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Samson Zhou

Publications and source records attributed to Samson Zhou.

At least 55 records · Page 3Linked to original sources
arXiv2023

Streaming Euclidean $k$-median and $k$-means with $o(\log n)$ Space

We consider the classic Euclidean $k$-median and $k$-means objective on data streams, where the goal is to provide a $(1+\varepsilon)$-approximation to the optimal $k$-median or $k$-means solution, while using as little memory as possible. Over the last 20 years, clustering in data streams has received a tremendous amount of attention and has been the test-bed for a large variety of new techniques, including coresets, the merge-and-reduce framework, bicriteria approximation, sensitivity sampling, and so on. Despite this intense effort to obtain smaller sketches for these problems, all known techniques require storing at least $Ω(\log(nΔ))$ words of memory, where $n$ is the size of the input and $Δ$ is the aspect ratio. A natural question is if one can beat this logarithmic dependence on $n$ and $Δ$. In this paper, we break this barrier by first giving an insertion-only streaming algorithm that achieves a $(1+\varepsilon)$-approximation to the more general $(k,z)$-clustering problem, using $\tilde{\mathcal{O}}\left(\frac{dk}{\varepsilon^2}\right)\cdot(2^{z\log z})\cdot\min\left(\frac{1}{\varepsilon^z},k\right)\cdot\text{poly}(\log\log(nΔ))$ words of memory. Our techniques can also be used to achieve two-pass algorithms for $k$-median and $k$-means clustering on dynamic streams using $\tilde{\mathcal{O}}\left(\frac{1}{\varepsilon^2}\right)\cdot\text{poly}(d,k,\log\log(nΔ))$ words of memory.

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arXiv2023

How to Make Your Approximation Algorithm Private: A Black-Box Differentially-Private Transformation for Tunable Approximation Algorithms of Functions with Low Sensitivity

We develop a framework for efficiently transforming certain approximation algorithms into differentially-private variants, in a black-box manner. Specifically, our results focus on algorithms A that output an approximation to a function f of the form $(1-a)f(x)-k \leq A(x) \leq (1+a)f(x)+k$, where $k \in \mathbb{R}_{\geq 0}$ denotes additive error and $a \in [0,1)$ denotes multiplicative error can be``tuned" to small-enough values while incurring only a polynomial blowup in the running time/space. We show that such algorithms can be made DP without sacrificing accuracy, as long as the function f has small global sensitivity. We achieve these results by applying the smooth sensitivity framework developed by Nissim, Raskhodnikova, and Smith (STOC 2007). Our framework naturally applies to transform non-private FPRAS and FPTAS algorithms into $ε$-DP approximation algorithms where the former case requires an additional postprocessing step. We apply our framework in the context of sublinear-time and sublinear-space algorithms, while preserving the nature of the algorithm in meaningful ranges of the parameters. Our results include the first (to the best of our knowledge) $ε$-edge DP sublinear-time algorithm for estimating the number of triangles, the number of connected components, and the weight of a minimum spanning tree of a graph. In the area of streaming algorithms, our results include $ε$-DP algorithms for estimating Lp-norms, distinct elements, and weighted minimum spanning tree for both insertion-only and turnstile streams. Our transformation also provides a private version of the smooth histogram framework, which is commonly used for converting streaming algorithms into sliding window variants, and achieves a multiplicative approximation to many problems, such as estimating Lp-norms, distinct elements, and the length of the longest increasing subsequence.

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arXiv2023

Differentially Private Aggregation via Imperfect Shuffling

In this paper, we introduce the imperfect shuffle differential privacy model, where messages sent from users are shuffled in an almost uniform manner before being observed by a curator for private aggregation. We then consider the private summation problem. We show that the standard split-and-mix protocol by Ishai et. al. [FOCS 2006] can be adapted to achieve near-optimal utility bounds in the imperfect shuffle model. Specifically, we show that surprisingly, there is no additional error overhead necessary in the imperfect shuffle model.

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arXiv2023

Private Data Stream Analysis for Universal Symmetric Norm Estimation

We study how to release summary statistics on a data stream subject to the constraint of differential privacy. In particular, we focus on releasing the family of symmetric norms, which are invariant under sign-flips and coordinate-wise permutations on an input data stream and include $L_p$ norms, $k$-support norms, top-$k$ norms, and the box norm as special cases. Although it may be possible to design and analyze a separate mechanism for each symmetric norm, we propose a general parametrizable framework that differentially privately releases a number of sufficient statistics from which the approximation of all symmetric norms can be simultaneously computed. Our framework partitions the coordinates of the underlying frequency vector into different levels based on their magnitude and releases approximate frequencies for the "heavy" coordinates in important levels and releases approximate level sizes for the "light" coordinates in important levels. Surprisingly, our mechanism allows for the release of an arbitrary number of symmetric norm approximations without any overhead or additional loss in privacy. Moreover, our mechanism permits $(1+α)$-approximation to each of the symmetric norms and can be implemented using sublinear space in the streaming model for many regimes of the accuracy and privacy parameters.

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arXiv2023

Fast $(1+\varepsilon)$-Approximation Algorithms for Binary Matrix Factorization

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 $\mathbf{A}$ as a product of low-rank factors $\mathbf{U}\in\{0,1\}^{n\times k}$ and $\mathbf{V}\in\{0,1\}^{k\times d}$. Equivalently, we want to find $\mathbf{U}$ and $\mathbf{V}$ that minimize the Frobenius loss $\|\mathbf{U}\mathbf{V} - \mathbf{A}\|_F^2$. Before this work, the state-of-the-art for this problem was the approximation algorithm of Kumar et. al. [ICML 2019], which achieves a $C$-approximation for some constant $C\ge 576$. We give the first $(1+\varepsilon)$-approximation algorithm using running time singly exponential in $k$, where $k$ is typically a small integer. Our techniques generalize to other common variants of the BMF problem, admitting bicriteria $(1+\varepsilon)$-approximation algorithms for $L_p$ loss functions and the setting where matrix operations are performed in $\mathbb{F}_2$. Our approach can be implemented in standard big data models, such as the streaming or distributed models.

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arXiv2023

Robust Algorithms on Adaptive Inputs from Bounded Adversaries

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. We also study robust algorithms in the standard centralized setting, where an adversary queries an algorithm in an adaptive manner, but the number of interactions between the adversary and the algorithm is bounded. We first recall a unified framework of [HKM+20, BKM+22, ACSS23] for answering $Q$ adaptive queries that incurs $\widetilde{\mathcal{O}}(\sqrt{Q})$ overhead in space, which is roughly a quadratic improvement over the naïve implementation, and only incurs a logarithmic overhead in query time. Although the general framework has diverse applications in machine learning and data science, such as adaptive distance estimation, kernel density estimation, linear regression, range queries, and point queries and serves as a preliminary benchmark, we demonstrate even better algorithmic improvements for (1) reducing the pre-processing time for adaptive distance estimation and (2) permitting an unlimited number of adaptive queries for kernel density estimation. Finally, we complement our theoretical results with additional empirical evaluations.

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arXiv2023

Nearly Optimal Distinct Elements and Heavy Hitters on Sliding Windows

We study the distinct elements and $\ell_p$-heavy hitters problems in the sliding window model, where only the most recent $n$ elements in the data stream form the underlying set. We first introduce the composable histogram, a simple twist on the exponential (Datar et al., SODA 2002) and smooth histograms (Braverman and Ostrovsky, FOCS 2007) that may be of independent interest. We then show that the composable histogram along with a careful combination of existing techniques to track either the identity or frequency of a few specific items suffices to obtain algorithms for both distinct elements and $\ell_p$-heavy hitters that are nearly optimal in both $n$ and $ε$. Applying our new composable histogram framework, we provide an algorithm that outputs a $(1+ε)$-approximation to the number of distinct elements in the sliding window model and uses $\mathcal{O}\left(\frac{1}{ε^2}\log n\log\frac{1}ε\log\log n+\frac{1}ε\log^2 n\right)$ bits of space. For $\ell_p$-heavy hitters, we provide an algorithm using space $\mathcal{O}\left(\frac{1}{ε^p}\log^3 n\left(\log\log n+\log\frac{1}ε\right)\right)$ for $0<p\le 2$, improving upon the best-known algorithm for $\ell_2$-heavy hitters (Braverman et al., COCOON 2014), which has space complexity $\mathcal{O}\left(\frac{1}{ε^4}\log^3 n\right)$. We also show lower bounds of $Ω\left(\frac{1}ε\log^2 n+\frac{1}{ε^2}\log n\right)$ for distinct elements and $Ω\left(\frac{1}{ε^p}\log^2 n\right)$ for $\ell_p$-heavy hitters.

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arXiv2023

Near Optimal Linear Algebra in the Online and Sliding Window Models

We initiate the study of numerical linear algebra in the sliding window model, where only the most recent $W$ updates in a stream form the underlying data set. We first introduce a unified row-sampling based framework that gives randomized algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and $\ell_1$-subspace embeddings in the sliding window model, which often use nearly optimal space and achieve nearly input sparsity runtime. Our algorithms are based on "reverse online" versions of offline sampling distributions such as (ridge) leverage scores, $\ell_1$ sensitivities, and Lewis weights to quantify both the importance and the recency of a row. Our row-sampling framework rather surprisingly implies connections to the well-studied online model; our structural results also give the first sample optimal (up to lower order terms) online algorithm for low-rank approximation/projection-cost preservation. Using this powerful primitive, we give online algorithms for column/row subset selection and principal component analysis that resolves the main open question of Bhaskara et. al.,(FOCS 2019). We also give the first online algorithm for $\ell_1$-subspace embeddings. We further formalize the connection between the online model and the sliding window model by introducing an additional unified framework for deterministic algorithms using a merge and reduce paradigm and the concept of online coresets. Our sampling based algorithms in the row-arrival online model yield online coresets, giving deterministic algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and $\ell_1$-subspace embeddings in the sliding window model that use nearly optimal space.

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arXiv2023

Provable Data Subset Selection For Efficient Neural Network Training

Radial basis function neural networks (\emph{RBFNN}) are {well-known} for their capability to approximate any continuous function on a closed bounded set with arbitrary precision given enough hidden neurons. In this paper, we introduce the first algorithm to construct coresets for \emph{RBFNNs}, i.e., small weighted subsets that approximate the loss of the input data on any radial basis function network and thus approximate any function defined by an \emph{RBFNN} on the larger input data. In particular, we construct coresets for radial basis and Laplacian loss functions. We then use our coresets to obtain a provable data subset selection algorithm for training deep neural networks. Since our coresets approximate every function, they also approximate the gradient of each weight in a neural network, which is a particular function on the input. We then perform empirical evaluations on function approximation and dataset subset selection on popular network architectures and data sets, demonstrating the efficacy and accuracy of our coreset construction.

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arXiv2023

Streaming Algorithms for Learning with Experts: Deterministic Versus Robust

In the online learning with experts problem, an algorithm must make a prediction about an outcome on each of $T$ days (or times), given a set of $n$ experts who make predictions on each day (or time). The algorithm is given feedback on the outcomes of each day, including the cost of its prediction and the cost of the expert predictions, and the goal is to make a prediction with the minimum cost, specifically compared to the best expert in the set. Recent work by Srinivas, Woodruff, Xu, and Zhou (STOC 2022) introduced the study of the online learning with experts problem under memory constraints. However, often the predictions made by experts or algorithms at some time influence future outcomes, so that the input is adaptively chosen. Whereas deterministic algorithms would be robust to adaptive inputs, existing algorithms all crucially use randomization to sample a small number of experts. In this paper, we study deterministic and robust algorithms for the experts problem. We first show a space lower bound of $\widetildeΩ\left(\frac{nM}{RT}\right)$ for any deterministic algorithm that achieves regret $R$ when the best expert makes $M$ mistakes. Our result shows that the natural deterministic algorithm, which iterates through pools of experts until each expert in the pool has erred, is optimal up to polylogarithmic factors. On the positive side, we give a randomized algorithm that is robust to adaptive inputs that uses $\widetilde{O}\left(\frac{n}{R\sqrt{T}}\right)$ space for $M=O\left(\frac{R^2 T}{\log^2 n}\right)$, thereby showing a smooth space-regret trade-off.

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arXiv2023

Differentially Private $L_2$-Heavy Hitters in the Sliding Window Model

The data management of large companies often prioritize more recent data, as a source of higher accuracy prediction than outdated data. For example, the Facebook data policy retains user search histories for $6$ months while the Google data retention policy states that browser information may be stored for up to $9$ months. These policies are captured by the sliding window model, in which only the most recent $W$ statistics form the underlying dataset. In this paper, we consider the problem of privately releasing the $L_2$-heavy hitters in the sliding window model, which include $L_p$-heavy hitters for $p\le 2$ and in some sense are the strongest possible guarantees that can be achieved using polylogarithmic space, but cannot be handled by existing techniques due to the sub-additivity of the $L_2$ norm. Moreover, existing non-private sliding window algorithms use the smooth histogram framework, which has high sensitivity. To overcome these barriers, we introduce the first differentially private algorithm for $L_2$-heavy hitters in the sliding window model by initiating a number of $L_2$-heavy hitter algorithms across the stream with significantly lower threshold. Similarly, we augment the algorithms with an approximate frequency tracking algorithm with significantly higher accuracy. We then use smooth sensitivity and statistical distance arguments to show that we can add noise proportional to an estimation of the $L_2$ norm. To the best of our knowledge, our techniques are the first to privately release statistics that are related to a sub-additive function in the sliding window model, and may be of independent interest to future differentially private algorithmic design in the sliding window model.

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arXiv2023

On Differential Privacy and Adaptive Data Analysis with Bounded Space

We study the space complexity of the two related fields of differential privacy and adaptive data analysis. Specifically, (1) Under standard cryptographic assumptions, we show that there exists a problem P that requires exponentially more space to be solved efficiently with differential privacy, compared to the space needed without privacy. To the best of our knowledge, this is the first separation between the space complexity of private and non-private algorithms. (2) The line of work on adaptive data analysis focuses on understanding the number of samples needed for answering a sequence of adaptive queries. We revisit previous lower bounds at a foundational level, and show that they are a consequence of a space bottleneck rather than a sampling bottleneck. To obtain our results, we define and construct an encryption scheme with multiple keys that is built to withstand a limited amount of key leakage in a very particular way.

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arXiv2023

Optimal Algorithms for Linear Algebra in the Current Matrix Multiplication Time

We study fundamental problems in linear algebra, such as finding a maximal linearly independent subset of rows or columns (a basis), solving linear regression, or computing a subspace embedding. For these problems, we consider input matrices $\mathbf{A}\in\mathbb{R}^{n\times d}$ with $n > d$. The input can be read in $\text{nnz}(\mathbf{A})$ time, which denotes the number of nonzero entries of $\mathbf{A}$. In this paper, we show that beyond the time required to read the input matrix, these fundamental linear algebra problems can be solved in $d^ω$ time, i.e., where $ω\approx 2.37$ is the current matrix-multiplication exponent. To do so, we introduce a constant-factor subspace embedding with the optimal $m=\mathcal{O}(d)$ number of rows, and which can be applied in time $\mathcal{O}\left(\frac{\text{nnz}(\mathbf{A})}α\right) + d^{2 + α}\text{poly}(\log d)$ for any trade-off parameter $α>0$, tightening a recent result by Chepurko et. al. [SODA 2022] that achieves an $\exp(\text{poly}(\log\log n))$ distortion with $m=d\cdot\text{poly}(\log\log d)$ rows in $\mathcal{O}\left(\frac{\text{nnz}(\mathbf{A})}α+d^{2+α+o(1)}\right)$ time. Our subspace embedding uses a recently shown property of stacked Subsampled Randomized Hadamard Transforms (SRHT), which actually increase the input dimension, to "spread" the mass of an input vector among a large number of coordinates, followed by random sampling. To control the effects of random sampling, we use fast semidefinite programming to reweight the rows. We then use our constant-factor subspace embedding to give the first optimal runtime algorithms for finding a maximal linearly independent subset of columns, regression, and leverage score sampling. To do so, we also introduce a novel subroutine that iteratively grows a set of independent rows, which may be of independent interest.

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arXiv2022

Sub-quadratic Algorithms for Kernel Matrices via Kernel Density Estimation

Kernel matrices, as well as weighted graphs represented by them, are ubiquitous objects in machine learning, statistics and other related fields. The main drawback of using kernel methods (learning and inference using kernel matrices) is efficiency -- given $n$ input points, most kernel-based algorithms need to materialize the full $n \times n$ kernel matrix before performing any subsequent computation, thus incurring $Ω(n^2)$ runtime. Breaking this quadratic barrier for various problems has therefore, been a subject of extensive research efforts. We break the quadratic barrier and obtain $\textit{subquadratic}$ time algorithms for several fundamental linear-algebraic and graph processing primitives, including approximating the top eigenvalue and eigenvector, spectral sparsification, solving linear systems, local clustering, low-rank approximation, arboricity estimation and counting weighted triangles. We build on the recent Kernel Density Estimation framework, which (after preprocessing in time subquadratic in $n$) can return estimates of row/column sums of the kernel matrix. In particular, we develop efficient reductions from $\textit{weighted vertex}$ and $\textit{weighted edge sampling}$ on kernel graphs, $\textit{simulating random walks}$ on kernel graphs, and $\textit{importance sampling}$ on matrices to Kernel Density Estimation and show that we can generate samples from these distributions in $\textit{sublinear}$ (in the support of the distribution) time. Our reductions are the central ingredient in each of our applications and we believe they may be of independent interest. We empirically demonstrate the efficacy of our algorithms on low-rank approximation (LRA) and spectral sparsification, where we observe a $\textbf{9x}$ decrease in the number of kernel evaluations over baselines for LRA and a $\textbf{41x}$ reduction in the graph size for spectral sparsification.

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arXiv2022

Near-Linear Sample Complexity for $L_p$ Polynomial Regression

We study $L_p$ polynomial regression. Given query access to a function $f:[-1,1] \rightarrow \mathbb{R}$, the goal is to find a degree $d$ polynomial $\hat{q}$ such that, for a given parameter $\varepsilon > 0$, $$ \|\hat{q}-f\|_p\le (1+\varepsilon) \cdot \min_{q:\text{deg}(q)\le d}\|q-f\|_p. $$ Here $\|\cdot\|_p$ is the $L_p$ norm, $\|g\|_p = (\int_{-1}^1 |g(t)|^p dt)^{1/p}$. We show that querying $f$ at points randomly drawn from the Chebyshev measure on $[-1,1]$ is a near-optimal strategy for polynomial regression in all $L_p$ norms. In particular, to find $\hat q$, it suffices to sample $O(d\, \frac{\text{polylog}\,d}{\text{poly}\,\varepsilon})$ points from $[-1,1]$ with probabilities proportional to this measure. While the optimal sample complexity for polynomial regression was well understood for $L_2$ and $L_\infty$, our result is the first that achieves sample complexity linear in $d$ and error $(1+\varepsilon)$ for other values of $p$ without any assumptions. Our result requires two main technical contributions. The first concerns $p\leq 2$, for which we provide explicit bounds on the $L_p$ Lewis weight function of the infinite linear operator underlying polynomial regression. Using tools from the orthogonal polynomial literature, we show that this function is bounded by the Chebyshev density. Our second key contribution is to take advantage of the structure of polynomials to reduce the $p>2$ case to the $p\leq 2$ case. By doing so, we obtain a better sample complexity than what is possible for general $p$-norm linear regression problems, for which $Ω(d^{p/2})$ samples are required.

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arXiv2022

Tight Bounds for Adversarially Robust Streams and Sliding Windows via Difference Estimators

In the adversarially robust streaming model, a stream of elements is presented to an algorithm and is allowed to depend on the output of the algorithm at earlier times during the stream. In the classic insertion-only model of data streams, Ben-Eliezer et. al. (PODS 2020, best paper award) show how to convert a non-robust algorithm into a robust one with a roughly $1/\varepsilon$ factor overhead. This was subsequently improved to a $1/\sqrt{\varepsilon}$ factor overhead by Hassidim et. al. (NeurIPS 2020, oral presentation), suppressing logarithmic factors. For general functions the latter is known to be best-possible, by a result of Kaplan et. al. (CRYPTO 2021). We show how to bypass this impossibility result by developing data stream algorithms for a large class of streaming problems, with no overhead in the approximation factor. Our class of streaming problems includes the most well-studied problems such as the $L_2$-heavy hitters problem, $F_p$-moment estimation, as well as empirical entropy estimation. We substantially improve upon all prior work on these problems, giving the first optimal dependence on the approximation factor. As in previous work, we obtain a general transformation that applies to any non-robust streaming algorithm and depends on the so-called twist number. However, the key technical innovation is that we apply the transformation to what we call a difference estimator for the streaming problem, rather than an estimator for the streaming problem itself. We then develop the first difference estimators for a wide range of problems. Our difference estimator methodology is not only applicable to the adversarially robust model, but to other streaming models where temporal properties of the data play a central role. (Abstract shortened to meet arXiv limit.)

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arXiv2022

Learning-Augmented Algorithms for Online Linear and Semidefinite Programming

Semidefinite programming (SDP) is a unifying framework that generalizes both linear programming and quadratically-constrained quadratic programming, while also yielding efficient solvers, both in theory and in practice. However, there exist known impossibility results for approximating the optimal solution when constraints for covering SDPs arrive in an online fashion. In this paper, we study online covering linear and semidefinite programs in which the algorithm is augmented with advice from a possibly erroneous predictor. We show that if the predictor is accurate, we can efficiently bypass these impossibility results and achieve a constant-factor approximation to the optimal solution, i.e., consistency. On the other hand, if the predictor is inaccurate, under some technical conditions, we achieve results that match both the classical optimal upper bounds and the tight lower bounds up to constant factors, i.e., robustness. More broadly, we introduce a framework that extends both (1) the online set cover problem augmented with machine-learning predictors, studied by Bamas, Maggiori, and Svensson (NeurIPS 2020), and (2) the online covering SDP problem, initiated by Elad, Kale, and Naor (ICALP 2016). Specifically, we obtain general online learning-augmented algorithms for covering linear programs with fractional advice and constraints, and initiate the study of learning-augmented algorithms for covering SDP problems. Our techniques are based on the primal-dual framework of Buchbinder and Naor (Mathematics of Operations Research, 34, 2009) and can be further adjusted to handle constraints where the variables lie in a bounded region, i.e., box constraints.

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arXiv2022

New Frameworks for Offline and Streaming Coreset Constructions

A coreset for a set of points is a small subset of weighted points that approximately preserves important properties of the original set. Specifically, if $P$ is a set of points, $Q$ is a set of queries, and $f:P\times Q\to\mathbb{R}$ is a cost function, then a set $S\subseteq P$ with weights $w:P\to[0,\infty)$ is an $ε$-coreset for some parameter $ε>0$ if $\sum_{s\in S}w(s)f(s,q)$ is a $(1+ε)$ multiplicative approximation to $\sum_{p\in P}f(p,q)$ for all $q\in Q$. Coresets are used to solve fundamental problems in machine learning under various big data models of computation. Many of the suggested coresets in the recent decade used, or could have used a general framework for constructing coresets whose size depends quadratically on what is known as total sensitivity $t$. In this paper we improve this bound from $O(t^2)$ to $O(t\log t)$. Thus our results imply more space efficient solutions to a number of problems, including projective clustering, $k$-line clustering, and subspace approximation. Moreover, we generalize the notion of sensitivity sampling for sup-sampling that supports non-multiplicative approximations, negative cost functions and more. The main technical result is a generic reduction to the sample complexity of learning a class of functions with bounded VC dimension. We show that obtaining an $(ν,α)$-sample for this class of functions with appropriate parameters $ν$ and $α$ suffices to achieve space efficient $ε$-coresets. Our result implies more efficient coreset constructions for a number of interesting problems in machine learning; we show applications to $k$-median/$k$-means, $k$-line clustering, $j$-subspace approximation, and the integer $(j,k)$-projective clustering problem.

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