained and composite convexminimization tasks. The minimization task is not known exactly; noisecontaminates the information about the composite loss function and the affineconstraints. S-FM-HSDM generates sequences of random variables that, undercertain conditions and with respect to a probability space, converge point-wiseto solutions of the noiseless minimization task. S-FM-HSDM enjoys desirableattributes of optimization techniques such as splitting of variables andconstant step size (learning rate). Furthermore, it provides a novel way ofexploiting the information about the affine constraints via fixed-point sets ofappropriate nonexpansive mappings. Among the offsprings of S-FM-HSDM, thehierarchical recursive least squares (HRLS) takes advantage of S-FM-HSDM'sversatility toward affine constraints and offers a novel twist to LS bygenerating sequences of estimates that converge to solutions of a hierarchicaloptimization task: Minimize a convex loss over the set of minimizers of theensemble LS loss. Numerical tests on a sparsity-aware LS task show that HRLScompares favorably to several state-of-the-art convex, as well as non-convex,stochastic-approximation and online-learning counterparts.

The Stochastic Fej\'er-Monotone Hybrid Steepest Descent Method and the Hierarchical RLS