We give a faster algorithm for computing an approximate John ellipsoid around $n$ points in $d$ dimensions. The best known prior algorithms are based on repeatedly computing the leverage scores of the points and reweighting them by these scores [CCLY19]. We show that this algorithm can be substantially sped up by delaying the computation of high accuracy leverage scores by using sampling, and then later computing multiple batches of high accuracy leverage scores via fast rectangular matrix multiplication. We also give low-space streaming algorithms for John ellipsoids using similar ideas.

A coreset of a dataset with $n$ examples and $d$ features is a weighted subset of examples that is sufficient for solving downstream data analytic tasks. Nearly optimal constructions of coresets for least squares and $\ell_p$ linear regression with a single response are known in prior work. However, for multiple $\ell_p$ regression where there can be $m$ responses, there are no known constructions with size sublinear in $m$. In this work, we construct coresets of size $\tilde O(\varepsilon^{-2}d)$ for $p<2$ and $\tilde O(\varepsilon^{-p}d^{p/2})$ for $p>2$ independently of $m$ (i.e., dimension-free) that approximate the multiple $\ell_p$ regression objective at every point in the domain up to $(1\pm\varepsilon)$ relative error. If we only need to preserve the minimizer subject to a subspace constraint, we improve these bounds by an $\varepsilon$ factor for all $p>1$. All of our bounds are nearly tight. We give two application of our results. First, we settle the number of uniform samples needed to approximate $\ell_p$ Euclidean power means up to a $(1+\varepsilon)$ factor, showing that $\tilde\Theta(\varepsilon^{-2})$ samples for $p = 1$, $\tilde\Theta(\varepsilon^{-1})$ samples for $1 < p < 2$, and $\tilde\Theta(\varepsilon^{1-p})$ samples for $p>2$ is tight, answering a question of Cohen-Addad, Saulpic, and Schwiegelshohn. Second, we show that for $1

Weighted low rank approximation (WLRA) is an important yet computationally challenging primitive with applications ranging from statistical analysis, model compression, and signal processing. To cope with the NP-hardness of this problem, prior work considers heuristics, bicriteria, or fixed parameter tractable algorithms to solve this problem. In this work, we introduce a new relaxed solution to WLRA which outputs a matrix that is not necessarily low rank, but can be stored using very few parameters and gives provable approximation guarantees when the weight matrix has low rank. Our central idea is to use the weight matrix itself to reweight a low rank solution, which gives an extremely simple algorithm with remarkable empirical performance in applications to model compression and on synthetic datasets. Our algorithm also gives nearly optimal communication complexity bounds for a natural distributed problem associated with this problem, for which we show matching communication lower bounds. Together, our communication complexity bounds show that the rank of the weight matrix provably parameterizes the communication complexity of WLRA. We also obtain the first relative error guarantees for feature selection with a weighted objective.

Neural network pruning is a key technique towards engineering large yet scalable, interpretable, and generalizable models. Prior work on the subject has developed largely along two orthogonal directions: (1) differentiable pruning for efficiently and accurately scoring the importance of parameters, and (2) combinatorial optimization for efficiently searching over the space of sparse models. We unite the two approaches, both theoretically and empirically, to produce a coherent framework for structured neural network pruning in which differentiable pruning guides combinatorial optimization algorithms to select the most important sparse set of parameters. Theoretically, we show how many existing differentiable pruning techniques can be understood as nonconvex regularization for group sparse optimization, and prove that for a wide class of nonconvex regularizers, the global optimum is unique, group-sparse, and provably yields an approximate solution to a sparse convex optimization problem. The resulting algorithm that we propose, SequentialAttention++, advances the state of the art in large-scale neural network block-wise pruning tasks on the ImageNet and Criteo datasets.

Sketching algorithms have recently proven to be a powerful approach both for designing low-space streaming algorithms as well as fast polynomial time approximation schemes (PTAS). In this work, we develop new techniques to extend the applicability of sketching-based approaches to the sparse dictionary learning and the Euclidean $k$-means clustering problems. In particular, we initiate the study of the challenging setting where the dictionary/clustering assignment for each of the $n$ input points must be output, which has surprisingly received little attention in prior work. On the fast algorithms front, we obtain a new approach for designing PTAS's for the $k$-means clustering problem, which generalizes to the first PTAS for the sparse dictionary learning problem. On the streaming algorithms front, we obtain new upper bounds and lower bounds for dictionary learning and $k$-means clustering. In particular, given a design matrix $\mathbf A\in\mathbb R^{n\times d}$ in a turnstile stream, we show an $\tilde O(nr/\epsilon^2 + dk/\epsilon)$ space upper bound for $r$-sparse dictionary learning of size $k$, an $\tilde O(n/\epsilon^2 + dk/\epsilon)$ space upper bound for $k$-means clustering, as well as an $\tilde O(n)$ space upper bound for $k$-means clustering on random order row insertion streams with a natural "bounded sensitivity" assumption. On the lower bounds side, we obtain a general $\tilde\Omega(n/\epsilon + dk/\epsilon)$ lower bound for $k$-means clustering, as well as an $\tilde\Omega(n/\epsilon^2)$ lower bound for algorithms which can estimate the cost of a single fixed set of candidate centers.

Feature selection is the problem of selecting a subset of features for a machine learning model that maximizes model quality subject to a budget constraint. We propose a feature selection algorithm called Sequential Attention that achieves state-of-the-art empirical results for neural networks. This algorithm is based on an efficient one-pass implementation of greedy forward selection and uses attention weights at each step as a proxy for feature importance. We give theoretical insights into our algorithm for linear regression by showing that an adaptation to this setting is equivalent to the classical Orthogonal Matching Pursuit (OMP) algorithm, and thus inherits all of its provable guarantees. Our theoretical and empirical analyses offer new explanations towards the effectiveness of attention and its connections to overparameterization, which may be of independent interest. Feature selection is a classic problem in machine learning and statistics where one is asked to find a subset of features from a larger set of features, such that the prediction quality of the model trained using the subset of features is maximized. Finding a small and high-quality feature subset is desirable for many reasons: improving model interpretability, reducing inference latency, decreasing model size, regularization, and removing redundant or noisy features to improve generalization. We direct the reader to Li et al. (2017b) for a comprehensive survey on the role of feature selection in machine learning. The widespread success of deep learning has prompted an intense study of feature selection algorithms for neural networks, especially in the supervised setting. While many methods have been proposed, we focus on a line of work that studies the use of attention for feature selection.

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.

We study active sampling algorithms for linear regression, which aim to query only a small number of entries of a target vector $b\in\mathbb{R}^n$ and output a near minimizer to $\min_{x\in\mathbb{R}^d}\|Ax-b\|$, where $A\in\mathbb{R}^{n \times d}$ is a design matrix and $\|\cdot\|$ is some loss function. For $\ell_p$ norm regression for any $0