arXiv subjects

Shyam Narayanan

Publications and source records attributed to Shyam Narayanan.

At least 19 recordsLinked to original sources

arXiv2025

On the Structure of Replicable Hypothesis Testers

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.

cs.DS↗

arXiv2024

Near-Optimal Trace Reconstruction for Mildly Separated Strings

In the trace reconstruction problem our goal is to learn an unknown string $x\in \{0,1\}^n$ given independent traces of $x$. A trace is obtained by independently deleting each bit of $x$ with some probability $δ$ and concatenating the remaining bits. It is a major open question whether the trace reconstruction problem can be solved with a polynomial number of traces when the deletion probability $δ$ is constant. The best known upper bound and lower bounds are respectively $\exp(\tilde O(n^{1/5}))$ and $\tilde Ω(n^{3/2})$ both by Chase [Cha21b,Cha21a]. Our main result is that if the string $x$ is mildly separated, meaning that the number of zeros between any two ones in $x$ is at least polylog$n$, and if $δ$ is a sufficiently small constant, then the trace reconstruction problem can be solved with $O(n \log n)$ traces and in polynomial time.

cs.DS↗

arXiv2024

Sample-Efficient Private Learning of Mixtures of Gaussians

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.

cs.LG↗

arXiv2024

Statistical-Computational Trade-offs for Density Estimation

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.

cs.DS↗

arXiv2024

Instance-Optimality in I/O-Efficient Sampling and Sequential Estimation

Suppose we have a memory storing $0$s and $1$s and we want to estimate the frequency of $1$s by sampling. We want to do this I/O-efficiently, exploiting that each read gives a block of $B$ bits at unit cost; not just one bit. If the input consists of uniform blocks: either all 1s or all 0s, then sampling a whole block at a time does not reduce the number of samples needed for estimation. On the other hand, if bits are randomly permuted, then getting a block of $B$ bits is as good as getting $B$ independent bit samples. However, we do not want to make any such assumptions on the input. Instead, our goal is to have an algorithm with instance-dependent performance guarantees which stops sampling blocks as soon as we know that we have a probabilistically reliable estimate. We prove our algorithms to be instance-optimal among algorithms oblivious to the order of the blocks, which we argue is the strongest form of instance optimality we can hope for. We also present similar results for I/O-efficiently estimating mean with both additive and multiplicative error, estimating histograms, quantiles, as well as the empirical cumulative distribution function. We obtain our above results on I/O-efficient sampling by reducing to corresponding problems in the so-called sequential estimation. In this setting, one samples from an unknown distribution until one can provide an estimate with some desired error probability. We then provide non-parametric instance-optimal results for several fundamental problems: mean and quantile estimation, as well as learning mixture distributions with respect to $\ell_\infty$ and the so-called Kolmogorov-Smirnov distance.

cs.DS↗

arXiv2024

Improved algorithms for learning quantum Hamiltonians, via flat polynomials

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].

quant-ph↗

arXiv2024

Robustness Implies Privacy in Statistical Estimation

We study the relationship between adversarial robustness and differential privacy in high-dimensional algorithmic statistics. We give the first black-box reduction from privacy to robustness which can produce private estimators with optimal tradeoffs among sample complexity, accuracy, and privacy for a wide range of fundamental high-dimensional parameter estimation problems, including mean and covariance estimation. We show that this reduction can be implemented in polynomial time in some important special cases. In particular, using nearly-optimal polynomial-time robust estimators for the mean and covariance of high-dimensional Gaussians which are based on the Sum-of-Squares method, we design the first polynomial-time private estimators for these problems with nearly-optimal samples-accuracy-privacy tradeoffs. Our algorithms are also robust to a nearly optimal fraction of adversarially-corrupted samples.

cs.DS↗

arXiv2024

Better and Simpler Lower Bounds for Differentially Private Statistical Estimation

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.

math.ST↗

arXiv2023

Query lower bounds for log-concave sampling

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.

math.ST↗

arXiv2023

SPAIC: A sub-$μ$W/Channel, 16-Channel General-Purpose Event-Based Analog Front-End with Dual-Mode Encoders

Low-power event-based analog front-ends (AFE) are a crucial component required to build efficient end-to-end neuromorphic processing systems for edge computing. Although several neuromorphic chips have been developed for implementing spiking neural networks (SNNs) and solving a wide range of sensory processing tasks, there are only a few general-purpose analog front-end devices that can be used to convert analog sensory signals into spikes and interfaced to neuromorphic processors. In this work, we present a novel, highly configurable analog front-end chip, denoted as SPAIC (signal-to-spike converter for analog AI computation), that offers a general-purpose dual-mode analog signal-to-spike encoding with delta modulation and pulse frequency modulation, with tunable frequency bands. The ASIC is designed in a 180 nm process. It supports and encodes a wide variety of signals spanning 4 orders of magnitude in frequency, and provides an event-based output that is compatible with existing neuromorphic processors. We validated the ASIC for its functions and present initial silicon measurement results characterizing the basic building blocks of the chip.

cs.AR↗

arXiv2023

Massively Parallel Algorithms for High-Dimensional Euclidean Minimum Spanning Tree

We study the classic Euclidean Minimum Spanning Tree (MST) problem in the Massively Parallel Computation (MPC) model. Given a set $X \subset \mathbb{R}^d$ of $n$ points, the goal is to produce a spanning tree for $X$ with weight within a small factor of optimal. Euclidean MST is one of the most fundamental hierarchical geometric clustering algorithms, and with the proliferation of enormous high-dimensional data sets, such as massive transformer-based embeddings, there is now a critical demand for efficient distributed algorithms to cluster such data sets. In low-dimensional space, where $d = O(1)$, Andoni, Nikolov, Onak, and Yaroslavtsev [STOC '14] gave a constant round MPC algorithm that obtains a high accuracy $(1+ε)$-approximate solution. However, the situation is much more challenging for high-dimensional spaces: the best-known algorithm to obtain a constant approximation requires $O(\log n)$ rounds. Recently Chen, Jayaram, Levi, and Waingarten [STOC '22] gave a $\tilde{O}(\log n)$ approximation algorithm in a constant number of rounds based on embeddings into tree metrics. However, to date, no known algorithm achieves both a constant number of rounds and approximation. In this paper, we make strong progress on this front by giving a constant factor approximation in $\tilde{O}(\log \log n)$ rounds of the MPC model. In contrast to tree-embedding-based approaches, which necessarily must pay $Ω(\log n)$-distortion, our algorithm is based on a new combination of graph-based distributed MST algorithms and geometric space partitions. Additionally, although the approximate MST we return can have a large depth, we show that it can be modified to obtain a $\tilde{O}(\log \log n)$-round constant factor approximation to the Euclidean Traveling Salesman Problem (TSP) in the MPC model. Previously, only a $O(\log n)$ round was known for the problem.

cs.DS↗

arXiv2023

A faster and simpler algorithm for learning shallow networks

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 $(d/\varepsilon)^{O(k^2)}$.

cs.LG↗

arXiv2023

The Full Landscape of Robust Mean Testing: Sharp Separations between Oblivious and Adaptive Contamination

We consider the question of Gaussian mean testing, a fundamental task in high-dimensional distribution testing and signal processing, subject to adversarial corruptions of the samples. We focus on the relative power of different adversaries, and show that, in contrast to the common wisdom in robust statistics, there exists a strict separation between adaptive adversaries (strong contamination) and oblivious ones (weak contamination) for this task. Specifically, we resolve both the information-theoretic and computational landscapes for robust mean testing. In the exponential-time setting, we establish the tight sample complexity of testing $\mathcal{N}(0,I)$ against $\mathcal{N}(αv, I)$, where $\|v\|_2 = 1$, with an $\varepsilon$-fraction of adversarial corruptions, to be \[ \tildeΘ\!\left(\max\left(\frac{\sqrt{d}}{α^2}, \frac{d\varepsilon^3}{α^4},\min\left(\frac{d^{2/3}\varepsilon^{2/3}}{α^{8/3}}, \frac{d \varepsilon}{α^2}\right)\right) \right) \,, \] while the complexity against adaptive adversaries is \[ \tildeΘ\!\left(\max\left(\frac{\sqrt{d}}{α^2}, \frac{d\varepsilon^2}{α^4} \right)\right) \,, \] which is strictly worse for a large range of vanishing $\varepsilon,α$. To the best of our knowledge, ours is the first separation in sample complexity between the strong and weak contamination models. In the polynomial-time setting, we close a gap in the literature by providing a polynomial-time algorithm against adaptive adversaries achieving the above sample complexity $\tildeΘ(\max({\sqrt{d}}/{α^2}, {d\varepsilon^2}/{α^4} ))$, and a low-degree lower bound (which complements an existing reduction from planted clique) suggesting that all efficient algorithms require this many samples, even in the oblivious-adversary setting.

cs.DS↗

arXiv2023

Improved Diversity Maximization Algorithms for Matching and Pseudoforest

In this work we consider the diversity maximization problem, where given a data set $X$ of $n$ elements, and a parameter $k$, the goal is to pick a subset of $X$ of size $k$ maximizing a certain diversity measure. [CH01] defined a variety of diversity measures based on pairwise distances between the points. A constant factor approximation algorithm was known for all those diversity measures except ``remote-matching'', where only an $O(\log k)$ approximation was known. In this work we present an $O(1)$ approximation for this remaining notion. Further, we consider these notions from the perpective of composable coresets. [IMMM14] provided composable coresets with a constant factor approximation for all but ``remote-pseudoforest'' and ``remote-matching'', which again they only obtained a $O(\log k)$ approximation. Here we also close the gap up to constants and present a constant factor composable coreset algorithm for these two notions. For remote-matching, our coreset has size only $O(k)$, and for remote-pseudoforest, our coreset has size $O(k^{1+\varepsilon})$ for any $\varepsilon > 0$, for an $O(1/\varepsilon)$-approximate coreset.

cs.DS↗

arXiv2023

Data Structures for Density Estimation

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.

cs.DS↗

arXiv2023

Learned Interpolation for Better Streaming Quantile Approximation with Worst-Case Guarantees

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.

cs.DS↗

arXiv2023

Krylov Methods are (nearly) Optimal for Low-Rank Approximation

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.

cs.DS↗

arXiv2022

Exponentially Improving the Complexity of Simulating the Weisfeiler-Lehman Test with Graph Neural Networks

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.

cs.LG↗