Shyam Narayanan

Justin Y. Chen, Talya Eden, Piotr Indyk et al.

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 2018) and (Jiang 2020) applied machine learning techniques in other data stream problems, using a trained oracle that can predict certain properties of the stream elements to improve on prior "classical" algorithms that did not use oracles. In this paper, we explore the power of a "heavy edge" oracle in multiple graph edge streaming models. In the adjacency list model, we present a one-pass triangle counting algorithm improving upon the previous space upper bounds without such an oracle. In the arbitrary order model, we present algorithms for both triangle and four cycle estimation with fewer passes and the same space complexity as in previous algorithms, and we show several of these bounds are optimal. We analyze our algorithms under several noise models, showing that the algorithms perform well even when the oracle errs. Our methodology expands upon prior work on "classical" streaming algorithms, as previous multi-pass and random order streaming algorithms can be seen as special cases of our algorithms, where the first pass or random order was used to implement the heavy edge oracle. Lastly, our experiments demonstrate advantages of the proposed method compared to state-of-the-art streaming algorithms.

Anders Aamand, Justin Y. Chen, Piotr Indyk et al.

Recent work shows that the expressive power of Graph Neural Networks (GNNs) in distinguishing non-isomorphic graphs is exactly the same as that of the Weisfeiler-Lehman (WL) graph test. In particular, they show that the WL test can be simulated by GNNs. However, those simulations involve neural networks for the 'combine' function of size polynomial or even exponential in the number of graph nodes $n$, as well as feature vectors of length linear in $n$. We present an improved simulation of the WL test on GNNs with \emph{exponentially} lower complexity. In particular, the neural network implementing the combine function in each node has only a polylogarithmic number of parameters in $n$, and the feature vectors exchanged by the nodes of GNN consists of only $O(\log n)$ bits. We also give logarithmic lower bounds for the feature vector length and the size of the neural networks, showing the (near)-optimality of our construction.

Vincent Cohen-Addad, Hossein Esfandiari, Vahab Mirrokni et al.

Motivated by data analysis and machine learning applications, we consider the popular high-dimensional Euclidean $k$-median and $k$-means problems. We propose a new primal-dual algorithm, inspired by the classic algorithm of Jain and Vazirani and the recent algorithm of Ahmadian, Norouzi-Fard, Svensson, and Ward. Our algorithm achieves an approximation ratio of $2.406$ and $5.912$ for Euclidean $k$-median and $k$-means, respectively, improving upon the 2.633 approximation ratio of Ahmadian et al. and the 6.1291 approximation ratio of Grandoni, Ostrovsky, Rabani, Schulman, and Venkat. Our techniques involve a much stronger exploitation of the Euclidean metric than previous work on Euclidean clustering. In addition, we introduce a new method of removing excess centers using a variant of independent sets over graphs that we dub a "nested quasi-independent set". In turn, this technique may be of interest for other optimization problems in Euclidean and $\ell_p$ metric spaces.

Anders Aamand, Alexandr Andoni, Justin Y. Chen et al.

We study statistical/computational tradeoffs for the following density estimation problem: given $k$ distributions $v_1, \ldots, v_k$ over a discrete domain of size $n$, and sampling access to a distribution $p$, identify $v_i$ that is "close" to $p$. Our main result is the first data structure that, given a sublinear (in $n$) number of samples from $p$, identifies $v_i$ in time sublinear in $k$. We also give an improved version of the algorithm of Acharya et al. (2018) that reports $v_i$ in time linear in $k$. The experimental evaluation of the latter algorithm shows that it achieves a significant reduction in the number of operations needed to achieve a given accuracy compared to prior work.

Nicholas Schiefer, Justin Y. Chen, Piotr Indyk et al.

An $\varepsilon$-approximate quantile sketch over a stream of $n$ inputs approximates the rank of any query point $q$ - that is, the number of input points less than $q$ - up to an additive error of $\varepsilon n$, generally with some probability of at least $1 - 1/\mathrm{poly}(n)$, while consuming $o(n)$ space. While the celebrated KLL sketch of Karnin, Lang, and Liberty achieves a provably optimal quantile approximation algorithm over worst-case streams, the approximations it achieves in practice are often far from optimal. Indeed, the most commonly used technique in practice is Dunning's t-digest, which often achieves much better approximations than KLL on real-world data but is known to have arbitrarily large errors in the worst case. We apply interpolation techniques to the streaming quantiles problem to attempt to achieve better approximations on real-world data sets than KLL while maintaining similar guarantees in the worst case.

Sinho Chewi, Jaume de Dios Pont, Jerry Li et al.

Log-concave sampling has witnessed remarkable algorithmic advances in recent years, but the corresponding problem of proving lower bounds for this task has remained elusive, with lower bounds previously known only in dimension one. In this work, we establish the following query lower bounds: (1) sampling from strongly log-concave and log-smooth distributions in dimension $d\ge 2$ requires $Ω(\log κ)$ queries, which is sharp in any constant dimension, and (2) sampling from Gaussians in dimension $d$ (hence also from general log-concave and log-smooth distributions in dimension $d$) requires $\widetilde Ω(\min(\sqrtκ\log d, d))$ queries, which is nearly sharp for the class of Gaussians. Here $κ$ denotes the condition number of the target distribution. Our proofs rely upon (1) a multiscale construction inspired by work on the Kakeya conjecture in geometric measure theory, and (2) a novel reduction that demonstrates that block Krylov algorithms are optimal for this problem, as well as connections to lower bound techniques based on Wishart matrices developed in the matrix-vector query literature.

Sitan Chen, Shyam Narayanan

We revisit the well-studied problem of learning a linear combination of $k$ ReLU activations given labeled examples drawn from the standard $d$-dimensional Gaussian measure. Chen et al. [CDG+23] recently gave the first algorithm for this problem to run in $\text{poly}(d,1/\varepsilon)$ time when $k = O(1)$, where $\varepsilon$ is the target error. More precisely, their algorithm runs in time $(d/\varepsilon)^{\mathrm{quasipoly}(k)}$ and learns over multiple stages. Here we show that a much simpler one-stage version of their algorithm suffices, and moreover its runtime is only $(d/\varepsilon)^{O(k^2)}$.

Ainesh Bakshi, Shyam Narayanan

We consider the problem of rank-$1$ low-rank approximation (LRA) in the matrix-vector product model under various Schatten norms: $$ \min_{\|u\|_2=1} \|A (I - u u^\top)\|_{\mathcal{S}_p} , $$ where $\|M\|_{\mathcal{S}_p}$ denotes the $\ell_p$ norm of the singular values of $M$. Given $\varepsilon>0$, our goal is to output a unit vector $v$ such that $$ \|A(I - vv^\top)\|_{\mathcal{S}_p} \leq (1+\varepsilon) \min_{\|u\|_2=1}\|A(I - u u^\top)\|_{\mathcal{S}_p}. $$ Our main result shows that Krylov methods (nearly) achieve the information-theoretically optimal number of matrix-vector products for Spectral ($p=\infty$), Frobenius ($p=2$) and Nuclear ($p=1$) LRA. In particular, for Spectral LRA, we show that any algorithm requires $Ω\left(\log(n)/\varepsilon^{1/2}\right)$ matrix-vector products, exactly matching the upper bound obtained by Krylov methods [MM15, BCW22]. Our lower bound addresses Open Question 1 in [Woo14], providing evidence for the lack of progress on algorithms for Spectral LRA and resolves Open Question 1.2 in [BCW22]. Next, we show that for any fixed constant $p$, i.e. $1\leq p =O(1)$, there is an upper bound of $O\left(\log(1/\varepsilon)/\varepsilon^{1/3}\right)$ matrix-vector products, implying that the complexity does not grow as a function of input size. This improves the $O\left(\log(n/\varepsilon)/\varepsilon^{1/3}\right)$ bound recently obtained in [BCW22], and matches their $Ω\left(1/\varepsilon^{1/3}\right)$ lower bound, to a $\log(1/\varepsilon)$ factor.

Shyam Narayanan

We provide improved differentially private algorithms for identity testing of high-dimensional distributions. Specifically, for $d$-dimensional Gaussian distributions with known covariance $Σ$, we can test whether the distribution comes from $\mathcal{N}(μ^*, Σ)$ for some fixed $μ^*$ or from some $\mathcal{N}(μ, Σ)$ with total variation distance at least $α$ from $\mathcal{N}(μ^*, Σ)$ with $(\varepsilon, 0)$-differential privacy, using only \[\tilde{O}\left(\frac{d^{1/2}}{α^2} + \frac{d^{1/3}}{α^{4/3} \cdot \varepsilon^{2/3}} + \frac{1}{α\cdot \varepsilon}\right)\] samples if the algorithm is allowed to be computationally inefficient, and only \[\tilde{O}\left(\frac{d^{1/2}}{α^2} + \frac{d^{1/4}}{α\cdot \varepsilon}\right)\] samples for a computationally efficient algorithm. We also provide a matching lower bound showing that our computationally inefficient algorithm has optimal sample complexity. We also extend our algorithms to various related problems, including mean testing of Gaussians with bounded but unknown covariance, uniformity testing of product distributions over $\{-1, 1\}^d$, and tolerant testing. Our results improve over the previous best work of Canonne et al.~\cite{CanonneKMUZ20} for both computationally efficient and inefficient algorithms, and even our computationally efficient algorithm matches the optimal \emph{non-private} sample complexity of $O\left(\frac{\sqrt{d}}{α^2}\right)$ in many standard parameter settings. In addition, our results show that, surprisingly, private identity testing of $d$-dimensional Gaussians can be done with fewer samples than private identity testing of discrete distributions over a domain of size $d$ \cite{AcharyaSZ18}, which refutes a conjectured lower bound of~\cite{CanonneKMUZ20}.

Shyam Narayanan

We provide optimal lower bounds for two well-known parameter estimation (also known as statistical estimation) tasks in high dimensions with approximate differential privacy. First, we prove that for any $α\le O(1)$, estimating the covariance of a Gaussian up to spectral error $α$ requires $\tildeΩ\left(\frac{d^{3/2}}{α\varepsilon} + \frac{d}{α^2}\right)$ samples, which is tight up to logarithmic factors. This result improves over previous work which established this for $α\le O\left(\frac{1}{\sqrt{d}}\right)$, and is also simpler than previous work. Next, we prove that estimating the mean of a heavy-tailed distribution with bounded $k$th moments requires $\tildeΩ\left(\frac{d}{α^{k/(k-1)} \varepsilon} + \frac{d}{α^2}\right)$ samples. Previous work for this problem was only able to establish this lower bound against pure differential privacy, or in the special case of $k = 2$. Our techniques follow the method of fingerprinting and are generally quite simple. Our lower bound for heavy-tailed estimation is based on a black-box reduction from privately estimating identity-covariance Gaussians. Our lower bound for covariance estimation utilizes a Bayesian approach to show that, under an Inverse Wishart prior distribution for the covariance matrix, no private estimator can be accurate even in expectation, without sufficiently many samples.

Shyam Narayanan

We give an improved algorithm for learning a quantum Hamiltonian given copies of its Gibbs state, that can succeed at any temperature. Specifically, we improve over the work of Bakshi, Liu, Moitra, and Tang [BLMT24], by reducing the sample complexity and runtime dependence to singly exponential in the inverse-temperature parameter, as opposed to doubly exponential. Our main technical contribution is a new flat polynomial approximation to the exponential function, with significantly lower degree than the flat polynomial approximation used in [BLMT24].

Anders Aamand, Alexandr Andoni, Justin Y. Chen et al.

We study the density estimation problem defined as follows: given $k$ distributions $p_1, \ldots, p_k$ over a discrete domain $[n]$, as well as a collection of samples chosen from a ``query'' distribution $q$ over $[n]$, output $p_i$ that is ``close'' to $q$. Recently~\cite{aamand2023data} gave the first and only known result that achieves sublinear bounds in {\em both} the sampling complexity and the query time while preserving polynomial data structure space. However, their improvement over linear samples and time is only by subpolynomial factors. Our main result is a lower bound showing that, for a broad class of data structures, their bounds cannot be significantly improved. In particular, if an algorithm uses $O(n/\log^c k)$ samples for some constant $c>0$ and polynomial space, then the query time of the data structure must be at least $k^{1-O(1)/\log \log k}$, i.e., close to linear in the number of distributions $k$. This is a novel \emph{statistical-computational} trade-off for density estimation, demonstrating that any data structure must use close to a linear number of samples or take close to linear query time. The lower bound holds even in the realizable case where $q=p_i$ for some $i$, and when the distributions are flat (specifically, all distributions are uniform over half of the domain $[n]$). We also give a simple data structure for our lower bound instance with asymptotically matching upper bounds. Experiments show that the data structure is quite efficient in practice.

Anders Aamand, Maryam Aliakbarpour, Justin Y. Chen et al.

A hypothesis testing algorithm is replicable if, when run on two different samples from the same distribution, it produces the same output with high probability. This notion, defined by by Impagliazzo, Lei, Pitassi, and Sorell [STOC'22], can increase trust in testing procedures and is deeply related to algorithmic stability, generalization, and privacy. We build general tools to prove lower and upper bounds on the sample complexity of replicable testers, unifying and quantitatively improving upon existing results. We identify a set of canonical properties, and prove that any replicable testing algorithm can be modified to satisfy these properties without worsening accuracy or sample complexity. A canonical replicable algorithm computes a deterministic function of its input (i.e., a test statistic) and thresholds against a uniformly random value in $[0,1]$. It is invariant to the order in which the samples are received, and, if the testing problem is ``symmetric,'' then the algorithm is also invariant to the labeling of the domain elements, resolving an open question by Liu and Ye [NeurIPS'24]. We prove new lower bounds for uniformity, identity, and closeness testing by reducing to the case where the replicable algorithm satisfies these canonical properties. We systematize and improve upon a common strategy for replicable algorithm design based on test statistics with known expectation and bounded variance. Our framework allow testers which have been extensively analyzed in the non-replicable setting to be made replicable with minimal overhead. As direct applications of our framework, we obtain constant-factor optimal bounds for coin testing and closeness testing and get replicability for free in a large parameter regime for uniformity testing. We also give state-of-the-art bounds for replicable Gaussian mean testing, and, unlike prior work, our algorithm runs in polynomial time.

Hassan Ashtiani, Mahbod Majid, Shyam Narayanan

We study the problem of learning mixtures of Gaussians with approximate differential privacy. We prove that roughly $kd^2 + k^{1.5} d^{1.75} + k^2 d$ samples suffice to learn a mixture of $k$ arbitrary $d$-dimensional Gaussians up to low total variation distance, with differential privacy. Our work improves over the previous best result [AAL24b] (which required roughly $k^2 d^4$ samples) and is provably optimal when $d$ is much larger than $k^2$. Moreover, we give the first optimal bound for privately learning mixtures of $k$ univariate (i.e., $1$-dimensional) Gaussians. Importantly, we show that the sample complexity for privately learning mixtures of univariate Gaussians is linear in the number of components $k$, whereas the previous best sample complexity [AAL21] was quadratic in $k$. Our algorithms utilize various techniques, including the inverse sensitivity mechanism [AD20b, AD20a, HKMN23], sample compression for distributions [ABDH+20], and methods for bounding volumes of sumsets.

Hossein Esfandiari, Vahab Mirrokni, Shyam Narayanan

In this work, we study high-dimensional mean estimation under user-level differential privacy, and design an $(\varepsilon,δ)$-differentially private mechanism using as few users as possible. In particular, we provide a nearly optimal trade-off between the number of users and the number of samples per user required for private mean estimation, even when the number of users is as low as $O(\frac{1}{\varepsilon}\log\frac{1}δ)$. Interestingly, this bound on the number of \emph{users} is independent of the dimension (though the number of \emph{samples per user} is allowed to depend polynomially on the dimension), unlike the previous work that requires the number of users to depend polynomially on the dimension. This resolves a problem first proposed by Amin et al. Moreover, our mechanism is robust against corruptions in up to $49\%$ of the users. Finally, our results also apply to optimal algorithms for privately learning discrete distributions with few users, answering a question of Liu et al., and a broader range of problems such as stochastic convex optimization and a variant of stochastic gradient descent via a reduction to differentially private mean estimation.

Shyam Narayanan, Sandeep Silwal, Piotr Indyk et al.

Random dimensionality reduction is a versatile tool for speeding up algorithms for high-dimensional problems. We study its application to two clustering problems: the facility location problem, and the single-linkage hierarchical clustering problem, which is equivalent to computing the minimum spanning tree. We show that if we project the input pointset $X$ onto a random $d = O(d_X)$-dimensional subspace (where $d_X$ is the doubling dimension of $X$), then the optimum facility location cost in the projected space approximates the original cost up to a constant factor. We show an analogous statement for minimum spanning tree, but with the dimension $d$ having an extra $\log \log n$ term and the approximation factor being arbitrarily close to $1$. Furthermore, we extend these results to approximating solutions instead of just their costs. Lastly, we provide experimental results to validate the quality of solutions and the speedup due to the dimensionality reduction. Unlike several previous papers studying this approach in the context of $k$-means and $k$-medians, our dimension bound does not depend on the number of clusters but only on the intrinsic dimensionality of $X$.

Hossein Esfandiari, Vahab Mirrokni, Shyam Narayanan

Recently, due to an increasing interest for transparency in artificial intelligence, several methods of explainable machine learning have been developed with the simultaneous goal of accuracy and interpretability by humans. In this paper, we study a recent framework of explainable clustering first suggested by Dasgupta et al.~\cite{dasgupta2020explainable}. Specifically, we focus on the $k$-means and $k$-medians problems and provide nearly tight upper and lower bounds. First, we provide an $O(\log k \log \log k)$-approximation algorithm for explainable $k$-medians, improving on the best known algorithm of $O(k)$~\cite{dasgupta2020explainable} and nearly matching the known $Ω(\log k)$ lower bound~\cite{dasgupta2020explainable}. In addition, in low-dimensional spaces $d \ll \log k$, we show that our algorithm also provides an $O(d \log^2 d)$-approximate solution for explainable $k$-medians. This improves over the best known bound of $O(d \log k)$ for low dimensions~\cite{laber2021explainable}, and is a constant for constant dimensional spaces. To complement this, we show a nearly matching $Ω(d)$ lower bound. Next, we study the $k$-means problem in this context and provide an $O(k \log k)$-approximation algorithm for explainable $k$-means, improving over the $O(k^2)$ bound of Dasgupta et al. and the $O(d k \log k)$ bound of \cite{laber2021explainable}. To complement this we provide an almost tight $Ω(k)$ lower bound, improving over the $Ω(\log k)$ lower bound of Dasgupta et al. Given an approximate solution to the classic $k$-means and $k$-medians, our algorithm for $k$-medians runs in time $O(kd \log^2 k )$ and our algorithm for $k$-means runs in time $ O(k^2 d)$.

Talya Eden, Piotr Indyk, Shyam Narayanan et al.

We consider the problem of estimating the number of distinct elements in a large data set (or, equivalently, the support size of the distribution induced by the data set) from a random sample of its elements. The problem occurs in many applications, including biology, genomics, computer systems and linguistics. A line of research spanning the last decade resulted in algorithms that estimate the support up to $ \pm \varepsilon n$ from a sample of size $O(\log^2(1/\varepsilon) \cdot n/\log n)$, where $n$ is the data set size. Unfortunately, this bound is known to be tight, limiting further improvements to the complexity of this problem. In this paper we consider estimation algorithms augmented with a machine-learning-based predictor that, given any element, returns an estimation of its frequency. We show that if the predictor is correct up to a constant approximation factor, then the sample complexity can be reduced significantly, to \[ \ \log (1/\varepsilon) \cdot n^{1-Θ(1/\log(1/\varepsilon))}. \] We evaluate the proposed algorithms on a collection of data sets, using the neural-network based estimators from {Hsu et al, ICLR'19} as predictors. Our experiments demonstrate substantial (up to 3x) improvements in the estimation accuracy compared to the state of the art algorithm.

Josh Alman, Timothy Chu, Gary Miller et al.

In this paper, we develop a new technique which we call representation theory of the real hyperrectangle, which describes how to compute the eigenvectors and eigenvalues of certain matrices arising from hyperrectangles. We show that these matrices arise naturally when analyzing a number of different algorithmic tasks such as kernel methods, neural network training, natural language processing, and the design of algorithms using the polynomial method. We then use our new technique along with these connections to prove several new structural results in these areas, including: $\bullet$ A function is a positive definite Manhattan kernel if and only if it is a completely monotone function. These kernels are widely used across machine learning; one example is the Laplace kernel which is widely used in machine learning for chemistry. $\bullet$ A function transforms Manhattan distances to Manhattan distances if and only if it is a Bernstein function. This completes the theory of Manhattan to Manhattan metric transforms initiated by Assouad in 1980. $\bullet$ A function applied entry-wise to any square matrix of rank $r$ always results in a matrix of rank $< 2^{r-1}$ if and only if it is a polynomial of sufficiently low degree. This gives a converse to a key lemma used by the polynomial method in algorithm design. Our work includes a sophisticated combination of techniques from different fields, including metric embeddings, the polynomial method, and group representation theory.

Shyam Narayanan, Jelani Nelson

Let $\varepsilon\in(0,1)$ and $X\subset\mathbb R^d$ be arbitrary with $|X|$ having size $n>1$. The Johnson-Lindenstrauss lemma states there exists $f:X\rightarrow\mathbb R^m$ with $m = O(\varepsilon^{-2}\log n)$ such that $$ \forall x\in X\ \forall y\in X, \|x-y\|_2 \le \|f(x)-f(y)\|_2 \le (1+\varepsilon)\|x-y\|_2 . $$ We show that a strictly stronger version of this statement holds, answering one of the main open questions of [MMMR18]: "$\forall y\in X$" in the above statement may be replaced with "$\forall y\in\mathbb R^d$", so that $f$ not only preserves distances within $X$, but also distances to $X$ from the rest of space. Previously this stronger version was only known with the worse bound $m = O(\varepsilon^{-4}\log n)$. Our proof is via a tighter analysis of (a specific instantiation of) the embedding recipe of [MMMR18].