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In this paper we consider non-smooth convex optimization problems with(possibly) infinite intersection of constraints. In contrast to the classicalapproach, where the constraints are usually represented as intersection ofsimple sets, which are easy to project onto, in this paper we consider thateach constraint set is given as the level set of a convex but not necessarilydifferentiable function. For these settings we propose subgradient iterativealgorithms with random minibatch feasibility updates. At each iteration, ouralgorithms take a step aimed at only minimizing the objective function and thena subsequent step minimizing the feasibility violation of the observedminibatch of constraints. The feasibility updates are performed based on eitherparallel or sequential random observations of several constraint components. Weanalyze the convergence behavior of the proposed algorithms for the case whenthe objective function is strongly convex and with bounded subgradients, whilethe functional constraints are endowed with a bounded first-order black-boxoracle. For a diminishing stepsize, we prove sublinear convergence rates forthe expected distances of the weighted averages of the iterates from theconstraint set, as well as for the expected suboptimality of the functionvalues along the weighted averages. Our convergence rates are known to beoptimal for subgradient methods on this class of problems. Moreover, the ratesdepend explicitly on the minibatch size and show when minibatching helps asubgradient scheme with random feasibility updates.

Random minibatch subgradient algorithms for convex problems with functional constraints