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- W4297914911 abstract "We study active sampling algorithms for linear regression, which aim to query only a few entries of a target vector $binmathbb R^n$ and output a near minimizer to $min_{xinmathbb R^d} |Ax-b|$, for a design matrix $Ainmathbb R^{n times d}$ and loss $|cdot|$. For $p$ norm regression for any $0<p<infty$, we give an algorithm based on Lewis weight sampling outputting a $(1+epsilon)$-approximate solution using just $tilde O(d/epsilon^2)$ queries to $b$ for $pin(0,1)$, $tilde{O}(d/epsilon)$ queries for $1<p<2$, and $tilde{O}(d^{p/2}/epsilon^p)$ queries for $2<p<infty$. For $0<p<2$, our bounds are optimal up to log factors, settling the query complexity for this range. For $2<p<infty$, our dependence on $d$ is optimal, while our dependence on $epsilon$ is off by at most $epsilon$, up to log factors. Our result resolves an open question of [CD21], who gave near optimal bounds for the $1$ norm, but required $d^2/epsilon^2$ samples for $ell_p$ regression with $1<p<2$, and gave no bounds for $2<p<infty$ or $0<p<1$. We also give the first total sensitivity bound of $O(d^{max{1,p/2}}log^2n)$ for loss functions of degree $p$ polynomial growth, improving a result of [TMF20]. By combining this with our techniques for $ell_p$ regression, we obtain an active regression algorithm making $tilde O(d^{1+max{1,p/2}}/mathrm{poly}(epsilon))$ queries for such loss functions, including the Tukey and Huber losses, answering another question of [CD21]. For the Huber loss, we further improve our bound to $tilde O(d^{4-2sqrt2}/mathrm{poly}(epsilon))$ samples. Our sensitivity bounds also have many applications, including Orlicz norm subspace embeddings, robust subspace approximation, and dimension reduction for smoothed $p$-norms. Finally, our active sampling results give the first sublinear time algorithms for Kronecker product regression under every $p$ norm." @default.
- W4297914911 created "2022-10-01" @default.
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- W4297914911 date "2021-11-08" @default.
- W4297914911 modified "2023-09-29" @default.
- W4297914911 title "Active Linear Regression for $ell_p$ Norms and Beyond" @default.
- W4297914911 doi "https://doi.org/10.48550/arxiv.2111.04888" @default.
- W4297914911 hasPublicationYear "2021" @default.
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