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 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: (1) 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]. (2) We prove new lower bounds for uniformity, identity, and closeness testing by reducing to the case where the replicable algorithm satisfies these canonical properties. (3) 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 combined with existing analyses of non-replicable testers, we obtain constant-factor optimal bounds for coin testing and closeness testing and get replicability for free for uniformity testing in a large parameter regime. As replicable coin testing can be used as a black-box to turn any tester into a replicable tester, our results directly imply improved replicable sampling bounds for a myriad of applications beyond the ones specifically studied in this paper. (4) We also give new algorithms and lower bounds for replicable Gaussian mean testing, improving over the previous best sample complexity of Bun, Gaboardi, Hopkins, Impagliazzo, Lei, Pitassi, Sivakumar, and Sorrell [STOC’23].

Abstract. 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 [Formula: see text] against [Formula: see text], where [Formula: see text], with an [Formula: see text]-fraction of oblivious adversarial corruptions, to be [Formula: see text], while the complexity against adaptive adversarial corruptions is [Formula: see text], which is strictly worse for a large range of vanishing [Formula: see text]. 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 [Formula: see text], 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.

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 (2024) 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 Bakshi et al.

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.

Long-term monitoring of biomedical signals is essential for modern data-driven clinical practices. This demonstration will present a low-power event-based pipeline for encoding and processing analog streaming signals in real time. This setup consists of an Analog Front-End (AFE) signal encoder directly integrated with a DYnamic Neuromorphic Asynchronous Processor (DYNAP-SE2). This system can produce a stream of spikes directly from analog sensors, for input to a Spiking Neural Network (SNN), implemented on the processor. By encoding correlated inputs and amplifying underlying partial synchronization patterns, we demonstrate the utilization of this setup in identifying evolving epileptic networks and detecting seizures.

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$\alpha \le O(1)$, estimating the covariance of a Gaussian up to spectral error$\alpha $requires$\tilde {\Omega }\left ({{\frac {d^{3/2}}{\alpha \varepsilon } + \frac {d}{\alpha ^{2}}}}\right)$samples, which is tight up to logarithmic factors. This result improves over previous work which established this for$\alpha \le O\left ({{\frac {1}{\sqrt {d}}}}\right)$, and is also simpler than previous work. Next, we prove that estimating the mean of a heavy-tailed distribution with bounded kth moments requires$\tilde {\Omega }\left ({{\frac {d}{\alpha ^{k/(k-1)} \varepsilon } + \frac {d}{\alpha ^{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.

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. recently gave the first algorithm for this problem to run in $\mathrm{poly}(d,1/\epsilon)$ time when $k = O(1)$, where $\epsilon$ is the target error. More precisely, their algorithm runs in time $(d/\epsilon)^{\mathrm{quasipoly}(k)}$ 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 $(d k/\epsilon)^{O(k^2)}$.

Suppose we have a memory storing 0s and 1s and we want to estimate the frequency of 1s 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 Os, 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$B$indendent 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. Sequential estimation has been considered extensively in statistics over the past century. However, the focus has been mostly on parametric estimation, making stringent assumptions on the distribution of the input, and thus not useful for our reduction. In this paper, we make no assumptions on the input distribution (apart from its support being a bounded set). Namely, we 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. All our algorithms are simple, natural, and practical, and some are even known from other contexts, e.g., from statistics in the parameterized setting. The main technical difficulty is in analyzing them and proving that they are instance optimal.

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 Aamand et al. gave the first and only known result that achieves sublinear bounds in 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 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.

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 (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 learning mixtures of univariate Gaussians is linear in the number of components $k$, whereas the previous best sample complexity was quadratic in $k$. Our algorithms utilize various techniques, including the inverse sensitivity mechanism, sample compression for distributions, and methods for bounding volumes of sumsets.

We introduce a new linear-algebraic tool based on group representation theory, and use it to address three key problems in machine learning. 1. Past researchers have proposed fast attention algorithms for LLMs by approximating or replace softmax attention with other functions, such as low-degree polynomials. The key property of these functions is that, when applied entry-wise to the matrix $QK^{\top}$, the result is a low rank matrix when $Q$ and $K$ are $n \times d$ matrices and $n \gg d$. This suggests a natural question: what are all functions $f$ with this property? If other $f$ exist and are quickly computable, they can be used in place of softmax for fast subquadratic attention algorithms. It was previously known that low-degree polynomials have this property. We prove that low-degree polynomials are the only piecewise continuous functions with this property. This suggests that the low-rank fast attention only works for functions approximable by polynomials. Our work gives a converse to the polynomial method in algorithm design. 2. We prove the first full classification of all positive definite kernels that are functions of Manhattan or $\ell_1$ distance. Our work generalizes an existing theorem at the heart of all kernel methods in machine learning: the classification of all positive definite kernels that are functions of Euclidean distance. 3. The key problem in metric transforms, a mathematical theory used in geometry and machine learning, asks what functions transform pairwise distances in semi-metric space $M$ to semi-metric space $N$ for specified $M$ and $N$. We provide the first full classification of functions that transform Manhattan distances to Manhattan distances. Our work generalizes the foundational work of Schoenberg, which fully classifies functions that transform Euclidean to Euclidean distances. We additionally prove results about stable-rank preserving functions that are potentially useful in algorithmic design, and more. Our core new tool is called the representation theory of the hyperrectangle.

We study the classic Euclidean Minimum Spanning Tree (MST) problem in the Massively Parallel Computation (MPC) model. Given a set X ⊂ ℝd 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.

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 \(\Omega (\log \kappa)\) 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 \(\widetilde{\Omega }(\min (\sqrt \kappa \log d, d))\) queries, which is nearly sharp for the class of Gaussians. Here, \(\kappa\) 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.

In classical statistics and distribution testing, it is often assumed that elements can be sampled exactly from some distribution 𝒫, and that when an element x is sampled, the probability 𝒫(x) of sampling x is also known. In this setting, 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? In this paper, we investigate these questions for the problem of estimating the sum of a multiset of N real values x_1, …, x_N. This problem is well-studied in the statistical literature in the case 𝒫 = 𝒬, where the Hansen-Hurwitz estimator [Annals of Mathematical Statistics, 1943] is frequently used. We assume that for some (known) distribution 𝒫, values are sampled from a distribution 𝒬 that is pointwise close to 𝒫. That is, there is a parameter γ < 1 such that for all x_i, (1 - γ) 𝒫(i) ≤ 𝒬(i) ≤ (1 + γ) 𝒫(i). For every positive integer k we define an estimator ζ_k for μ = ∑_i x_i whose bias is proportional to γ^k (where our ζ₁ 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 ∈ {0, 1}, for any ε > 0, we can estimate μ to within additive error ε N using m = Θ(N^{1-1/k}/ε^{2/k}) samples, where k = ⌈lg ε/lg γ⌉. We then show that this sample complexity is essentially optimal. Interestingly, our upper and lower 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.

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.

We consider the problem of rank-1 low-rank approximation (LRA) in the matrix-vector product model under various Schatten norms:\begin{equation*}\min _{\|u\|_{2}=1}\left\|A\left(I-u u^{\top}\right)\right\|_{\mathcal{S}_{p}}\end{equation*}where $\|M\|_{\mathcal{S}_{p}}$ denotes the $\ell_{p}$ norm of the singular values of M. Given $\varepsilon\gt 0$, our goal is to output a unit vector v such that \begin{equation*}\left\|A\left(I-v v^{\top}\right)\right\|_{\mathcal{S}_{p}} \leqslant\left(1+\varepsilon\right) \min _{\|u\|_{2}=1}\left\|A\left(I-u u^{\top}\right)\right\|_{\mathcal{S}_{p}}\end{equation*}Our main result shows that Krylov methods (nearly) achieve the information-theoretically optimal1number 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 $\Omega\left(\log (n) / \varepsilon^{1 / 2}\right)$ matrix-vector products, exactly matching the upper bound obtained by Krylov methods [40]. Our lower bound addresses Open Question 1 in [59], providing evidence for the lack of progress on algorithms for Spectral LRA and resolves Open Question 1.2 in [5]. Next, we show that for any fixed constant p, i.e. $1 \leqslant p=O(1)$, 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 [5], and matches their $\Omega\left(1 / \varepsilon^{1 / 3}\right)$ lower bound, to a $\log (1 / \varepsilon)$ factor.1For Spectral LRA, the upper and lower bounds match up to a fixed universal constant. For Frobenius and Nuclear LRA, they match up to a $\log (1 / \varepsilon)$ factor.

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 $\mathcal{N}(\alpha v, I)$, where $\|v\|_{2}=1$, with an $\varepsilon$-fraction of adversarial corruptions, to be $\tilde{\Theta}\left(\max \left(\frac{\sqrt{d}}{\alpha^{2}}, \frac{d \varepsilon^{3}}{\alpha^{4}}, \min \left(\frac{d^{2 / 3} \varepsilon^{2 / 3}}{\alpha^{8 / 3}}, \frac{d \varepsilon}{\alpha^{2}}\right)\right)\right)$ while the complexity against adaptive adversaries is $\tilde{\Theta}\left(\max \left(\frac{\sqrt{d}}{\alpha^{2}}, \frac{d \varepsilon^{2}}{\alpha^{4}}\right)\right)$ which is strictly worse for a large range of vanishing $\varepsilon, \alpha$. 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 $\tilde{\Theta}\left(\max \left(\sqrt{d} / \alpha^{2}, d \varepsilon^{2} / \alpha^{4}\right)\right)$, 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.

Log-concave sampling has witnessed remarkable algorithmic advances in recent years, but the corresponding problem of proving $\lt$bold$\gt$lower bounds$\lt$/bold$\gt$ 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 $d \geq 2$ requires $\Omega(\log \kappa)$ queries, which is sharp in any constant dimension, and (2) sampling from Gaussians in dimension d (hence also from general logconcave and log-smooth distributions in dimension d) requires $\widetilde{\Omega}(\min (\sqrt{\kappa} \log d, d))$ queries, which is nearly sharp for the class of Gaussians. Here $\kappa$ 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.

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.

Range counting (e.g., counting the number of data points falling into a given query ball) under differential privacy has been studied extensively. However, the current algorithms for this problem are subject to the following dichotomy. One class of algorithms suffers from an additive error that is a fixed polynomial in the number of points. Another class of algorithms allows for polylogarithmic additive error, but the error grows exponentially in the dimension. To achieve the latter, the problem is relaxed to allow a “fuzzy” definition of the range boundary, e.g., a count of the points in a ball of radius $r$ might also include points in a ball of radius $cr$ for some $c>1$. In this paper we present an efficient algorithm that offers a sweet spot between these two classes. The algorithm has an additive error that is an arbitrary small power of the data set size, depending on how fuzzy the range boundary is, as well as a small ($1+o(1)$) multiplicative error. Crucially, the amount of noise added has no dependence on the dimension. Our algorithm introduces a variant of Locality-Sensitive Hashing, utilizing it in a novel manner.

In this paper, we initiate the study of Euclidean clustering with Distance-based privacy. Distance-based privacy is motivated by the fact that it is often only needed to protect the privacy of exact, rather than approximate, locations. We provide constant-approximate algorithms for $k$-means and $k$-median clustering, with additive error depending only on the attacker's precision bound $\rho$, rather than the radius $\Lambda$ of the space. In addition, we empirically demonstrate that our algorithm performs significantly better than previous differentially private clustering algorithms, as well as naive distance-based private clustering baselines.

We study the problem of releasing the weights of all-pairs shortest paths in a weighted undirected graph with differential privacy (DP). In this setting, the underlying graph is fixed and two graphs are neighbors if their edge weights differ by at most 1 in the ℓ1-distance. We give an algorithm with additive error Õ(n2/3/ε) in the ε-DP case and an algorithm with additive error in the (ε,δ)-DP case, where n denotes the number of vertices. This positively answers a question of Sealfon [Sea16, Sea20], who asked whether a o(n)- error algorithm exists. We also show that an additive error of Ω(n1/6) is necessary for any sufficiently small ε,δ > 0. Furthermore, we show that if the graph is promised to have reasonably bounded weights, one can improve the error further to roughly in the ε-DP case and roughly in the (ε, δ)-DP case. Previously, it was only known how to obtain Õ(n2/3/ε1/3) additive error in the ε-DP case and additive error in the (ε,δ)-DP case for bounded-weight graphs [Sea16]. Finally, we consider a relaxation where a multiplicative approximation is allowed. We show that, with a multiplicative approximation factor k, the additive error can be reduced to Õ(n1/2+O(1/k)/ε) in the ε-DP case and Õ(n1/3+O(1/k)/ε) in the (ε,δ)-DP case.

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.

We provide improved differentially private algorithms for identity testing of high-dimensional distributions. Specifically, for $d$-dimensional Gaussian distributions with known covariance $\Sigma$, we can test whether the distribution comes from $\mathcal{N}(\mu^*, \Sigma)$ for some fixed $\mu^*$ or from some $\mathcal{N}(\mu, \Sigma)$ with total variation distance at least $\alpha$ from $\mathcal{N}(\mu^*, \Sigma)$ with $(\varepsilon, 0)$-differential privacy, using only \[\tilde{O}\left(\frac{d^{1/2}}{\alpha^2} + \frac{d^{1/3}}{\alpha^{4/3} \cdot \varepsilon^{2/3}} + \frac{1}{\alpha \cdot \varepsilon}\right)\]{samples} if the algorithm is allowed to be computationally inefficient, and only \[\tilde{O}\left(\frac{d^{1/2}}{\alpha^2} + \frac{d^{1/4}}{\alpha \cdot \varepsilon}\right)\]{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. (2020) for both computationally efficient and inefficient algorithms, and even our computationally efficient algorithm matches the optimal \emph{non-private} sample complexity of $O\left(\frac{\sqrt{d}}{\alpha^2}\right)$ 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 $d$ (Acharya et al., 2018), which refutes a conjectured lower bound of Canonne et al. (2020).

The 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 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 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. We show that, for a game consisting of t rounds, the optimal backlog is Θ (b (t)) where b(t) = t (if t ≤ log n) t^{1/3} log^{2/3} ({n^3}/t + 1) (if log n < t ≤ n^3) n (if n ^ 3 < t). An important consequence is that the optimal backlog is Θ(n) if and only if t ≥ Ω(n³). 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.

In this work, we study high-dimensional mean estimation under user-level differential privacy, and design an $(\varepsilon,\delta)$-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 $O(\frac{1}{\varepsilon}\log\frac{1}{\delta})$. Interestingly, this bound on the number of users is independent of the dimension (though the number of 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. (2019). 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. (2020), 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.

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%.

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 behavior. In this work, we present an on-chip 120 dB programmable range pulse generator, that can generate pulse widths ranging from 10 ns to 10 ms ± 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 180 nm CMOS technology.

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 {\em exponentially} lower complexity. In particular, the neural network implementing the combine function in each node has only $\mathrm{polylog}(n)$ parameters, 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.

We study the differentially private (DP) $k$-means and $k$-median clustering problems of $n$ points in $d$-dimensional Euclidean space in the massively parallel computation (MPC) model. We provide two near-optimal algorithms where the near-optimality is in three aspects: they both achieve (1). $O(1)$ parallel computation rounds, (2). near-linear in $n$ and polynomial in $k$ total computational work (i.e., near-linear running time when $n$ is a sufficient polynomial in $k$), (3). $O(1)$ relative approximation and $\text{poly}(k, d)$ additive error. Note that $\Omega(1)$ relative approximation is provably necessary even for any polynomial-time non-private algorithm, and $\Omega(k)$ additive error is a provable lower bound for any polynomial-time DP $k$-means/median algorithm. Our two algorithms provide a tradeoff between the relative approximation and the additive error: the first has $O(1)$ relative approximation and $\sim (k^{2.5} + k^{1.01} \sqrt{d})$ additive error, and the second one achieves $(1+\gamma)$ relative approximation to the optimal non-private algorithm for an arbitrary small constant $\gamma>0$ and with $\text{poly}(k, d)$ additive error for a larger polynomial dependence on $k$ and $d$. To achieve our result, we develop a general framework which partitions the data and reduces the DP clustering problem for the entire dataset to the DP clustering problem for each part. To control the blow-up of the additive error introduced by each part, we develop a novel charging argument which might be of independent interest.

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. [11]. Specifically, we focus on the k-means and k-median problems and provide nearly tight upper and lower bounds. First, we provide an O(log k log log k)-approximation algorithm for explainable k-median, improving on the best known algorithm of O(k) [11] and nearly matching the known Ω(log k) lower bound [11]. In addition, in low-dimensional spaces d ≪ log k, we show that our algorithm also provides an O(d log2 d)-approximate solution for explainable k-median. This improves over the best known bound of O(d log k) for low dimensions [19], 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(k2) bound of Dasgupta et al. and the O(dk log k) bound of [19]. 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-median, our algorithm for k-median runs in time O(kd log2 k) and our algorithm for k-means runs in time O(k2 d).

Frequency estimation, also known as the Point Query problem, is one of the most fundamental problems in streaming algorithms. Given a stream S of elements from some universe U = {1 … n}, the goal is to compute, in a single pass, a short “sketch” of S so that for any element i ∊ U, one can estimate the number xi of times i occurs in S based on the sketch alone. Two state of the art solutions to this problems are Count-Min and Count-Sketch algorithms. They are based on linear sketches, which means that the data elements can be deleted as well as inserted and sketches for two different streams can be combined via addition. However, the guarantees offered by Count-Min and Count-Sketch are incomparable. The frequency estimator x produced by Count-Min sketch, using O(1/∊·log n) dimensions, guarantees that with high probability, and holds deterministically. Also, Count-Min works under the assumption that x ≥ 0. On the other hand, Count-Sketch, using O(1/∊2 · log n) dimensions, guarantees that 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 ℓ2 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. Specifically, we show that: Any linear sketch satisfying the ℓp norm error guarantee with probability at least 2/3 and having the no-underestimation property must be of dimension of at least Ω(n1–1/p/∊), even if the sketched vectors are non-negative. This bound is tight, as we also give a linear sketch of dimension O(n1–1/p/∊) satisfying these properties. Any streaming algorithm satisfying the ℓp norm error guarantee with probability at least 2/3 and having the no-underestimation property must use at least Ω(n1–1/p/∊) bits. This holds even for algorithms that only allow insertions and make any constant number of passes over the stream. This bound is tight up to a logarithmic factor. We also study the complementary problem, where the sketch is required to not over-estimate, i.e., should hold always. We show that any linear sketch satisfying this property and having the ℓp error guarantee with probability at least 2/3 must be of dimension at least Ω(n1–1/p/∊). We also show that this bound is tight up to polylogarithmic factors, by providing an appropriate linear sketch.

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.

The 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$v_{1, \ldots, v_{n}}$, 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$O(n\ \text{lg}(np))$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.

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 {\em solutions} instead of just their {\em 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 $X$.

Trace reconstruction is the problem of learning an unknown string $x$ from independent traces of $x$, where traces are generated by independently deleting each bit of $x$ with some deletion probability $q$. In this paper, we initiate the study of Circular trace reconstruction, where the unknown string $x$ is circular and traces are now rotated by a random cyclic shift. Trace reconstruction is related to many computational biology problems studying DNA, which is a primary motivation for this problem as well, as many types of DNA are known to be circular. Our main results are as follows. First, we prove that we can reconstruct arbitrary circular strings of length $n$ using $\exp\big(\tilde{O}(n^{1/3})\big)$ traces for any constant deletion probability $q$, as long as $n$ is prime or the product of two primes. For $n$ of this form, this nearly matches what was the best known bound of $\exp\big(O(n^{1/3})\big)$ for standard trace reconstruction when this paper was initially released. We note, however, that Chase very recently improved the standard trace reconstruction bound to $\exp\big(\tilde{O}(n^{1/5})\big)$. Next, we prove that we can reconstruct random circular strings with high probability using $n^{O(1)}$ traces for any constant deletion probability $q$. Finally, we prove a lower bound of $\tildeΩ(n^3)$ traces for arbitrary circular strings, which is greater than the best known lower bound of $\tildeΩ(n^{3/2})$ in standard trace reconstruction.

Recently, there has been significant work studying distribution testing under the Conditional Sampling model. In this model, a query specifies a subset S of the domain, and the output received is a sample drawn from the distribution conditioned on being in S. In this paper, we improve query complexity bounds for several classic distribution testing problems in this model. First, we prove that tolerant uniformity testing in the conditional sampling model can be solved using Õ(∊–2) queries, which is optimal and improves upon the Õ(∊–20)-query algorithm of Canonne et al. [CRS15]. This bound even holds under a restricted version of the conditional sampling model called the Pair Conditional Sampling model. Next, we prove that tolerant identity testing in the conditional sampling model can be solved in Õ(∊–4) queries, which is the first known bound independent of the support size of the distribution for this problem. Next, we use our algorithm for tolerant uniformity testing to get an Õ(∊–4)-query algorithm for monotonicity testing in the conditional sampling model, improving on the Õ(∊–22)-query algorithm of Canonne [Can15]. Finally, we study (non-tolerant) identity testing under the pair conditional sampling model, and provide a tight bound of for the query complexity, where the domain of the distribution has size N. This improves upon both the known upper and lower bounds in [CRS15].

The population recovery problem asks one to recover an unknown distribution over n-bit strings given access to independent noisy samples of strings drawn from the distribution. Recently, Ban et al. [BCF+ 19] studied the problem where the noise is induced through the deletion channel. This problem generalizes the famous trace reconstruction problem, where one wishes to learn a single string under the deletion channel. Ban et al. showed how to learn ℓ-sparse distributions over strings using exp (n1/2 · (log n)O(ℓ)) samples. In this work, we learn the distribution using only exp (Õ(n1/3) · ℓ2) samples, by developing a higher-moment analog of the algorithms of [DOS17a, NP17], which solve trace reconstruction in exp (Õ(n1/3)) samples. We also give the first algorithm with a runtime subexponential in n, solving population recovery in exp (Õ(n1/3) · ℓ3) samples and time. Notably, our dependence on n nearly matches the upper bound of [DOS17a, NP17] when ℓ = O(1), and we reduce the dependence on ℓ from doubly to singly exponential. Therefore, we are able to learn large mixtures of strings: while Ban et al.'s algorithm can only learn a mixture of O(log n/ log log n) strings with a subexponential number of samples, we are able to learn a mixture of no(1) strings in exp (n1/3+o(1)) samples and time.

A family of problems that have been studied in the context of various streaming algorithms are generalizations of the fact that the expected maximum distance of a 4-wise independent random walk on a line over n steps is O(sqrt{n}). For small values of k, there exist k-wise independent random walks that can be stored in much less space than storing n random bits, so these properties are often useful for lowering space bounds. In this paper, we show that for all of these examples, 4-wise independence is required by demonstrating a pairwise independent random walk with steps uniform in +/- 1 and expected maximum distance Omega(sqrt{n} lg n) from the origin. We also show that this bound is tight for the first and second moment, i.e. the expected maximum square distance of a 2-wise independent random walk is always O(n lg^2 n). Also, for any even k >= 4, we show that the kth moment of the maximum distance of any k-wise independent random walk is O(n^{k/2}). The previous two results generalize to random walks tracking insertion-only streams, and provide higher moment bounds than currently known. We also prove a generalization of Kolmogorov’s maximal inequality by showing an asymptotically equivalent statement that requires only 4-wise independent random variables with bounded second moments, which also generalizes a result of Błasiok.

Let ε∈(0,1) and X⊂d be arbitrary with |X| having size n>1. The Johnson-Lindenstrauss lemma states there exists f:X→m with m = O(ε−2logn) such that ∀ x∈ X ∀ y∈ X, ||x−y||2 ≤ ||f(x)−f(y)||2 ≤ (1+ε)||x−y||2 . We show that a strictly stronger version of this statement holds, answering one of the main open questions posed by Mahabadi et al. in STOC 2018: “∀ y∈ X” in the above statement may be replaced with “∀ y∈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(ε−4logn). Our proof is via a tighter analysis of (a specific instantiation of) the embedding recipe of Mahabadi et al.

The 1-center clustering with outliers problem asks about identifying a prototypical robust statistic that approximates the location of a cluster of points. Given some constant 0 < alpha < 1 and n points such that alpha n of them are in some (unknown) ball of radius r, the goal is to compute a ball of radius O(r) that also contains alpha n points. This problem can be formulated with the points in a normed vector space such as R^d or in a general metric space. The problem has a simple randomized solution: a randomly selected point is a correct solution with constant probability, and its correctness can be verified in linear time. However, the deterministic complexity of this problem was not known. In this paper, for any L^p vector space, we show an O(nd)-time solution with a ball of radius O(r) for a fixed alpha > 1/2, and for any normed vector space, we show an O(nd)-time solution with a ball of radius O(r) when alpha > 1/2 as well as an O(nd log^{(k)}(n))-time solution with a ball of radius O(r) for all alpha > 0, k in N, where log^{(k)}(n) represents the kth iterated logarithm, assuming distance computation and vector space operations take O(d) time. For an arbitrary metric space, we show for any C in N an O(n^{1+1/C})-time solution that finds a ball of radius 2Cr, assuming distance computation between any pair of points takes O(1)-time, and show that for any alpha, C, an O(n^{1+1/C})-time solution that finds a ball of radius ((2C-3)(1-alpha)-1)r cannot exist.