Jayadev Acharya

Jimmy Z. Di, Jack Douglas, Jayadev Acharya et al.

We introduce camouflaged data poisoning attacks, a new attack vector that arises in the context of machine unlearning and other settings when model retraining may be induced. An adversary first adds a few carefully crafted points to the training dataset such that the impact on the model's predictions is minimal. The adversary subsequently triggers a request to remove a subset of the introduced points at which point the attack is unleashed and the model's predictions are negatively affected. In particular, we consider clean-label targeted attacks (in which the goal is to cause the model to misclassify a specific test point) on datasets including CIFAR-10, Imagenette, and Imagewoof. This attack is realized by constructing camouflage datapoints that mask the effect of a poisoned dataset.

Jayadev Acharya, Yuhan Liu, Ziteng Sun

We study discrete distribution estimation under user-level local differential privacy (LDP). In user-level $\varepsilon$-LDP, each user has $m\ge1$ samples and the privacy of all $m$ samples must be preserved simultaneously. We resolve the following dilemma: While on the one hand having more samples per user should provide more information about the underlying distribution, on the other hand, guaranteeing the privacy of all $m$ samples should make the estimation task more difficult. We obtain tight bounds for this problem under almost all parameter regimes. Perhaps surprisingly, we show that in suitable parameter regimes, having $m$ samples per user is equivalent to having $m$ times more users, each with only one sample. Our results demonstrate interesting phase transitions for $m$ and the privacy parameter $\varepsilon$ in the estimation risk. Finally, connecting with recent results on shuffled DP, we show that combined with random shuffling, our algorithm leads to optimal error guarantees (up to logarithmic factors) under the central model of user-level DP in certain parameter regimes. We provide several simulations to verify our theoretical findings.

Jayadev Acharya, Clément L. Canonne, Ziteng Sun et al.

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.

Jayadev Acharya, Ayush Jain, Gautam Kamath et al.

We study the problem of robustly estimating the parameter $p$ of an Erdős-Rényi random graph on $n$ nodes, where a $γ$ fraction of nodes may be adversarially corrupted. After showing the deficiencies of canonical estimators, we design a computationally-efficient spectral algorithm which estimates $p$ up to accuracy $\tilde O(\sqrt{p(1-p)}/n + γ\sqrt{p(1-p)} /\sqrt{n}+ γ/n)$ for $γ< 1/60$. Furthermore, we give an inefficient algorithm with similar accuracy for all $γ<1/2$, the information-theoretic limit. Finally, we prove a nearly-matching statistical lower bound, showing that the error of our algorithms is optimal up to logarithmic factors.

Sourbh Bhadane, Aaron B. Wagner, Jayadev Acharya

We consider a linear autoencoder in which the latent variables are quantized, or corrupted by noise, and the constraint is Schur-concave in the set of latent variances. Although finding the optimal encoder/decoder pair for this setup is a nonconvex optimization problem, we show that decomposing the source into its principal components is optimal. If the constraint is strictly Schur-concave and the empirical covariance matrix has only simple eigenvalues, then any optimal encoder/decoder must decompose the source in this way. As one application, we consider a strictly Schur-concave constraint that estimates the number of bits needed to represent the latent variables under fixed-rate encoding, a setup that we call \emph{Principal Bit Analysis (PBA)}. This yields a practical, general-purpose, fixed-rate compressor that outperforms existing algorithms. As a second application, we show that a prototypical autoencoder-based variable-rate compressor is guaranteed to decompose the source into its principal components.

Jayadev Acharya, Ziteng Sun, Huanyu Zhang

We study robust testing and estimation of discrete distributions in the strong contamination model. We consider both the "centralized setting" and the "distributed setting with information constraints" including communication and local privacy (LDP) constraints. Our technique relates the strength of manipulation attacks to the earth-mover distance using Hamming distance as the metric between messages(samples) from the users. In the centralized setting, we provide optimal error bounds for both learning and testing. Our lower bounds under local information constraints build on the recent lower bound methods in distributed inference. In the communication constrained setting, we develop novel algorithms based on random hashing and an $\ell_1/\ell_1$ isometry.

Ayush Sekhari, Jayadev Acharya, Gautam Kamath et al.

We study the problem of unlearning datapoints from a learnt model. The learner first receives a dataset $S$ drawn i.i.d. from an unknown distribution, and outputs a model $\widehat{w}$ that performs well on unseen samples from the same distribution. However, at some point in the future, any training datapoint $z \in S$ can request to be unlearned, thus prompting the learner to modify its output model while still ensuring the same accuracy guarantees. We initiate a rigorous study of generalization in machine unlearning, where the goal is to perform well on previously unseen datapoints. Our focus is on both computational and storage complexity. For the setting of convex losses, we provide an unlearning algorithm that can unlearn up to $O(n/d^{1/4})$ samples, where $d$ is the problem dimension. In comparison, in general, differentially private learning (which implies unlearning) only guarantees deletion of $O(n/d^{1/2})$ samples. This demonstrates a novel separation between differential privacy and machine unlearning.

Jayadev Acharya, Clément L. Canonne, Cody Freitag et al.

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.

Jayadev Acharya, Peter Kairouz, Yuhan Liu et al.

We consider the problem of estimating sparse discrete distributions under local differential privacy (LDP) and communication constraints. We characterize the sample complexity for sparse estimation under LDP constraints up to a constant factor and the sample complexity under communication constraints up to a logarithmic factor. Our upper bounds under LDP are based on the Hadamard Response, a private coin scheme that requires only one bit of communication per user. Under communication constraints, we propose public coin schemes based on random hashing functions. Our tight lower bounds are based on the recently proposed method of chi squared contractions.

Jayadev Acharya, Clément L. Canonne, Ziteng Sun et al.

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 Cramér--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.

Jayadev Acharya, Clément L. Canonne, Yuhan Liu et al.

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.

Jayadev Acharya, Ziteng Sun, Huanyu Zhang

Le Cam's method, Fano's inequality, and Assouad's lemma are three widely used techniques to prove lower bounds for statistical estimation tasks. We propose their analogues under central differential privacy. Our results are simple, easy to apply and we use them to establish sample complexity bounds in several estimation tasks. We establish the optimal sample complexity of discrete distribution estimation under total variation distance and $\ell_2$ distance. We also provide lower bounds for several other distribution classes, including product distributions and Gaussian mixtures that are tight up to logarithmic factors. The technical component of our paper relates coupling between distributions to the sample complexity of estimation under differential privacy.

Jayadev Acharya, Sourbh Bhadane, Piotr Indyk et al.

We consider the task of estimating the entropy of $k$-ary distributions from samples in the streaming model, where space is limited. Our main contribution is an algorithm that requires $O\left(\frac{k \log (1/\varepsilon)^2}{\varepsilon^3}\right)$ samples and a constant $O(1)$ memory words of space and outputs a $\pm\varepsilon$ estimate of $H(p)$. Without space limitations, the sample complexity has been established as $S(k,\varepsilon)=Θ\left(\frac k{\varepsilon\log k}+\frac{\log^2 k}{\varepsilon^2}\right)$, which is sub-linear in the domain size $k$, and the current algorithms that achieve optimal sample complexity also require nearly-linear space in $k$. Our algorithm partitions $[0,1]$ into intervals and estimates the entropy contribution of probability values in each interval. The intervals are designed to trade off the bias and variance of these estimates.

Jayadev Acharya, Keith Bonawitz, Peter Kairouz et al.

Local differential privacy (LDP) is a strong notion of privacy for individual users that often comes at the expense of a significant drop in utility. The classical definition of LDP assumes that all elements in the data domain are equally sensitive. However, in many applications, some symbols are more sensitive than others. This work proposes a context-aware framework of local differential privacy that allows a privacy designer to incorporate the application's context into the privacy definition. For binary data domains, we provide a universally optimal privatization scheme and highlight its connections to Warner's randomized response (RR) and Mangat's improved response. Motivated by geolocation and web search applications, for $k$-ary data domains, we consider two special cases of context-aware LDP: block-structured LDP and high-low LDP. We study discrete distribution estimation and provide communication-efficient, sample-optimal schemes and information-theoretic lower bounds for both models. We show that using contextual information can require fewer samples than classical LDP to achieve the same accuracy.

Jayadev Acharya, Ananda Theertha Suresh

A primary concern of excessive reuse of test datasets in machine learning is that it can lead to overfitting. Multiclass classification was recently shown to be more resistant to overfitting than binary classification. In an open problem of COLT 2019, Feldman, Frostig, and Hardt ask to characterize the dependence of the amount of overfitting bias with the number of classes $m$, the number of accuracy queries $k$, and the number of examples in the dataset $n$. We resolve this problem and determine the amount of overfitting possible in multi-class classification. We provide computationally efficient algorithms that achieve overfitting bias of $\tildeΘ(\max\{\sqrt{{k}/{(mn)}}, k/n\})$, matching the known upper bounds.

Jayadev Acharya, Clément L. Canonne, Yanjun Han et al.

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.

Jayadev Acharya, Ziteng Sun

We consider the problems of distribution estimation and heavy hitter (frequency) estimation under privacy and communication constraints. While these constraints have been studied separately, optimal schemes for one are sub-optimal for the other. We propose a sample-optimal $\varepsilon$-locally differentially private (LDP) scheme for distribution estimation, where each user communicates only one bit, and requires no public randomness. We show that Hadamard Response, a recently proposed scheme for $\varepsilon$-LDP distribution estimation is also utility-optimal for heavy hitter estimation. Finally, we show that unlike distribution estimation, without public randomness where only one bit suffices, any heavy hitter estimation algorithm that communicates $o(\min \{\log n, \log k\})$ bits from each user cannot be optimal.

Jayadev Acharya, Clément L. Canonne, Himanshu Tyagi

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.

Jayadev Acharya, Christopher De Sa, Dylan J. Foster et al.

In distributed statistical learning, $N$ samples are split across $m$ machines and a learner wishes to use minimal communication to learn as well as if the examples were on a single machine. This model has received substantial interest in machine learning due to its scalability and potential for parallel speedup. However, in high-dimensional settings, where the number examples is smaller than the number of features ("dimension"), the speedup afforded by distributed learning may be overshadowed by the cost of communicating a single example. This paper investigates the following question: When is it possible to learn a $d$-dimensional model in the distributed setting with total communication sublinear in $d$? Starting with a negative result, we show that for learning $\ell_1$-bounded or sparse linear models, no algorithm can obtain optimal error until communication is linear in dimension. Our main result is that that by slightly relaxing the standard boundedness assumptions for linear models, we can obtain distributed algorithms that enjoy optimal error with communication logarithmic in dimension. This result is based on a family of algorithms that combine mirror descent with randomized sparsification/quantization of iterates, and extends to the general stochastic convex optimization model.

Jayadev Acharya, Clément L. Canonne, Himanshu Tyagi

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.

Jayadev Acharya, Clément L. Canonne, Cody Freitag et al.

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.

Jayadev Acharya, Arnab Bhattacharyya, Constantinos Daskalakis et al.

We consider testing and learning problems on causal Bayesian networks as defined by Pearl (Pearl, 2009). Given a causal Bayesian network $\mathcal{M}$ on a graph with $n$ discrete variables and bounded in-degree and bounded `confounded components', we show that $O(\log n)$ interventions on an unknown causal Bayesian network $\mathcal{X}$ on the same graph, and $\tilde{O}(n/ε^2)$ samples per intervention, suffice to efficiently distinguish whether $\mathcal{X}=\mathcal{M}$ or whether there exists some intervention under which $\mathcal{X}$ and $\mathcal{M}$ are farther than $ε$ in total variation distance. We also obtain sample/time/intervention efficient algorithms for: (i) testing the identity of two unknown causal Bayesian networks on the same graph; and (ii) learning a causal Bayesian network on a given graph. Although our algorithms are non-adaptive, we show that adaptivity does not help in general: $Ω(\log n)$ interventions are necessary for testing the identity of two unknown causal Bayesian networks on the same graph, even adaptively. Our algorithms are enabled by a new subadditivity inequality for the squared Hellinger distance between two causal Bayesian networks.

Jayadev Acharya, Clément L. Canonne, Himanshu Tyagi

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.

Jayadev Acharya, Gautam Kamath, Ziteng Sun et al.

We develop differentially private methods for estimating various distributional properties. Given a sample from a discrete distribution $p$, some functional $f$, and accuracy and privacy parameters $α$ and $\varepsilon$, the goal is to estimate $f(p)$ up to accuracy $α$, while maintaining $\varepsilon$-differential privacy of the sample. We prove almost-tight bounds on the sample size required for this problem for several functionals of interest, including support size, support coverage, and entropy. We show that the cost of privacy is negligible in a variety of settings, both theoretically and experimentally. Our methods are based on a sensitivity analysis of several state-of-the-art methods for estimating these properties with sublinear sample complexities.

Jayadev Acharya, Ziteng Sun, Huanyu Zhang

We study the problem of estimating $k$-ary distributions under $\varepsilon$-local differential privacy. $n$ samples are distributed across users who send privatized versions of their sample to a central server. All previously known sample optimal algorithms require linear (in $k$) communication from each user in the high privacy regime $(\varepsilon=O(1))$, and run in time that grows as $n\cdot k$, which can be prohibitive for large domain size $k$. We propose Hadamard Response (HR}, a local privatization scheme that requires no shared randomness and is symmetric with respect to the users. Our scheme has order optimal sample complexity for all $\varepsilon$, a communication of at most $\log k+2$ bits per user, and nearly linear running time of $\tilde{O}(n + k)$. Our encoding and decoding are based on Hadamard matrices, and are simple to implement. The statistical performance relies on the coding theoretic aspects of Hadamard matrices, ie, the large Hamming distance between the rows. An efficient implementation of the algorithm using the Fast Walsh-Hadamard transform gives the computational gains. We compare our approach with Randomized Response (RR), RAPPOR, and subset-selection mechanisms (SS), both theoretically, and experimentally. For $k=10000$, our algorithm runs about 100x faster than SS, and RAPPOR.

Jayadev Acharya, Ziteng Sun, Huanyu Zhang

We study the fundamental problems of identity testing (goodness of fit), and closeness testing (two sample test) of distributions over $k$ elements, under differential privacy. While the problems have a long history in statistics, finite sample bounds for these problems have only been established recently. In this work, we derive upper and lower bounds on the sample complexity of both the problems under $(\varepsilon, δ)$-differential privacy. We provide optimal sample complexity algorithms for identity testing problem for all parameter ranges, and the first results for closeness testing. Our closeness testing bounds are optimal in the sparse regime where the number of samples is at most $k$. Our upper bounds are obtained by privatizing non-private estimators for these problems. The non-private estimators are chosen to have small sensitivity. We propose a general framework to establish lower bounds on the sample complexity of statistical tasks under differential privacy. We show a bound on differentially private algorithms in terms of a coupling between the two hypothesis classes we aim to test. By constructing carefully chosen priors over the hypothesis classes, and using Le Cam's two point theorem we provide a general mechanism for proving lower bounds. We believe that the framework can be used to obtain strong lower bounds for other statistical tasks under privacy.

Jayadev Acharya, Hirakendu Das, Alon Orlitsky et al.

The advent of data science has spurred interest in estimating properties of distributions over large alphabets. Fundamental symmetric properties such as support size, support coverage, entropy, and proximity to uniformity, received most attention, with each property estimated using a different technique and often intricate analysis tools. We prove that for all these properties, a single, simple, plug-in estimator---profile maximum likelihood (PML)---performs as well as the best specialized techniques. This raises the possibility that PML may optimally estimate many other symmetric properties.

Jayadev Acharya, Ilias Diakonikolas, Jerry Li et al.

We study the fixed design segmented regression problem: Given noisy samples from a piecewise linear function $f$, we want to recover $f$ up to a desired accuracy in mean-squared error. Previous rigorous approaches for this problem rely on dynamic programming (DP) and, while sample efficient, have running time quadratic in the sample size. As our main contribution, we provide new sample near-linear time algorithms for the problem that -- while not being minimax optimal -- achieve a significantly better sample-time tradeoff on large datasets compared to the DP approach. Our experimental evaluation shows that, compared with the DP approach, our algorithms provide a convergence rate that is only off by a factor of $2$ to $4$, while achieving speedups of three orders of magnitude.

Jayadev Acharya, Constantinos Daskalakis, Gautam Kamath

Given samples from an unknown distribution $p$, is it possible to distinguish whether $p$ belongs to some class of distributions $\mathcal{C}$ versus $p$ being far from every distribution in $\mathcal{C}$? This fundamental question has received tremendous attention in statistics, focusing primarily on asymptotic analysis, and more recently in information theory and theoretical computer science, where the emphasis has been on small sample size and computational complexity. Nevertheless, even for basic properties of distributions such as monotonicity, log-concavity, unimodality, independence, and monotone-hazard rate, the optimal sample complexity is unknown. We provide a general approach via which we obtain sample-optimal and computationally efficient testers for all these distribution families. At the core of our approach is an algorithm which solves the following problem: Given samples from an unknown distribution $p$, and a known distribution $q$, are $p$ and $q$ close in $χ^2$-distance, or far in total variation distance? The optimality of our testers is established by providing matching lower bounds with respect to both $n$ and $\varepsilon$. Finally, a necessary building block for our testers and an important byproduct of our work are the first known computationally efficient proper learners for discrete log-concave and monotone hazard rate distributions.

Jayadev Acharya, Ilias Diakonikolas, Jerry Li et al.

We design a new, fast algorithm for agnostically learning univariate probability distributions whose densities are well approximated by piecewise polynomial functions. Let $f$ be the density function of an arbitrary univariate distribution, and suppose that $f$ is $\mathrm{OPT}$-close in $L_1$-distance to an unknown piecewise polynomial function with $t$ interval pieces and degree $d$. Our algorithm draws $n = O(t(d+1)/ε^2)$ samples from $f$, runs in time $\tilde{O}(n \cdot \mathrm{poly}(d))$, and with probability at least $9/10$ outputs an $O(t)$-piecewise degree-$d$ hypothesis $h$ that is $4 \cdot \mathrm{OPT} +ε$ close to $f$. Our general algorithm yields (nearly) sample-optimal and nearly-linear time estimators for a wide range of structured distribution families over both continuous and discrete domains in a unified way. For most of our applications, these are the first sample-optimal and nearly-linear time estimators in the literature. As a consequence, our work resolves the sample and computational complexities of a broad class of inference tasks via a single "meta-algorithm". Moreover, we experimentally demonstrate that our algorithm performs very well in practice. Our algorithm consists of three "levels": (i) At the top level, we employ an iterative greedy algorithm for finding a good partition of the real line into the pieces of a piecewise polynomial. (ii) For each piece, we show that the sub-problem of finding a good polynomial fit on the current interval can be solved efficiently with a separation oracle method. (iii) We reduce the task of finding a separating hyperplane to a combinatorial problem and give an efficient algorithm for this problem. Combining these three procedures gives a density estimation algorithm with the claimed guarantees.

Jayadev Acharya, Clément L. Canonne, Gautam Kamath

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.

Jayadev Acharya, Constantinos Daskalakis

A Poisson Binomial distribution over $n$ variables is the distribution of the sum of $n$ independent Bernoullis. We provide a sample near-optimal algorithm for testing whether a distribution $P$ supported on $\{0,...,n\}$ to which we have sample access is a Poisson Binomial distribution, or far from all Poisson Binomial distributions. The sample complexity of our algorithm is $O(n^{1/4})$ to which we provide a matching lower bound. We note that our sample complexity improves quadratically upon that of the naive "learn followed by tolerant-test" approach, while instance optimal identity testing [VV14] is not applicable since we are looking to simultaneously test against a whole family of distributions.

Jayadev Acharya, Alon Orlitsky, Ananda Theertha Suresh et al.

It was recently shown that estimating the Shannon entropy $H({\rm p})$ of a discrete $k$-symbol distribution ${\rm p}$ requires $Θ(k/\log k)$ samples, a number that grows near-linearly in the support size. In many applications $H({\rm p})$ can be replaced by the more general Rényi entropy of order $α$, $H_α({\rm p})$. We determine the number of samples needed to estimate $H_α({\rm p})$ for all $α$, showing that $α< 1$ requires a super-linear, roughly $k^{1/α}$ samples, noninteger $α>1$ requires a near-linear $k$ samples, but, perhaps surprisingly, integer $α>1$ requires only $Θ(k^{1-1/α})$ 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 $α$ 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 Rényi entropies that are hard to distinguish.

Jayadev Acharya, Ashkan Jafarpour, Alon Orlitsky et al.

The Poisson-sampling technique eliminates dependencies among symbol appearances in a random sequence. It has been used to simplify the analysis and strengthen the performance guarantees of randomized algorithms. Applying this method to universal compression, we relate the redundancies of fixed-length and Poisson-sampled sequences, use the relation to derive a simple single-letter formula that approximates the redundancy of any envelope class to within an additive logarithmic term. As a first application, we consider i.i.d. distributions over a small alphabet as a step-envelope class, and provide a short proof that determines the redundancy of discrete distributions over a small al- phabet up to the first order terms. We then show the strength of our method by applying the formula to tighten the existing bounds on the redundancy of exponential and power-law classes, in particular answering a question posed by Boucheron, Garivier and Gassiat.

Jayadev Acharya, Ashkan Jafarpour, Alon Orlitsky et al.

Statistical and machine-learning algorithms are frequently applied to high-dimensional data. In many of these applications data is scarce, and often much more costly than computation time. We provide the first sample-efficient polynomial-time estimator for high-dimensional spherical Gaussian mixtures. For mixtures of any $k$ $d$-dimensional spherical Gaussians, we derive an intuitive spectral-estimator that uses $\mathcal{O}_k\bigl(\frac{d\log^2d}{ε^4}\bigr)$ samples and runs in time $\mathcal{O}_{k,ε}(d^3\log^5 d)$, both significantly lower than previously known. The constant factor $\mathcal{O}_k$ is polynomial for sample complexity and is exponential for the time complexity, again much smaller than what was previously known. We also show that $Ω_k\bigl(\frac{d}{ε^2}\bigr)$ samples are needed for any algorithm. Hence the sample complexity is near-optimal in the number of dimensions. We also derive a simple estimator for one-dimensional mixtures that uses $\mathcal{O}\bigl(\frac{k \log \frac{k}ε }{ε^2} \bigr)$ samples and runs in time $\widetilde{\mathcal{O}}\left(\bigl(\frac{k}ε\bigr)^{3k+1}\right)$. Our other technical contributions include a faster algorithm for choosing a density estimate from a set of distributions, that minimizes the $\ell_1$ distance to an unknown underlying distribution.