通过 $\ell_2$ 增广获得 $\ell_p$ 敏感度采样的最优界限
Optimal bounds for $\ell_p$ sensitivity sampling via $\ell_2$ augmentation
AI总结：
本文通过用 $\ell_2$ 敏感度增广 $\ell_p$ 敏感度，将 $p\in[1,2]$ 时的 $\ell_p$ 子空间嵌入采样复杂度改进至最优的线性界限，并应用于逻辑回归问题。
摘要：
数据子采样是用一个小的代表性代理来近似海量数据集的最自然方法之一。特别是，敏感度采样受到了广泛关注，它根据称为敏感度的个体重要性度量来对点进行采样。在非常一般的设置中，该框架将数据大小减少到大约 VC 维 $d$ 乘以总敏感度 $\mathfrak S$，同时对近似质量提供强 $(1\pm\varepsilon)$ 保证。Woodruff & Yasuda (2023c) 最近的工针对 $\ell_p$ 子空间嵌入这一重要问题，将一般的 $\tilde O(\varepsilon^{-2}\mathfrak Sd)$ 界限大幅改进为适用于 $p\in[1,2]$ 的 $\tilde O(\varepsilon^{-2}\mathfrak S^{2/p})$。他们的结果被 Chen & Derezinski (2021) 工作中隐含给出的更早的 $\tilde O(\varepsilon^{-2}\mathfrak Sd^{1-p/2})$ 界限所包含。我们表明，当根据普通 $\ell_p$ 敏感度进行采样时，他们的结果是紧的。我们观察到，通过用 $\ell_2$ 敏感度增广 $\ell_p$ 敏感度，我们获得了更好的界限，将上述结果改进为适用于所有 $p \in [1,2]$ 的最优线性 $\tilde O(\varepsilon^{-2}(\mathfrak S+d)) = \tilde O(\varepsilon^{-2}d)$ 采样复杂度。特别是，这对 $p \in [1,2]$ 肯定地解决了 Woodruff & Yasuda (2023c) 的一个开放问题，并将敏感度子采样带入了以前仅知使用 Lewis 权重 (Cohen & Peng, 2015) 才可能实现的范畴。作为主要结果的应用，我们还获得了逻辑回归的 $\tilde O(\varepsilon^{-2}μd)$ 敏感度采样界限，其中 $μ$ 是该问题的自然复杂度度量。这改进了 Mai et al. (2021) 先前基于 Lewis 权重子采样的 $\tilde O(\varepsilon^{-2}μ^2 d)$ 界限。
英文摘要：
Data subsampling is one of the most natural methods to approximate a massively large data set by a small representative proxy. In particular, sensitivity sampling received a lot of attention, which samples points proportional to an individual importance measure called sensitivity. This framework reduces in very general settings the size of data to roughly the VC dimension $d$ times the total sensitivity $\mathfrak S$ while providing strong $(1\pm\varepsilon)$ guarantees on the quality of approximation. The recent work of Woodruff & Yasuda (2023c) improved substantially over the general $\tilde O(\varepsilon^{-2}\mathfrak Sd)$ bound for the important problem of $\ell_p$ subspace embeddings to $\tilde O(\varepsilon^{-2}\mathfrak S^{2/p})$ for $p\in[1,2]$. Their result was subsumed by an earlier $\tilde O(\varepsilon^{-2}\mathfrak Sd^{1-p/2})$ bound which was implicitly given in the work of Chen & Derezinski (2021). We show that their result is tight when sampling according to plain $\ell_p$ sensitivities. We observe that by augmenting the $\ell_p$ sensitivities by $\ell_2$ sensitivities, we obtain better bounds improving over the aforementioned results to optimal linear $\tilde O(\varepsilon^{-2}(\mathfrak S+d)) = \tilde O(\varepsilon^{-2}d)$ sampling complexity for all $p \in [1,2]$. In particular, this resolves an open question of Woodruff & Yasuda (2023c) in the affirmative for $p \in [1,2]$ and brings sensitivity subsampling into the regime that was previously only known to be possible using Lewis weights (Cohen & Peng, 2015). As an application of our main result, we also obtain an $\tilde O(\varepsilon^{-2}μd)$ sensitivity sampling bound for logistic regression, where $μ$ is a natural complexity measure for this problem. This improves over the previous $\tilde O(\varepsilon^{-2}μ^2 d)$ bound of Mai et al. (2021) which was based on Lewis weights subsampling.
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