Projected Stochastic Primal-Dual Method for Constrained Online Learning with Kernels

We consider the problem of stochastic optimization with nonlinear constraints, where the decision variable is not vector-valued but instead a function belonging to a reproducing Kernel Hilbert Space (RKHS). Currently, there exist solutions to only special cases of this problem. To solve this constrained problem with kernels, we first generalize the Representer Theorem to a class of saddle-point problems defined over RKHS. Furthermore, we develop a primal-dual method which executes alternating projected primal/dual stochastic descent/ascent on the dual-augmented Lagrangian of this problem. The primal projection sets are low-dimensional subspaces of the ambient function space which are greedily constructed using matching pursuit. By tuning the projection-induced error to the algorithm step-size, we are able to establish mean convergence both in primal objective sub-optimality and constraint violation, respectively to the \mathcal{O}(\sqrt{T}) and \mathcal{O}(T^{3/4}) neighborhoods, where T is the total number of iterations. We evaluate the proposed method through numerical tests for the application of risk-aware supervised learning.