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. Then, we develop a primal-dual method which that executes alternating projected primal/dual stochastic gradient descent/ascent on the dual-augmented Lagrangian of the 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 in both primal objective sub-optimality and constraint violation, to respective <inline-formula><tex-math notation="LaTeX">${\mathcal O}(\sqrt{T})$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">${\mathcal O}(T^{3/4})$</tex-math></inline-formula> neighborhoods. Here, <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula> is the final iteration index and the constant step-size is chosen as <inline-formula><tex-math notation="LaTeX">$1/\sqrt{T}$</tex-math></inline-formula> with <inline-formula><tex-math notation="LaTeX">$1/T$</tex-math></inline-formula> approximation budget. Finally, we demonstrate experimentally the effectiveness of the proposed method for risk-aware supervised learning.