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We study high-dimensional sparse estimation under three natural constraints: communication constraints, local privacy constraints, and linear measurements (compressive sensing). Without sparsity assumptions, it has been established that interactivity cannot improve the minimax rates of estimation under these information constraints. The question of whether interactivity helps with natural inference tasks has been a topic of active research. We settle this question in the affirmative for the prototypical problems of high-dimensional sparse mean estimation and compressive sensing, by demonstrating a gap between interactive and noninteractive protocols. We further establish that the gap increases when we have more structured sparsity: for block sparsity this gap can be as large as polynomial in the dimensionality. Thus, the more structured the sparsity is, the greater is the advantage of interaction. Proving the lower bounds requires a careful breaking of a sum of correlated random variables into independent components using Baranyai's theorem on decomposition of hypergraphs, which might be of independent interest.

We consider density estimation for Besov spaces when each sample is quantized to only a limited number of bits. We provide a noninteractive adaptive estimator that exploits the sparsity of wavelet bases, along with a simulate-and-infer technique from parametric estimation under communication constraints. We show that our estimator is nearly rate-optimal by deriving minimax lower bounds that hold even when interactive protocols are allowed. Interestingly, while our wavelet-based estimator is almost rate-optimal for Sobolev spaces as well, it is unclear whether the standard Fourier basis, which arise naturally for those spaces, can be used to achieve the same performance.

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.

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., Comment: To appear in the Special Issue on Privacy and Security of Information Systems of the IEEE Journal on Selected Areas in Information Theory (JSAIT), 2021. Journal version of the AISTATS'19 paper "Test without Trust: Optimal Locally Private Distribution Testing" (arXiv:1808.02174), which it extends and supersedes

We consider distributed parameter estimation using interactive protocols subject to local information constraints such as bandwidth limitations, local differential privacy, and restricted measurements. We provide a unified framework enabling us to derive a variety of (tight) minimax lower bounds for different parametric families of distributions, both continuous and discrete, under any $\ell_p$ loss. Our lower bound framework is versatile and yields "plug-and-play" bounds that are widely applicable to a large range of estimation problems, and, for the prototypical case of the Gaussian family, circumvents limitations of previous techniques. In particular, our approach recovers bounds obtained using data processing inequalities and Cram\'er--Rao bounds, two other alternative approaches for proving lower bounds in our setting of interest. Further, for the families considered, we complement our lower bounds with matching upper bounds., Comment: Streamline some statements; add the low-privacy corollary (high value of the privacy parameter) for Corollary 1, along with the implications for the applications considered; add the upper bound for the low-privacy regimes for Bernoulli (Theorem 3) and Gaussian (Theorem 4); slightly improve Lemma 5 by relaxing the independence assumption

We study the role of interactivity in distributed statistical inference under information constraints, e.g., communication constraints and local differential privacy. We focus on the tasks of goodness-of-fit testing and estimation of discrete distributions. From prior work, these tasks are well understood under noninteractive protocols. Extending these approaches directly for interactive protocols is difficult due to correlations that can build due to interactivity; in fact, gaps can be found in prior claims of tight bounds of distribution estimation using interactive protocols. We propose a new approach to handle this correlation and establish a unified method to establish lower bounds for both tasks. As an application, we obtain optimal bounds for both estimation and testing under local differential privacy and communication constraints. We also provide an example of a natural testing problem where interactivity helps., Comment: Modifying the proof and statement of Proposition 24 to address an issue in the variance analysis

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.

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., Comment: To appear in IEEE Transactions on Information Theory. An abridged version of this work appeared in the 2019 International Conference on Machine Learning (ICML)

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., Comment: To appear in IEEE Transactions on Information Theory

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.

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 $\Theta(k^2/2^\ell\epsilon^2)$ for learning the unknown distribution to total variation distance $\epsilon$. For the prototypical testing problem of identity testing, simulate-and-infer works with $O(k^{3/2}/2^\ell\epsilon^2)$ 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 $O(k/\sqrt{2^\ell}\epsilon^2)$ 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 $\chi^2$ distance of product distributions due to communication constraints and may be of independent interest., Comment: This work is superseded by the more recent "Inference under Information Constraints II: Communication Constraints and Shared Randomness" (arXiv:1905.08302), by the same authors

It was recently shown that estimating the Shannon entropy $H({\rm p})$ of a discrete $k$-symbol distribution ${\rm p}$ requires $\Theta(k/\log k)$ samples, a number that grows near-linearly in the support size. In many applications $H({\rm p})$ can be replaced by the more general R\'enyi entropy of order $\alpha$, $H_\alpha({\rm p})$. We determine the number of samples needed to estimate $H_\alpha({\rm p})$ for all $\alpha$, showing that $\alpha < 1$ requires a super-linear, roughly $k^{1/\alpha}$ samples, noninteger $\alpha>1$ requires a near-linear $k$ samples, but, perhaps surprisingly, integer $\alpha>1$ requires only $\Theta(k^{1-1/\alpha})$ samples. Furthermore, developing on a recently established connection between polynomial approximation and estimation of additive functions of the form $\sum_{x} f({\rm p}_x)$, we reduce the sample complexity for noninteger values of $\alpha$ by a factor of $\log k$ compared to the empirical estimator. The estimators achieving these bounds are simple and run in time linear in the number of samples. Our lower bounds provide explicit constructions of distributions with different R\'enyi entropies that are hard to distinguish.