We consider the problem of stochastic optimizationwith nonlinear constraints, where the decision variable is notvector-valued but instead a function belonging to a reproducingKernel Hilbert Space (RKHS). Currently, there exist solutions toonly special cases of this problem. To solve this constrained problem with kernels, we first generalize the Representer Theoremto a class of saddle-point problems defined over RKHS. Then,we develop a primal-dual method which executes alternatingprojected primal/dual stochastic gradient descent/ascent on thedual-augmented Lagrangian of the problem. The primal projection sets are low-dimensional subspaces of the ambient functionspace, which are greedily constructed using matching pursuit. Bytuning the projection-induced error to the algorithm step-size, weare able to establish mean convergence in both primal objectivesub-optimality and constraint violation, to respective O(√T) andO(T3/4) neighborhoods. Here T is the final iteration index andthe constant step-size is chosen as 1/√T with 1/T approximationbudget. Finally, we demonstrate experimentally the effectivenessof the proposed method for risk-aware supervised learning
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