Graph-based Adversarial Online Kernel Learning with Adaptive Embedding

Traditional online learning for vertex classification adapts graph Laplacian regularization into ridge regression, which hardly resolve robustness issue against adversarial examples. To tackle the problem, we propose a more general min-max optimization framework for adversarial online kernel learning (OKL). The derived online algorithm can achieve a min-max regret compared with the optimal model found offline. Nonetheless, optimizing in the reproducing kernel Hilbert space suffers from expensive computational costs. While first-order methods accumulate an optimal $\mathcal{O}(\sqrt{T})$ regret, they only require $\mathcal{O}(t)$ time and space per trial. Second-order methods converge to an optimum much faster with $\mathcal{O}(\log T)$, but suffer an expensive $\mathcal{O}(t^{2})$ per-trial cost. This paper adopts selective sampling to make the OKL scaling to large datasets through constructing a small and accurate embedded space for vertex representation, so that OKL could be performed more efficiently on this sketched kernel space. To achieve this goal, we introduce a novel confidence-aware dictionary selection strategy and a model-based update routine, where for an expected sampling probability p, the computational cost can be reduced by a factor of $p^{2}$ to $\mathcal{O}(p^{2}t^{2})$ space and time per trial, while the regret remains comparable. Numerical experiments are conducted on real-world benchmark datasets to illustrate the efficacy of our method.