Second-Order Kernel Online Convex Optimization with Adaptive Sketching

Kernel online convex optimization (KOCO) is a framework combining the expressiveness of nonparametric kernel models with the regret guarantees of online learning. First-order KOCO methods such as functional gradient descent require onlyO(t) time and space per iteration, and, when the only information on the losses is their convexity, achieve a minimax optimal O( √ T ) regret. Nonetheless, many common losses in kernel problems, such as squared loss, logistic loss, and squared hinge loss posses stronger curvature that can be exploited. In this case, second-order KOCO methods achieveO(log(Det(K))) regret, which we show scales as O(deff log T ), where deff is the effective dimension of the problem and is usually much smaller than O( √ T ). The main drawback of second-order methods is their much higher O(t) space and time complexity. In this paper, we introduce kernel online Newton step (KONS), a new second-order KOCO method that also achievesO(deff log T ) regret. To address the computational complexity of second-order methods, we introduce a new matrix sketching algorithm for the kernel matrix Kt, and show that for a chosen parameter γ ≤ 1 our Sketched-KONS reduces the space and time complexity by a factor of γ toO(tγ) space and time per iteration, while incurring only 1/γ times more regret.