We study linear and kernel classification in the streaming model. For linear classification, we improve upon the algorithm of (Tai, et al. 2018), which solves the $\ell_1$ point query problem on the optimal weight vector $w_* \in \mathbb{R}^d$ in sublinear space. We first give an algorithm solving the more difficult $\ell_2$ point query problem on $w_*$, also in sublinear space. We also give an algorithm which solves the $\ell_2$ heavy hitter problem on $w_*$, in sublinear space and running time. Finally, we give an algorithm which can $\textit{deterministically}$ solve the $\ell_1$ point query problem on $w_*$, with sublinear space improving upon that of (Tai, et al. 2018). For kernel classification, if $w_* \in \mathbb{R}^{d^p}$ is the optimal weight vector classifying points in the stream according to their $p^{th}$-degree polynomial kernel, then we give an algorithm solving the $\ell_2$ point query problem on $w_*$ in $\text{poly}(\frac{p \log d}{\varepsilon})$ space, and an algorithm solving the $\ell_2$ heavy hitter problem in $\text{poly}(\frac{p \log d}{\varepsilon})$ space and running time. Note that our space and running time are polynomial in $p$, making our algorithms well-suited to high-degree polynomial kernels and the Gaussian kernel (approximated by the polynomial kernel of degree $p = \Theta(\log T)$).

In differential privacy, $\textit{continual observation}$ refers to problems in which we wish to continuously release a function of a dataset that is revealed one element at a time. The challenge is to maintain a good approximation while keeping the combined output over all time steps differentially private. In the special case of $\textit{continual counting}$ we seek to approximate a sum of binary input elements. This problem has received considerable attention lately, in part due to its relevance in implementations of differentially private stochastic gradient descent. $\textit{Factorization mechanisms}$ are the leading approach to continual counting, but the best such mechanisms do not work well in $\textit{streaming}$ settings since they require space proportional to the size of the input. In this paper, we present a simple approach to approximating factorization mechanisms in low space via $\textit{binning}$, where adjacent matrix entries with similar values are changed to be identical in such a way that a matrix-vector product can be maintained in sublinear space. Our approach has provable sublinear space guarantees for a class of lower triangular matrices whose entries are monotonically decreasing away from the diagonal. We show empirically that even with very low space usage we are able to closely match, and sometimes surpass, the performance of asymptotically optimal factorization mechanisms. Recently, and independently of our work, Dvijotham et al. have also suggested an approach to implementing factorization mechanisms in a streaming setting. Their work differs from ours in several respects: It only addresses factorization into $\textit{Toeplitz}$ matrices, only considers $\textit{maximum}$ error, and uses a different technique based on rational function approximation that seems less versatile than our binning approach.

We study linear and kernel classification in the streaming model. For linear classification, we improve upon the algorithm of (Tai, et al. 2018), which solves the \ell_1 point query problem on the optimal weight vector w_* \in \mathbb{R} d in sublinear space. We first give an algorithm solving the more difficult \ell_2 point query problem on w_*, also in sublinear space. We also give an algorithm which solves the \ell_2 heavy hitter problem on w_*, in sublinear space and running time. Finally, we give an algorithm which can \textit{deterministically} solve the \ell_1 point query problem on w_*, with sublinear space improving upon that of (Tai, et al. 2018).