93 accepted papers
In the data selection problem, the objective is to choose a small, representative subset of data that can be used to efficiently train a machine learning model. Sener and Savarese [ICLR 2018] showed that, given an embedding representation of the data and suitable geometric assumptions, heuristics ba…
In this paper, we study the distributed experts problem, where $n$ experts are distributed across $s$ servers for $T$ timesteps. The loss of each expert at each time $t$ is the $\ell_p$ norm of the vector that consists of the losses of the expert at each of the $s$ servers at time $t$. The goal is t…
We introduce and study the problem of consistent low-rank approximation, in which rows of an input matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$ arrive sequentially and the goal is to provide a sequence of subspaces that well-approximate the optimal rank-$k$ approximation to the submatrix $\mathbf{A}…
We study the problem of constructing $(1+\varepsilon)$-coresets for Euclidean $(k,z)$-clustering in the distributed setting, where $n$ data points are partitioned across $s$ sites. We focus on two prominent communication models: the coordinator model and the blackboard model. In the coordinator mode…
Motivated by the prevalence and success of machine learning, a line of recent work has studied learning-augmented algorithms in the streaming model. These results have shown that for natural and practical oracles implemented with machine learning models, we can obtain streaming algorithms with impro…
We study the problem of graph and hypergraph sparsification in insertion-only data streams. The input is a hypergraph $H=(V, E, w)$ with $n$ nodes, $m$ hyperedges, and rank $r$, and the goal is to compute a hypergraph $\widehat{H}$ that preserves the energy of each vector $x \in \mathbb{R}^n$ in $H$…
We study the Nearest Neighbor Search (NNS) problem in a high-dimensional setting where data originates from a low-dimensional subspace and is corrupted by Gaussian noise. Specifically, we consider a semi-random model where $n$ points from an unknown $k$-dimensional subspace of $\mathbb{R}^d$ ($k \ll…
We study beyond worst-case dimensionality reduction for $s$-sparse vectors (vectors with at most $s$ non-zero coordinates). Our work is divided into two parts, each focusing on a different facet of beyond worst-case analysis: \noindent (a) We first consider average-case guarantees for embedding $s…
A central problem related to transformers can be stated as follows: given two $n \times d$ matrices $Q$ and $K$, and a non-negative function $f$, define the matrix $A$ as follows: (1) apply the function $f$ to each entry of the $n \times n$ matrix $Q K^T$, and then (2) normalize each of the row sums…
In the maximum coverage problem we are given $d$ subsets from a universe $[n]$, and the goal is to output $k$ subsets such that their union covers the largest possible number of distinct items. We present the first algorithm for maximum coverage in the turnstile streaming model, where updates which…
$k$-anonymity is a widely-used privacy-preserving concept that ensures each record in a dataset is indistinguishable from at least $k-1$ other records. In this paper, we revisit $k$-anonymity by suppression and give an $O(k)$-approximation algorithm with a nearly-linear runtime of $\tilde{O}(nd + n^…
Recently differential privacy has been used for a number of streaming, data structure, and dynamic graph problems as a means of hiding the internal randomness of the data structure, so that multiple possibly adaptive queries can be made without sacrificing the correctness of the responses. Although…
We study the problem of distributed distinct element estimation, where $\alpha$ servers each receive a subset of a universe $[n]$ and aim to compute a $(1+\varepsilon)$-approximation to the number of distinct elements using minimal communication. While prior work establishes a worst-case bound of $\…
We propose an algorithm with improved query-complexity for the problem of hypothesis selection under local differential privacy constraints. Given a set of $k$ probability distributions $Q$, we describe an algorithm that satisfies local differential privacy, performs $\tilde{O}(k^{3/2})$ non-adaptiv…
Robustness to outliers is important in machine learning. Many classical problems, including subspace embedding, clustering, and low-rank approximation, lack scalable, outlier-resilient algorithms. This paper considers machine learning problems of the form $\min_{x\in \mathbb{R}^d} F(x)$, where $F(x)…
We initiate the study of one-pass streaming algorithms for underdetermined $\ell_p$ linear regression problems of the form $$ \min_{\mathbf A\mathbf x = \mathbf b} \lVert\mathbf x\rVert_p \,, \qquad \text{where } \mathbf A \in \mathbb R^{n \times d} \text{ with } n \ll d \,, $$ wh…
We study the computational model where we can access a matrix $\mathbf{A}$ only by computing matrix-vector products $\mathbf{A}\mathrm{x}$ for vectors of the form $\mathrm{x} = \mathrm{x}_1 \otimes \cdots \otimes \mathrm{x}_q$. We prove exponential lower bounds on the number of queries needed to…
Fast changing states or volatile environments pose a significant challenge to online optimization, which needs to perform rapid adaptation under limited observation. In this paper, we give query and regret optimal bandit algorithms under the strict notion of strongly adaptive regret, which measures…
In the adversarial streaming model, the input is a sequence of adaptive updates that defines an underlying dataset and the goal is to approximate, collect, or compute some statistic while using space sublinear in the size of the dataset. In 2022, Ben-Eliezer, Eden, and Onak showed a dense-sparse tra…
When rows of an $n \times d$ matrix $A$ are given in a stream, we study algorithms for approximating the top eigenvector of $A^T A$ (equivalently, the top right singular vector of $A$). We consider worst case inputs $A$ but assume that the rows are presented to the streaming algorithm in a uniformly…
In this work, we study the experts problem in the distributed setting where an expert's cost needs to be aggregated across multiple servers. Our study considers various communication models such as the message-passing model and the broadcast model, along with multiple aggregation functions, such as…
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. Howeve…
We study the data selection problem, whose aim is to select a small representative subset of data that can be used to efficiently train a machine learning model. We present a new data selection approach based on $k$-means clustering and sensitivity sampling. Assuming access to an embedding represent…
Graph Transformers excel in long-range dependency modeling, but generally require quadratic memory complexity in the number of nodes in an input graph, and hence have trouble scaling to large graphs. Sparse attention variants such as Exphormer can help, but may require high-degree augmentations to t…
We introduce a new approach for applying sampling-based sketches to two and three mode tensors. We illustrate our technique to construct sketches for the classical problems of $\ell_0$ sampling and producing $\ell_1$ embeddings. In both settings we achieve sketches that can be applied to a rank one…
Recently, the question of adversarially robust streaming, where the stream is allowed to depend on the randomness of the streaming algorithm, has gained a lot of attention. In this work, we consider a strong white-box adversarial model (Ajtai et al. PODS 2022), in which the adversary has access to a…
We study streaming algorithms for the $\ell_p$ subspace approximation problem. Given points $a_1, \ldots, a_n$ as an insertion-only stream and a rank parameter $k$, the $\ell_p$ subspace approximation problem is to find a $k$-dimensional subspace $V$ such that $(\sum_{i=1}^n d(a_i, V)^p)^{1/p}$ is m…
We present an approximate attention mechanism named `HyperAttention` to address the computational challenges posed by the growing complexity of long contexts used in Large Language Models (LLMs). Recent work suggests that in the worst-case scenario, the quadratic time is necessary unless the entrie…
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 (Cohen et al., 2019). We show that this algorithm can b…
In the Generalized Mastermind problem, there is an unknown subset $H$ of the hypercube 0,1$^d$ containing $n$ points. The goal is to learn $H$ by making a few queries to an oracle which given a point $q$ in 0,1$^d$, returns the point in $H$ nearest to $q$. We give a two-round adaptive algorithm for…
Low-rank approximation and column subset selection are two fundamental and related problems that are applied across a wealth of machine learning applications. In this paper, we study the question of socially fair low-rank approximation and socially fair column subset selection, where the goal is to…
We study the problem of residual error estimation for matrix and vector norms using a linear sketch. Such estimates can be used, for example, to quickly assess how useful a more expensive low-rank approximation computation will be. The matrix case concerns the Frobenius norm and the task is to appro…
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 parameterized…
We improve upon previous oblivious sketching and turnstile streaming results for $\ell_1$ and logistic regression, giving a much smaller sketching dimension achieving $O(1)$-approximation and yielding an efficient optimization problem in the sketch space. Namely, we achieve for any constant $c>0$ a…
Recent works in dimensionality reduction for regression tasks have introduced the notion of sensitivity, an estimate of the importance of a specific datapoint in a dataset, offering provable guarantees on the quality of the approximation after removing low-sensitivity datapoints via subsampling. How…
We introduce efficient $(1+\varepsilon)$-approximation algorithms for the binary matrix factorization (BMF) problem, where the inputs are a matrix $\mathbf{A}\in\{0,1\}^{n\times d}$, a rank parameter $k>0$, as well as an accuracy parameter $\varepsilon>0$, and the goal is to approximate $\mathbf{A}$…
Inspired by fast algorithms in natural language processing, we study low rank approximation in the entrywise transformed setting where we want to find a good rank $k$ approximation to $f(U \cdot V)$, where $U, V^\top \in \mathbb{R}^{n \times r}$ are given, $r = O(\log(n))$, and $f(x)$ is a general s…
We study streaming algorithms in the white-box adversarial stream model, where the internal state of the streaming algorithm is revealed to an adversary who adaptively generates the stream updates, but the algorithm obtains fresh randomness unknown to the adversary at each time step. We incorporate…
We consider sketching algorithms which first compress data by multiplication with a random sketch matrix, and then apply the sketch to quickly solve an optimization problem, e.g., low-rank approximation and regression. In the learning-based sketching paradigm proposed by Indyk et al., the sketch mat…
We study lower bounds on adaptive sensing algorithms for recovering low rank matrices using linear measurements. Given an $n \times n$ matrix $A$, a general linear measurement $S(A)$, for an $n \times n$ matrix $S$, is just the inner product of $S$ and $A$, each treated as $n^2$-dimensional vectors.…
Clustering is an important technique for identifying structural information in large-scale data analysis, where the underlying dataset may be too large to store. In many applications, recent data can provide more accurate information and thus older data past a certain time is expired. The sliding wi…
In the online learning with experts problem, an algorithm makes predictions about an outcome on each of $T$ days, given a set of $n$ experts who make predictions on each day. The algorithm is given feedback on the outcomes of each day, including the cost of its prediction and the cost of the expert…
We study oblivious sketching for $k$-sparse linear regression under various loss functions. In particular, we are interested in a distribution over sketching matrices $S\in\mathbb{R}^{m\times n}$ that does not depend on the inputs $A\in\mathbb{R}^{n\times d}$ and $b\in\mathbb{R}^n$, such that, given…
We study dynamic algorithms robust to adaptive input generated from sources with bounded capabilities, such as sparsity or limited interaction. For example, we consider robust linear algebraic algorithms when the updates to the input are sparse but given by an adversary with access to a query oracle…
In large scale machine learning, *random sampling* is a popular way to approximate datasets by a small representative subset of examples. In particular, *sensitivity sampling* is an intensely studied technique which provides provable guarantees on the quality of approximation, while reducing the num…
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 dictiona…
A common method in training neural networks is to initialize all the weights to be independent Gaussian vectors. We observe that by instead initializing the weights into independent pairs, where each pair consists of two identical Gaussian vectors, we can significantly improve the convergence analys…
We study the $\ell_p$ regression problem, which requires finding $\mathbf{x}\in\mathbb R^{d}$ that minimizes $\|\mathbf{A}\mathbf{x}-\mathbf{b}\|_p$ for a matrix $\mathbf{A}\in\mathbb R^{n \times d}$ and response vector $\mathbf{b}\in\mathbb R^{n}$. There has been recent interest in developing subsa…
A treap is a classic randomized binary search tree data structure that is easy to implement and supports O(log n) expected time access. However, classic treaps do not take advantage of the input distribution or patterns in the input. Given recent advances in algorithms with predictions, we propose p…
$k$-means clustering is a well-studied problem due to its wide applicability. Unfortunately, there exist strong theoretical limits on the performance of any algorithm for the $k$-means problem on worst-case inputs. To overcome this barrier, we consider a scenario where ``advice'' is provided to help…
We propose an input sparsity time sampling algorithm that can spectrally approximate the Gram matrix corresponding to the q-fold column-wise tensor product of q matrices using a nearly optimal number of samples, improving upon all previously known methods by poly(q) factors. Furthermore, for the imp…
We consider the problem of minimizing the number of matrix-vector queries needed for accurate trace estimation in the dynamic setting where our underlying matrix is changing slowly, such as during an optimization process. Specifically, for any $m$ matrices $\mathbf{A}_1,...,\mathbf{A}_m$ with consec…
We create classical (non-quantum) dynamic data structures supporting queries for recommender systems and least-squares regression that are comparable to their quantum analogues. De-quantizing such algorithms has received a flurry of attention in recent years; we obtain sharper bounds for these probl…
We give a sketching-based iterative algorithm that computes a $1+\varepsilon$ approximate solution for the ridge regression problem $\min_x \|Ax-b\|_2^2 +\lambda\|x\|_2^2$ where $A \in R^{n \times d}$ with $d \ge n$. Our algorithm, for a constant number of iterations (requiring a constant number of…
We propose data-driven one-pass streaming algorithms for estimating the number of triangles and four cycles, two fundamental problems in graph analytics that are widely studied in the graph data stream literature. Recently, Hsu et al. (2019) and Jiang et al. (2020) applied machine learning technique…
We give a dimensionality reduction procedure to approximate the sum of distances of a given set of $n$ points in $R^d$ to any “shape” that lies in a $k$-dimensional subspace. Here, by “shape” we mean any set of points in $R^d$. Our algorithm takes an input in the form of an $n \times d$ matrix $A$,…
Kernel methods are fundamental in machine learning, and faster algorithms for kernel approximation provide direct speedups for many core tasks in machine learning. The polynomial kernel is especially important as other kernels can often be approximated by the polynomial kernel via a Taylor series ex…
Recently, data-driven and learning-based algorithms for low rank matrix approximation were shown to outperform classical data-oblivious algorithms by wide margins in terms of accuracy. Those algorithms are based on the optimization of sparse sketching matrices, which lead to large savings in time a…
Sketching is a powerful dimensionality reduction technique for accelerating algorithms for data analysis. A crucial step in sketching methods is to compute a subspace embedding (SE) for a large matrix $A \in \mathbb{R}^{N \times d}$. SE’s are the primary tool for obtaining extremely efficient soluti…
We consider the problem of learning a latent $k$-vertex simplex $K\in\mathbb{R}^d$, given $\mathbf{A}\in\mathbb{R}^{d\times n}$, which can be viewed as $n$ data points that are formed by randomly perturbing some latent points in $K$, possibly beyond $K$. A large class of latent variable models, such…
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 t…
What guarantees are possible for solving logistic regression in one pass over a data stream? To answer this question, we present the first data oblivious sketch for logistic regression. Our sketch can be computed in input sparsity time over a turnstile data stream and reduces the size of a $d$-dimen…
Matrix trace estimation is ubiquitous in machine learning applications and has traditionally relied on Hutchinson's method, which requires $O(\log(1/\delta)/\epsilon^2)$ matrix-vector product queries to achieve a $(1 \pm \epsilon)$-multiplicative approximation to $\text{trace}(A)$ with failure proba…
In applications such as natural language processing or computer vision, one is given a large $n \times n$ matrix $A = (a_{i,j})$ and would like to compute a matrix decomposition, e.g., a low rank approximation, of a function $f(A) = (f(a_{i,j}))$ applied entrywise to $A$. A very important special ca…
We give the first single-pass streaming algorithm for Column Subset Selection with respect to the entrywise $\ell_p$-norm with $1 \leq p < 2$. We study the $\ell_p$ norm loss since it is often considered more robust to noise than the standard Frobenius norm. Given an input matrix $A \in \mathbb{R}^{…
We give the first input-sparsity time algorithms for the rank-$k$ low rank approximation problem in every Schatten norm. Specifically, for a given $n\times n$ matrix $A$, our algorithm computes $Y,Z\in \R^{n\times k}$, which, with high probability, satisfy $\|A-YZ^T\|_p \leq (1+\eps)\|A-A_k\|_p$, wh…
To accelerate kernel methods, we propose a near input sparsity time method for sampling the high-dimensional space implicitly defined by a kernel transformation. Our main contribution is an importance sampling method for subsampling the feature space of a degree $q$ tensoring of data points in almos…
We consider the ridge regression problem, for which we are given an nxd matrix A of examples and a corresponding nxd’ matrix B of labels, as well as a ridge parameter $\lambda \geq 0$, and would like to output an $X’ \in R^{d \times d’}$ for which $$\|AX’-B\|_F^2 + \lambda \|X’\|_F^2 \leq (1+\epsilo…
We introduce a new scalable approximation for Gaussian processes with provable guarantees which holds simultaneously over its entire parameter space. Our approximation is obtained from an improved sample complexity analysis for sparse spectrum Gaussian processes (SSGPs). In particular, our analysis…
Weighted sampling is a fundamental tool in data analysis and machine learning pipelines. Samples are used for efficient estimation of statistics or as sparse representations of the data. When weight distributions are skewed, as is often the case in practice, without-replacement (WOR) sampling is mu…
We study the column subset selection problem with respect to the entrywise $\ell_1$-norm loss. It is known that in the worst case, to obtain a good rank-$k$ approximation to a matrix, one needs an arbitrarily large $n^{\Omega(1)}$ number of columns to obtain a $(1+\epsilon)$-approximation to an $n \…
We consider the following conditional linear regression problem: the task is to identify both (i) a $k$-DNF condition $c$ and (ii) a linear rule $f$ such that the probability of $c$ is (approximately) at least some given bound $\mu$, and minimizing the $l_p$ loss of $f$ at predicting the target $z$…
We give the first dimensionality reduction methods for the overconstrained Tukey regression problem. The Tukey loss function $\|y\|_M = \sum_i M(y_i)$ has $M(y_i) \approx |y_i|^p$ for residual errors $y_i$ smaller than a prescribed threshold $\tau$, but $M(y_i)$ becomes constant for errors $|y_i| >…
We take an unorthodox view of voting by expanding the design space to include both the elicitation rule, whereby voters map their (cardinal) preferences to votes, and the aggregation rule, which transforms the reported votes into collective decisions. Intuitively, there is a tradeoff between the com…
We give faster approximation algorithms for well-studied variants of Binary Matrix Factorization (BMF), where we are given a binary $m \times n$ matrix $A$ and would like to find binary rank-$k$ matrices $U, V$ to minimize the Frobenius norm of $U \cdot V - A$. In the first setting, $U \cdot V$ deno…
We study the Kronecker product regression problem, in which the design matrix is a Kronecker product of two or more matrices. Formally, given $A_i \in \R^{n_i \times d_i}$ for $i=1,2,\dots,q$ where $n_i \gg d_i$ for each $i$, and $b \in \R^{n_1 n_2 \cdots n_q}$, let $\mathcal{A} = A_i \otimes A_2 \…
The classical low rank approximation problem is to find a rank $k$ matrix $UV$ (where $U$ has $k$ columns and $V$ has $k$ rows) that minimizes the Frobenius norm of $A - UV$. Although this problem can be solved efficiently, we study an NP-hard variant of this problem that involves weights and regula…
We revisit the classic randomized sketch of a tensor product of $q$ vectors $x_i\in\mathbb{R}^n$. The $i$-th coordinate $(Sx)_i$ of the sketch is equal to $\prod_{j = 1}^q \langle u^{i, j}, x^j \rangle / \sqrt{m}$, where $u^{i,j}$ are independent random sign vectors. Kar and Karnick (JMLR, 2012) sho…
Kernel methods generalize machine learning algorithms that only depend on the pairwise inner products of the dataset by replacing inner products with kernel evaluations, a function that passes input points through a nonlinear feature map before taking the inner product in a higher dimensional space.…
In the total least squares problem, one is given an $m \times n$ matrix $A$, and an $m \times d$ matrix $B$, and one seeks to ``correct'' both $A$ and $B$, obtaining matrices $\hat{A}$ and $\hat{B}$, so that there exists an $X$ satisfying the equation $\hat{A}X = \hat{B}$. Typically the problem is o…
There are a number of approximation algorithms for NP-hard versions of low rank approximation, such as finding a rank-$k$ matrix $B$ minimizing the sum of absolute values of differences to a given $n$-by-$n$ matrix $A$, $\min_{\textrm{rank-}k~B}\|A-B\|_1$, or more generally finding a rank-$k$ matrix…
Work on approximate linear algebra has led to efficient distributed and streaming algorithms for problems such as approximate matrix multiplication, low rank approximation, and regression, primarily for the Euclidean norm $\ell_2$. We study other $\ell_p$ norms, which are more robust for $p < 2$, an…
A central problem in mining massive data streams is characterizing which functions of an underlying frequency vector can be approximated efficiently. Given the prevalence of large scale linear algebra problems in machine learning, recently there has been considerable effort in extending this data st…
Coresets are one of the central methods to facilitate the analysis of large data. We continue a recent line of research applying the theory of coresets to logistic regression. First, we show the negative result that no strongly sublinear sized coresets exist for logistic regression. To deal with int…
We study robust subspace estimation in the streaming and distributed settings. Given a set of n data points {a
TensorSketch is an oblivious linear sketch introduced in (Pagh, 2013) and later used in (Pham and Pagh, 2013) in the context of SVMs for polynomial kernels. It was shown in (Avron et al., 2014) that TensorSketch provides a subspace embedding, and therefore can be used for canonical correlation analy…
Let $\PP=\{ p_1, p_2, \ldots p_n \}$ and $\QQ = \{ q_1, q_2 \ldots q_m \}$ be two point sets in an arbitrary metric space. Let $\AA$ represent the $m\times n$ pairwise distance matrix with $\AA_{i,j} = d(p_i, q_j)$. Such distance matrices are commonly computed in software packages and have applicati…
We study the $\ell_0$-Low Rank Approximation Problem, where the goal is, given an $m \times n$ matrix $A$, to output a rank-$k$ matrix $A'$ for which $\|A'-A\|_0$ is minimized. Here, for a matrix $B$, $\|B\|_0$ denotes the number of its non-zero entries. This NP-hard variant of low rank a…
Low-rank approximation is a common tool used to accelerate kernel methods: the $n \times n$ kernel matrix $K$ is approximated via a rank-$k$ matrix $\tilde K$ which can be stored in much less space and processed more quickly. In this work we study the limits of computationally efficient low-rank ker…
We study the least squares regression problem $\min_{\Theta \in \RR^{p_1 \times \cdots \times p_D}} \| \cA(\Theta) - b \|_2^2$, where $\Theta$ is a low-rank tensor, defined as $\Theta = \sum_{r=1}^{R} \theta_1^{(r)} \circ \cdots \circ \theta_D^{(r)}$, for vectors $\theta_d^{(r)} \in \mathbb{R}^{p_d…
Clustering large datasets is a fundamental problem with a number of applications in machine learning. Data is often collected on different sites and clustering needs to be performed in a distributed manner with low communication. We would like the quality of the clustering in the distributed setting…
Have you ever wanted to multiply an n \times d matrix X, with n ≫d, on the left by an m \times n matrix \tilde G of i.i.d. Gaussian random variables, but could not afford to do it because it was too slow? In this work we propose a new randomized m \times n matrix T, for which one can compute T ⋅X in…
A recent work (Wang et. al., NIPS 2015) gives the fastest known algorithms for orthogonal tensor decomposition with provable guarantees. Their algorithm is based on computing sketches of the input tensor, which requires reading the entire input. We show in a number of cases one can achieve the same…