arXiv subjects

Shyam Narayanan

Publications and source records attributed to Shyam Narayanan.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2022

Estimating the Effective Support Size in Constant Query Complexity

Estimating the support size of a distribution is a well-studied problem in statistics. Motivated by the fact that this problem is highly non-robust (as small perturbations in the distributions can drastically affect the support size) and thus hard to estimate, Goldreich [ECCC 2019] studied the query complexity of estimating the $ε$-\emph{effective support size} $\text{Ess}_ε$ of a distribution ${P}$, which is equal to the smallest support size of a distribution that is $ε$-far in total variation distance from ${P}$. In his paper, he shows an algorithm in the dual access setting (where we may both receive random samples and query the sampling probability $p(x)$ for any $x$) for a bicriteria approximation, giving an answer in $[\text{Ess}_{(1+β)ε},(1+γ) \text{Ess}_ε]$ for some values $β, γ> 0$. However, his algorithm has either super-constant query complexity in the support size or super-constant approximation ratio $1+γ= ω(1)$. He then asked if this is necessary, or if it is possible to get a constant-factor approximation in a number of queries independent of the support size. We answer his question by showing that not only is complexity independent of $n$ possible for $γ>0$, but also for $γ=0$, that is, that the bicriteria relaxation is not necessary. Specifically, we show an algorithm with query complexity $O(\frac{1}{β^3 ε^3})$. That is, for any $0 < ε, β< 1$, we output in this complexity a number $\tilde{n} \in [\text{Ess}_{(1+β)ε},\text{Ess}_ε]$. We also show that it is possible to solve the approximate version with approximation ratio $1+γ$ in complexity $O\left(\frac{1}{β^2 ε} + \frac{1}{βεγ^2}\right)$. Our algorithm is very simple, and has $4$ short lines of pseudocode.

cs.DS↗

arXiv2022

Sampling an Edge in Sublinear Time Exactly and Optimally

Sampling edges from a graph in sublinear time is a fundamental problem and a powerful subroutine for designing sublinear-time algorithms. Suppose we have access to the vertices of the graph and know a constant-factor approximation to the number of edges. An algorithm for pointwise $\varepsilon$-approximate edge sampling with complexity $O(n/\sqrt{\varepsilon m})$ has been given by Eden and Rosenbaum [SOSA 2018]. This has been later improved by Tětek and Thorup [STOC 2022] to $O(n \log(\varepsilon^{-1})/\sqrt{m})$. At the same time, $Ω(n/\sqrt{m})$ time is necessary. We close the problem, by giving an algorithm with complexity $O(n/\sqrt{m})$ for the task of sampling an edge exactly uniformly.

cs.DS↗

arXiv2022

Bias Reduction for Sum Estimation

In classical statistics and distribution testing, it is often assumed that elements can be sampled from some distribution $P$, and that when an element $x$ is sampled, the probability $P$ of sampling $x$ is also known. Recent work in distribution testing has shown that many algorithms are robust in the sense that they still produce correct output if the elements are drawn from any distribution $Q$ that is sufficiently close to $P$. This phenomenon raises interesting questions: under what conditions is a "noisy" distribution $Q$ sufficient, and what is the algorithmic cost of coping with this noise? We investigate these questions for the problem of estimating the sum of a multiset of $N$ real values $x_1, \ldots, x_N$. This problem is well-studied in the statistical literature in the case $P = Q$, where the Hansen-Hurwitz estimator is frequently used. We assume that for some known distribution $P$, values are sampled from a distribution $Q$ that is pointwise close to $P$. For every positive integer $k$ we define an estimator $ζ_k$ for $μ= \sum_i x_i$ whose bias is proportional to $γ^k$ (where our $ζ_1$ reduces to the classical Hansen-Hurwitz estimator). As a special case, we show that if $Q$ is pointwise $γ$-close to uniform and all $x_i \in \{0, 1\}$, for any $ε> 0$, we can estimate $μ$ to within additive error $εN$ using $m = Θ({N^{1-\frac{1}{k}} / ε^{2/k}})$ samples, where $k = \left\lceil (\log ε)/(\log γ)\right\rceil$. We show that this sample complexity is essentially optimal. Our bounds show that the sample complexity need not vary uniformly with the desired error parameter $ε$: for some values of $ε$, perturbations in its value have no asymptotic effect on the sample complexity, while for other values, any decrease in its value results in an asymptotically larger sample complexity.

cs.DS↗

arXiv2022

Private High-Dimensional Hypothesis Testing

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}.

cs.DS↗

arXiv2022

Optimal Time-Backlog Tradeoffs for the Variable-Processor Cup Game

The \emph{$ p$-processor cup game} is a classic and widely studied scheduling problem that captures the setting in which a $p$-processor machine must assign tasks to processors over time in order to ensure that no individual task ever falls too far behind. The problem is formalized as a multi-round game in which two players, a filler (who assigns work to tasks) and an emptier (who schedules tasks) compete. The emptier's goal is to minimize backlog, which is the maximum amount of outstanding work for any task. Recently, Kuszmaul and Westover (ITCS, 2021) proposed the \emph{variable-processor cup game}, which considers the same problem, except that the amount of resources available to the players (i.e., the number $p$ of processors) fluctuates between rounds of the game. They showed that this seemingly small modification fundamentally changes the dynamics of the game: whereas the optimal backlog in the fixed $p$-processor game is $Θ(\log n)$, independent of $p$, the optimal backlog in the variable-processor game is $Θ(n)$. The latter result was only known to apply to games with \emph{exponentially many} rounds, however, and it has remained an open question what the optimal tradeoff between time and backlog is for shorter games. This paper establishes a tight trade-off curve between time and backlog in the variable-processor cup game. Importantly, we prove that for a game consisting of $t$ rounds, the optimal backlog is $Θ(n)$ if and only if $t \ge Ω(n^3)$. Our techniques also allow for us to resolve several other open questions concerning how the variable-processor cup game behaves in beyond-worst-case-analysis settings.

cs.DS↗

arXiv2022

Tight and Robust Private Mean Estimation with Few Users

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.

cs.DS↗

arXiv2022

Improved Approximations for Euclidean $k$-means and $k$-median, via Nested Quasi-Independent Sets

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.

cs.DS↗

arXiv2022

All-Pairs Shortest Path Distances with Differential Privacy: Improved Algorithms for Bounded and Unbounded Weights

We revisit the problem of privately releasing the all-pairs shortest path distances of a weighted undirected graph up to low additive error, which was first studied by Sealfon [Sea16]. In this paper, we improve significantly on Sealfon's results, both for arbitrary weighted graphs and for bounded-weight graphs on $n$ nodes. Specifically, we provide an approximate-DP algorithm that outputs all-pairs shortest path distances up to maximum additive error $\tilde{O}(\sqrt{n})$, and a pure-DP algorithm that outputs all pairs shortest path distances up to maximum additive error $\tilde{O}(n^{2/3})$ (where we ignore dependencies on $\varepsilon, δ$). This improves over the previous best result of $\tilde{O}(n)$ additive error for both approximate-DP and pure-DP [Sea16], and partially resolves an open question posed by Sealfon [Sea16, Sea20]. We also show that if the graph is promised to have reasonably bounded weights, one can improve the error further to roughly $n^{\sqrt{2}-1+o(1)}$ in the approximate-DP setting and roughly $n^{(\sqrt{17}-3)/2 + o(1)}$ in the pure-DP setting. Previously, it was only known how to obtain $\tilde{O}(n^{1/2})$ additive error in the approximate-DP setting and $\tilde{O}(n^{2/3})$ additive error in the pure-DP setting for bounded-weight graphs [Sea16].

cs.DS↗

arXiv2022

Triangle and Four Cycle Counting with Predictions in Graph Streams

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.

cs.DS↗

arXiv2022

A 120dB Programmable-Range On-Chip Pulse Generator for Characterizing Ferroelectric Devices

Novel non-volatile memory devices based on ferroelectric thin films represent a promising emerging technology that is ideally suited for neuromorphic applications. The physical switching mechanism in such films is the nucleation and growth of ferroelectric domains. Since this has a strong dependence on both pulse width and voltage amplitude, it is important to use precise pulsing schemes for a thorough characterization of their behaviour. In this work, we present an on-chip 120 dB programmable range pulse generator, that can generate pulse widths ranging from 10ns to 10ms $\pm$2.5% which eliminates the RLC bottleneck in the device characterisation setup. We describe the pulse generator design and show how the pulse width can be tuned with high accuracy, using Digital to Analog converters. Finally, we present experimental results measured from the circuit, fabricated using a standard 180nm CMOS technology.

cs.ET↗

arXiv2022

Stochastic dendrites enable online learning in mixed-signal neuromorphic processing systems

The stringent memory and power constraints required in edge-computing sensory-processing applications have made event-driven neuromorphic systems a promising technology. On-chip online learning provides such systems the ability to learn the statistics of the incoming data and to adapt to their changes. Implementing online learning on event driven-neuromorphic systems requires (i) a spike-based learning algorithm that calculates the weight updates using only local information from streaming data, (ii) mapping these weight updates onto limited bit precision memory and (iii) doing so in a robust manner that does not lead to unnecessary updates as the system is reaching its optimal output. Recent neuroscience studies have shown how dendritic compartments of cortical neurons can solve these problems in biological neural networks. Inspired by these studies we propose spike-based learning circuits to implement stochastic dendritic online learning. The circuits are embedded in a prototype spiking neural network fabricated using a 180nm process. Following an algorithm-circuits co-design approach we present circuits and behavioral simulation results that demonstrate the learning rule features. We validate the proposed method using behavioral simulations of a single-layer network with 4-bit precision weights applied to the MNIST benchmark and demonstrating results that reach accuracy levels above 85%.

cs.ET↗

arXiv2022

Bounds on expected propagation time of probabilistic zero forcing

Probabilistic zero forcing is a coloring game played on a graph where the goal is to color every vertex blue starting with an initial blue vertex set. As long as the graph is connected, if at least one vertex is blue then eventually all of the vertices will be colored blue. The most studied parameter in probabilistic zero forcing is the expected propagation time starting from a given vertex of $G.$ In this paper we improve on upper bounds for the expected propagation time by Geneson and Hogben and Chan et al. in terms of a graph's order and radius. In particular, for a connected graph $G$ of order $n$ and radius $r,$ we prove the bound $\text{ept}(G) = O(r\log(n/r)).$ We also show using Doob's Optional Stopping Theorem and a combinatorial object known as a cornerstone that $\text{ept}(G) \le n/2 + O(\log n).$ Finally, we derive an explicit lower bound $\text{ept}(G)\ge \log_2 \log_2 n.$

math.CO↗

arXiv2021

Stochastic and Worst-Case Generalized Sorting Revisited

The \emph{generalized sorting problem} is a restricted version of standard comparison sorting where we wish to sort $n$ elements but only a subset of pairs are allowed to be compared. Formally, there is some known graph $G = (V, E)$ on the $n$ elements $v_1, \dots, v_n$, and the goal is to determine the true order of the elements using as few comparisons as possible, where all comparisons $(v_i, v_j)$ must be edges in $E$. We are promised that if the true ordering is $x_1 < x_2 < \cdots < x_n$ for $\{x_i\}$ an unknown permutation of the vertices $\{v_i\}$, then $(x_i, x_{i+1}) \in E$ for all $i$: this Hamiltonian path ensures that sorting is actually possible. In this work, we improve the bounds for generalized sorting on both random graphs and worst-case graphs. For Erdős-Renyi random graphs $G(n, p)$ (with the promised Hamiltonian path added to ensure sorting is possible), we provide an algorithm for generalized sorting with an expected $O(n \log (np))$ comparisons, which we prove to be optimal for query complexity. This strongly improves over the best known algorithm of Huang, Kannan, and Khanna (FOCS 2011), which uses $\tilde{O}(\min(n \sqrt{np}, n/p^2))$ comparisons. For arbitrary graphs $G$ with $n$ vertices and $m$ edges (again with the promised Hamiltonian path), we provide an algorithm for generalized sorting with $\tilde{O}(\sqrt{mn})$ comparisons. This improves over the best known algorithm of Huang et al., which uses $\min(m, \tilde{O}(n^{3/2}))$ comparisons.

cs.DS↗

arXiv2021

Frequency Estimation with One-Sided Error

Frequency estimation is one of the most fundamental problems in streaming algorithms. Given a stream $S$ of elements from some universe $U=\{1 \ldots n\}$, the goal is to compute, in a single pass, a short sketch of $S$ so that for any element $i \in U$, one can estimate the number $x_i$ of times $i$ occurs in $S$ based on the sketch alone. Two state of the art solutions to this problems are the Count-Min and Count-Sketch algorithms. The frequency estimator $\tilde{x}$ produced by Count-Min, using $O(1/\varepsilon \cdot \log n)$ dimensions, guarantees that $\|\tilde{x}-x\|_{\infty} \le \varepsilon \|x\|_1$ with high probability, and $\tilde{x} \ge x$ holds deterministically. Also, Count-Min works under the assumption that $x \ge 0$. On the other hand, Count-Sketch, using $O(1/\varepsilon^2 \cdot \log n)$ dimensions, guarantees that $\|\tilde{x}-x\|_{\infty} \le \varepsilon \|x\|_2$ with high probability. A natural question is whether it is possible to design the best of both worlds sketching method, with error guarantees depending on the $\ell_2$ norm and space comparable to Count-Sketch, but (like Count-Min) also has the no-underestimation property. Our main set of results shows that the answer to the above question is negative. We show this in two incomparable computational models: linear sketching and streaming algorithms. We also study the complementary problem, where the sketch is required to not over-estimate, i.e., $\tilde{x} \le x$ should hold always.

cs.DS↗

arXiv2021

Metric Transforms and Low Rank Matrices via Representation Theory of the Real Hyperrectangle

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.

cs.CG↗

arXiv2021

Almost Tight Approximation Algorithms for Explainable Clustering

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)$.

cs.LG↗

arXiv2021

Randomized Dimensionality Reduction for Facility Location and Single-Linkage Clustering

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 $X$.

cs.DS↗