arXiv subjects

Clément L. Canonne

Publications and source records attributed to Clément L. Canonne.

At least 55 records · Page 3Linked to original sources

arXiv2021

Identity testing under label mismatch

Testing whether the observed data conforms to a purported model (probability distribution) is a basic and fundamental statistical task, and one that is by now well understood. However, the standard formulation, identity testing, fails to capture many settings of interest; in this work, we focus on one such natural setting, identity testing under promise of permutation. In this setting, the unknown distribution is assumed to be equal to the purported one, up to a relabeling (permutation) of the model: however, due to a systematic error in the reporting of the data, this relabeling may not be the identity. The goal is then to test identity under this assumption: equivalently, whether this systematic labeling error led to a data distribution statistically far from the reference model.

math.ST↗

arXiv2021

Information-constrained optimization: can adaptive processing of gradients help?

We revisit first-order optimization under local information constraints such as local privacy, gradient quantization, and computational constraints limiting access to a few coordinates of the gradient. In this setting, the optimization algorithm is not allowed to directly access the complete output of the gradient oracle, but only gets limited information about it subject to the local information constraints. We study the role of adaptivity in processing the gradient output to obtain this limited information from it.We consider optimization for both convex and strongly convex functions and obtain tight or nearly tight lower bounds for the convergence rate, when adaptive gradient processing is allowed. Prior work was restricted to convex functions and allowed only nonadaptive processing of gradients. For both of these function classes and for the three information constraints mentioned above, our lower bound implies that adaptive processing of gradients cannot outperform nonadaptive processing in most regimes of interest. We complement these results by exhibiting a natural optimization problem under information constraints for which adaptive processing of gradient strictly outperforms nonadaptive processing.

math.OC↗

arXiv2021

Random Restrictions of High-Dimensional Distributions and Uniformity Testing with Subcube Conditioning

We give a nearly-optimal algorithm for testing uniformity of distributions supported on $\{-1,1\}^n$, which makes $\tilde O (\sqrt{n}/\varepsilon^2)$ queries to a subcube conditional sampling oracle (Bhattacharyya and Chakraborty (2018)). The key technical component is a natural notion of random restriction for distributions on $\{-1,1\}^n$, and a quantitative analysis of how such a restriction affects the mean vector of the distribution. Along the way, we consider the problem of mean testing with independent samples and provide a nearly-optimal algorithm.

cs.DS↗

arXiv2021

Inference under Information Constraints III: Local Privacy Constraints

We study goodness-of-fit and independence testing of discrete distributions in a setting where samples are distributed across multiple users. The users wish to preserve the privacy of their data while enabling a central server to perform the tests. Under the notion of local differential privacy, we propose simple, sample-optimal, and communication-efficient protocols for these two questions in the noninteractive setting, where in addition users may or may not share a common random seed. In particular, we show that the availability of shared (public) randomness greatly reduces the sample complexity. Underlying our public-coin protocols are privacy-preserving mappings which, when applied to the samples, minimally contract the distance between their respective probability distributions.

cs.DS↗

arXiv2020

Inference under Information Constraints I: Lower Bounds from Chi-Square Contraction

Multiple players are each given one independent sample, about which they can only provide limited information to a central referee. Each player is allowed to describe its observed sample to the referee using a channel from a family of channels $\mathcal{W}$, which can be instantiated to capture both the communication- and privacy-constrained settings and beyond. The referee uses the messages from players to solve an inference problem for the unknown distribution that generated the samples. We derive lower bounds for sample complexity of learning and testing discrete distributions in this information-constrained setting. Underlying our bounds is a characterization of the contraction in chi-square distances between the observed distributions of the samples when information constraints are placed. This contraction is captured in a local neighborhood in terms of chi-square and decoupled chi-square fluctuations of a given channel, two quantities we introduce. The former captures the average distance between distributions of channel output for two product distributions on the input, and the latter for a product distribution and a mixture of product distribution on the input. Our bounds are tight for both public- and private-coin protocols. Interestingly, the sample complexity of testing is order-wise higher when restricted to private-coin protocols.

cs.DS↗

arXiv2020

Inference under Information Constraints II: Communication Constraints and Shared Randomness

A central server needs to perform statistical inference based on samples that are distributed over multiple users who can each send a message of limited length to the center. We study problems of distribution learning and identity testing in this distributed inference setting and examine the role of shared randomness as a resource. We propose a general-purpose simulate-and-infer strategy that uses only private-coin communication protocols and is sample-optimal for distribution learning. This general strategy turns out to be sample-optimal even for distribution testing among private-coin protocols. Interestingly, we propose a public-coin protocol that outperforms simulate-and-infer for distribution testing and is, in fact, sample-optimal. Underlying our public-coin protocol is a random hash that when applied to the samples minimally contracts the chi-squared distance of their distribution to the uniform distribution.

cs.DS↗

arXiv2020

Testing Data Binnings

Motivated by the question of data quantization and "binning," we revisit the problem of identity testing of discrete probability distributions. Identity testing (a.k.a. one-sample testing), a fundamental and by now well-understood problem in distribution testing, asks, given a reference distribution (model) $\mathbf{q}$ and samples from an unknown distribution $\mathbf{p}$, both over $[n]=\{1,2,\dots,n\}$, whether $\mathbf{p}$ equals $\mathbf{q}$, or is significantly different from it. In this paper, we introduce the related question of 'identity up to binning,' where the reference distribution $\mathbf{q}$ is over $k \ll n$ elements: the question is then whether there exists a suitable binning of the domain $[n]$ into $k$ intervals such that, once "binned," $\mathbf{p}$ is equal to $\mathbf{q}$. We provide nearly tight upper and lower bounds on the sample complexity of this new question, showing both a quantitative and qualitative difference with the vanilla identity testing one, and answering an open question of Canonne (2019). Finally, we discuss several extensions and related research directions.

cs.DS↗

arXiv2020

A short note on learning discrete distributions

The goal of this short note is to provide simple proofs for the "folklore facts" on the sample complexity of learning a discrete probability distribution over a known domain of size $k$ to various distances $\varepsilon$, with error probability $δ$.

math.ST↗

arXiv2019

Finding monotone patterns in sublinear time

We study the problem of finding monotone subsequences in an array from the viewpoint of sublinear algorithms. For fixed $k \in \mathbb{N}$ and $\varepsilon > 0$, we show that the non-adaptive query complexity of finding a length-$k$ monotone subsequence of $f \colon [n] \to \mathbb{R}$, assuming that $f$ is $\varepsilon$-far from free of such subsequences, is $Θ((\log n)^{\lfloor \log_2 k \rfloor})$. Prior to our work, the best algorithm for this problem, due to Newman, Rabinovich, Rajendraprasad, and Sohler (2017), made $(\log n)^{O(k^2)}$ non-adaptive queries; and the only lower bound known, of $Ω(\log n)$ queries for the case $k = 2$, followed from that on testing monotonicity due to Ergün, Kannan, Kumar, Rubinfeld, and Viswanathan (2000) and Fischer (2004).

cs.DS↗

arXiv2019

Domain Compression and its Application to Randomness-Optimal Distributed Goodness-of-Fit

We study goodness-of-fit of discrete distributions in the distributed setting, where samples are divided between multiple users who can only release a limited amount of information about their samples due to various information constraints. Recently, a subset of the authors showed that having access to a common random seed (i.e., shared randomness) leads to a significant reduction in the sample complexity of this problem. In this work, we provide a complete understanding of the interplay between the amount of shared randomness available, the stringency of information constraints, and the sample complexity of the testing problem by characterizing a tight trade-off between these three parameters. We provide a general distributed goodness-of-fit protocol that as a function of the amount of shared randomness interpolates smoothly between the private- and public-coin sample complexities. We complement our upper bound with a general framework to prove lower bounds on the sample complexity of this testing problems under limited shared randomness. Finally, we instantiate our bounds for the two archetypal information constraints of communication and local privacy, and show that our sample complexity bounds are optimal as a function of all the parameters of the problem, including the amount of shared randomness. A key component of our upper bounds is a new primitive of domain compression, a tool that allows us to map distributions to a much smaller domain size while preserving their pairwise distances, using a limited amount of randomness.

cs.DS↗

arXiv2019

Learning from satisfying assignments under continuous distributions

What kinds of functions are learnable from their satisfying assignments? Motivated by this simple question, we extend the framework of De, Diakonikolas, and Servedio [DDS15], which studied the learnability of probability distributions over $\{0,1\}^n$ defined by the set of satisfying assignments to "low-complexity" Boolean functions, to Boolean-valued functions defined over continuous domains. In our learning scenario there is a known "background distribution" $\mathcal{D}$ over $\mathbb{R}^n$ (such as a known normal distribution or a known log-concave distribution) and the learner is given i.i.d. samples drawn from a target distribution $\mathcal{D}_f$, where $\mathcal{D}_f$ is $\mathcal{D}$ restricted to the satisfying assignments of an unknown low-complexity Boolean-valued function $f$. The problem is to learn an approximation $\mathcal{D}'$ of the target distribution $\mathcal{D}_f$ which has small error as measured in total variation distance. We give a range of efficient algorithms and hardness results for this problem, focusing on the case when $f$ is a low-degree polynomial threshold function (PTF). When the background distribution $\mathcal{D}$ is log-concave, we show that this learning problem is efficiently solvable for degree-1 PTFs (i.e.,~linear threshold functions) but not for degree-2 PTFs. In contrast, when $\mathcal{D}$ is a normal distribution, we show that this learning problem is efficiently solvable for degree-2 PTFs but not for degree-4 PTFs. Our hardness results rely on standard assumptions about secure signature schemes.

cs.DS↗

arXiv2019

Distributed Simulation and Distributed Inference

Independent samples from an unknown probability distribution $\bf p$ on a domain of size $k$ are distributed across $n$ players, with each player holding one sample. Each player can communicate $\ell$ bits to a central referee in a simultaneous message passing model of communication to help the referee infer a property of the unknown $\bf p$. What is the least number of players for inference required in the communication-starved setting of $\ell<\log k$? We begin by exploring a general "simulate-and-infer" strategy for such inference problems where the center simulates the desired number of samples from the unknown distribution and applies standard inference algorithms for the collocated setting. Our first result shows that for $\ell<\log k$ perfect simulation of even a single sample is not possible. Nonetheless, we present a Las Vegas algorithm that simulates a single sample from the unknown distribution using $O(k/2^\ell)$ samples in expectation. As an immediate corollary, we get that simulate-and-infer attains the optimal sample complexity of $Θ(k^2/2^\ellε^2)$ for learning the unknown distribution to total variation distance $ε$. For the prototypical testing problem of identity testing, simulate-and-infer works with $O(k^{3/2}/2^\ellε^2)$ samples, a requirement that seems to be inherent for all communication protocols not using any additional resources. Interestingly, we can break this barrier using public coins. Specifically, we exhibit a public-coin communication protocol that performs identity testing using $O(k/\sqrt{2^\ell}ε^2)$ samples. Furthermore, we show that this is optimal up to constant factors. Our theoretically sample-optimal protocol is easy to implement in practice. Our proof of lower bound entails showing a contraction in $χ^2$ distance of product distributions due to communication constraints and may be of independent interest.

cs.DS↗

arXiv2019

The Structure of Optimal Private Tests for Simple Hypotheses

Hypothesis testing plays a central role in statistical inference, and is used in many settings where privacy concerns are paramount. This work answers a basic question about privately testing simple hypotheses: given two distributions $P$ and $Q$, and a privacy level $\varepsilon$, how many i.i.d. samples are needed to distinguish $P$ from $Q$ subject to $\varepsilon$-differential privacy, and what sort of tests have optimal sample complexity? Specifically, we characterize this sample complexity up to constant factors in terms of the structure of $P$ and $Q$ and the privacy level $\varepsilon$, and show that this sample complexity is achieved by a certain randomized and clamped variant of the log-likelihood ratio test. Our result is an analogue of the classical Neyman-Pearson lemma in the setting of private hypothesis testing. We also give an application of our result to the private change-point detection. Our characterization applies more generally to hypothesis tests satisfying essentially any notion of algorithmic stability, which is known to imply strong generalization bounds in adaptive data analysis, and thus our results have applications even when privacy is not a primary concern.

cs.DS↗

arXiv2018

A Chasm Between Identity and Equivalence Testing with Conditional Queries

A recent model for property testing of probability distributions (Chakraborty et al., ITCS 2013, Canonne et al., SICOMP 2015) enables tremendous savings in the sample complexity of testing algorithms, by allowing them to condition the sampling on subsets of the domain. In particular, Canonne, Ron, and Servedio (SICOMP 2015) showed that, in this setting, testing identity of an unknown distribution $D$ (whether $D=D^\ast$ for an explicitly known $D^\ast$) can be done with a constant number of queries, independent of the support size $n$ -- in contrast to the required $Ω(\sqrt{n})$ in the standard sampling model. It was unclear whether the same stark contrast exists for the case of testing equivalence, where both distributions are unknown. While Canonne et al. established a $\mathrm{poly}(\log n)$-query upper bound for equivalence testing, very recently brought down to $\tilde O(\log\log n)$ by Falahatgar et al. (COLT 2015), whether a dependence on the domain size $n$ is necessary was still open, and explicitly posed by Fischer at the Bertinoro Workshop on Sublinear Algorithms (2014). We show that any testing algorithm for equivalence must make $Ω(\sqrt{\log\log n})$ queries in the conditional sampling model. This demonstrates a gap between identity and equivalence testing, absent in the standard sampling model (where both problems have sampling complexity $n^{Θ(1)}$). We also obtain results on the query complexity of uniformity testing and support-size estimation with conditional samples. We answer a question of Chakraborty et al. (ITCS 2013) showing that non-adaptive uniformity testing indeed requires $Ω(\log n)$ queries in the conditional model. For the related problem of support-size estimation, we provide both adaptive and non-adaptive algorithms, with query complexities $\mathrm{poly}(\log\log n)$ and $\mathrm{poly}(\log n)$, respectively.

cs.DS↗

arXiv2018

Test without Trust: Optimal Locally Private Distribution Testing

We study the problem of distribution testing when the samples can only be accessed using a locally differentially private mechanism and focus on two representative testing questions of identity (goodness-of-fit) and independence testing for discrete distributions. We are concerned with two settings: First, when we insist on using an already deployed, general-purpose locally differentially private mechanism such as the popular RAPPOR or the recently introduced Hadamard Response for collecting data, and must build our tests based on the data collected via this mechanism; and second, when no such restriction is imposed, and we can design a bespoke mechanism specifically for testing. For the latter purpose, we introduce the Randomized Aggregated Private Testing Optimal Response (RAPTOR) mechanism which is remarkably simple and requires only one bit of communication per sample. We propose tests based on these mechanisms and analyze their sample complexities. Each proposed test can be implemented efficiently. In each case (barring one), we complement our performance bounds for algorithms with information-theoretic lower bounds and establish sample optimality of our proposed algorithm. A peculiar feature that emerges is that our sample-optimal algorithm based on RAPTOR uses public-coins, and any test based on RAPPOR or Hadamard Response, which are both private-coin mechanisms, requires significantly more samples.

cs.DS↗

arXiv2018

Testing Conditional Independence of Discrete Distributions

We study the problem of testing \emph{conditional independence} for discrete distributions. Specifically, given samples from a discrete random variable $(X, Y, Z)$ on domain $[\ell_1]\times[\ell_2] \times [n]$, we want to distinguish, with probability at least $2/3$, between the case that $X$ and $Y$ are conditionally independent given $Z$ from the case that $(X, Y, Z)$ is $ε$-far, in $\ell_1$-distance, from every distribution that has this property. Conditional independence is a concept of central importance in probability and statistics with a range of applications in various scientific domains. As such, the statistical task of testing conditional independence has been extensively studied in various forms within the statistics and econometrics communities for nearly a century. Perhaps surprisingly, this problem has not been previously considered in the framework of distribution property testing and in particular no tester with sublinear sample complexity is known, even for the important special case that the domains of $X$ and $Y$ are binary. The main algorithmic result of this work is the first conditional independence tester with {\em sublinear} sample complexity for discrete distributions over $[\ell_1]\times[\ell_2] \times [n]$. To complement our upper bounds, we prove information-theoretic lower bounds establishing that the sample complexity of our algorithm is optimal, up to constant factors, for a number of settings. Specifically, for the prototypical setting when $\ell_1, \ell_2 = O(1)$, we show that the sample complexity of testing conditional independence (upper bound and matching lower bound) is \[ Θ\left({\max\left(n^{1/2}/ε^2,\min\left(n^{7/8}/ε,n^{6/7}/ε^{8/7}\right)\right)}\right)\,. \]

cs.DS↗

arXiv2017

Improved Bounds for Testing Forbidden Order Patterns

A sequence $f\colon\{1,\dots,n\}\to\mathbb{R}$ contains a permutation $π$ of length $k$ if there exist $i_1<\dots<i_k$ such that, for all $x,y$, $f(i_x)<f(i_y)$ if and only if $π(x)<π(y)$; otherwise, $f$ is said to be $π$-free. In this work, we consider the problem of testing for $π$-freeness with one-sided error, continuing the investigation of [Newman et al., SODA'17]. We demonstrate a surprising behavior for non-adaptive tests with one-sided error: While a trivial sampling-based approach yields an $\varepsilon$-test for $π$-freeness making $Θ(\varepsilon^{-1/k} n^{1-1/k})$ queries, our lower bounds imply that this is almost optimal for most permutations! Specifically, for most permutations $π$ of length $k$, any non-adaptive one-sided $\varepsilon$-test requires $\varepsilon^{-1/(k-Θ(1))}n^{1-1/(k-Θ(1))}$ queries; furthermore, the permutations that are hardest to test require $Θ(\varepsilon^{-1/(k-1)}n^{1-1/(k-1)})$ queries, which is tight in $n$ and $\varepsilon$. Additionally, we show two hierarchical behaviors here. First, for any $k$ and $l\leq k-1$, there exists some $π$ of length $k$ that requires $\tildeΘ_{\varepsilon}(n^{1-1/l})$ non-adaptive queries. Second, we show an adaptivity hierarchy for $π=(1,3,2)$ by proving upper and lower bounds for (one- and two-sided) testing of $π$-freeness with $r$ rounds of adaptivity. The results answer open questions of Newman et al. and [Canonne and Gur, CCC'17].

cs.DS↗

arXiv2017

Generalized Uniformity Testing

In this work, we revisit the problem of uniformity testing of discrete probability distributions. A fundamental problem in distribution testing, testing uniformity over a known domain has been addressed over a significant line of works, and is by now fully understood. The complexity of deciding whether an unknown distribution is uniform over its unknown (and arbitrary) support, however, is much less clear. Yet, this task arises as soon as no prior knowledge on the domain is available, or whenever the samples originate from an unknown and unstructured universe. In this work, we introduce and study this generalized uniformity testing question, and establish nearly tight upper and lower bound showing that -- quite surprisingly -- its sample complexity significantly differs from the known-domain case. Moreover, our algorithm is intrinsically adaptive, in contrast to the overwhelming majority of known distribution testing algorithms.

cs.DS↗