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Online Lewis Weight Sampling

The seminal work of Cohen and Peng introduced Lewis weight sampling to the theoretical computer science community, yielding fast row sampling algorithms for approximating $d$-dimensional subspaces of $\ell_p$ up to $(1+\epsilon)$ error. Several works have extended this important primitive to other settings, including the online coreset and sliding window models. However, these results are only for $p\in\{1,2\}$, and results for $p=1$ require a suboptimal $\tilde O(d^2/\epsilon^2)$ samples. In this work, we design the first nearly optimal $\ell_p$ subspace embeddings for all $p\in(0,\infty)$ in the online coreset and sliding window models. In both models, our algorithms store $\tilde O(d^{1\lor(p/2)}/\epsilon^2)$ rows. This answers a substantial generalization of the main open question of [BDMMUWZ2020], and gives the first results for all $p\notin\{1,2\}$. Towards our result, we give the first analysis of "one-shot'' Lewis weight sampling of sampling rows proportionally to their Lewis weights, with sample complexity $\tilde O(d^{p/2}/\epsilon^2)$ for $p>2$. Previously, this scheme was only known to have sample complexity $\tilde O(d^{p/2}/\epsilon^5)$, whereas $\tilde O(d^{p/2}/\epsilon^2)$ is known if a more sophisticated recursive sampling is used. The recursive sampling cannot be implemented online, thus necessitating an analysis of one-shot Lewis weight sampling. Our analysis uses a novel connection to online numerical linear algebra. As an application, we obtain the first one-pass streaming coreset algorithms for $(1+\epsilon)$ approximation of important generalized linear models, such as logistic regression and $p$-probit regression. Our upper bounds are parameterized by a complexity parameter $\mu$ introduced by [MSSW2018], and we show the first lower bounds showing that a linear dependence on $\mu$ is necessary.

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