  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 $\mathcal{O}(t)$ time and space per iteration, and, when the only
information on the losses is their convexity, achieve a minimax optimal
$\mathcal{O}(\sqrt{T})$ 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 $\mathcal{O}(\log(\text{Det}(\boldsymbol{K})))$ regret, which
we show scales as $\mathcal{O}(d_{\text{eff}}\log T)$, where $d_{\text{eff}}$
is the effective dimension of the problem and is usually much smaller than
$\mathcal{O}(\sqrt{T})$. The main drawback of second-order methods is their
much higher $\mathcal{O}(t^2)$ space and time complexity. In this paper, we
introduce kernel online Newton step (KONS), a new second-order KOCO method that
also achieves $\mathcal{O}(d_{\text{eff}}\log T)$ regret. To address the
computational complexity of second-order methods, we introduce a new matrix
sketching algorithm for the kernel matrix $\boldsymbol{K}_t$, and show that for
a chosen parameter $\gamma \leq 1$ our Sketched-KONS reduces the space and time
complexity by a factor of $\gamma^2$ to $\mathcal{O}(t^2\gamma^2)$ space and
time per iteration, while incurring only $1/\gamma$ times more regret.
