Second-Order Kernel Online Convex Optimization with Adaptive Sketching

Kernel online convex optimization (KOCO) is a framework combining the expressiveness of non-parametric kernel models with the regret guarantees of online learning. First-order KOCO methods such as functional gradient descent require only O(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 achieve O(log(Det(boldsymbolK))) regret, which we show scales as O(defflog 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(t2) space and time complexity. In this paper, we introduce kernel online Newton step (KONS), a new second-order KOCO method that also achieves O(defflog T) regret. To address the computational complexity of second-order methods, we introduce a new matrix sketching algorithm for the kernel matrix boldsymbolKt, and show that for a chosen parameter γ ≤ 1 our Sketched-KONS reduces the space and time complexity by a factor of γ2 to O(t2γ2) space and time per iteration, while incurring only 1/γ times more regret.