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In this work, we address optimization problems where the objective functionis a nonlinear function of an expected value, i.e., compositional stochastic{strongly convex programs}. We consider the case where the decision variable isnot vector-valued but instead belongs to a reproducing Kernel Hilbert Space(RKHS), motivated by risk-aware formulations of supervised learning and MarkovDecision Processes defined over continuous spaces.  We develop the first memory-efficient stochastic algorithm for this setting,which we call Compositional Online Learning with Kernels (COLK). COLK, at itscore a two-time-scale stochastic approximation method, addresses the fact that(i) compositions of expected value problems cannot be addressed by classicalstochastic gradient due to the presence of the inner expectation; and (ii) theRKHS-induced parameterization has complexity which is proportional to theiteration index which is mitigated through greedily constructed subspaceprojections. We establish almost sure convergence of COLK with attenuatingstep-sizes, and linear convergence in mean to a neighborhood with constantstep-sizes, as well as the fact that its complexity is at-worst finite. Theexperiments with robust formulations of supervised learning demonstrate thatCOLK reliably converges, attains consistent performance across training runs,and thus overcomes overfitting.

Nonparametric Compositional Stochastic Optimization for Risk-Sensitive Kernel Learning