Given samples from an unknown multivariate distribution p, is it possible to distinguish whether p is the product of its marginals versus p being far from every product distribution? Similarly, is it possible to distinguish whether p equals a given distribution q versus p and q being far from each other? These problems of testing independence and goodness-of-fit have received enormous attention in statistics, information theory, and theoretical computer science, with sample-optimal algorithms known in several interesting regimes of parameters [BFF+01, Pan08, VV17, ADK15, DK16]. Unfortunately, it has also been understood that these problems become intractable in large dimensions, necessitating exponential sample complexity.
Motivated by the exponential lower bounds for general distributions as well as the ubiquity of Markov Random Fields (MRFs) in the modeling of high-dimensional distributions, we initiate the study of distribution testing on structured multivariate distributions, and in particular the prototypical example of MRFs: the Ising Model. We demonstrate that, in this structured setting, we can avoid the curse of dimensionality, obtaining sample and time efficient testers for independence and goodness-of-fit. One of the key technical challenges we face along the way is bounding the variance of functions of the Ising model. more »« less

We propose a new setting for testing properties of distributions while receiving samples from several distributions, but few samples per distribution. Given samples from s distributions, p_1, p_2, …, p_s, we design testers for the following problems: (1) Uniformity Testing: Testing whether all the p_i’s are uniform or ε-far from being uniform in ℓ_1-distance (2) Identity Testing: Testing whether all the p_i’s are equal to an explicitly given distribution q or ε-far from q in ℓ_1-distance, and (3) Closeness Testing: Testing whether all the p_i’s are equal to a distribution q which we have sample access to, or ε-far from q in ℓ_1-distance. By assuming an additional natural condition about the source distributions, we provide sample optimal testers for all of these problems.

We propose a new setting for testing properties of distributions while receiving samples from several distributions, but few samples per distribution. Given samples from s distributions, p_1, p_2, …, p_s, we design testers for the following problems: (1) Uniformity Testing: Testing whether all the p_i’s are uniform or ε-far from being uniform in ℓ_1-distance (2) Identity Testing: Testing whether all the p_i’s are equal to an explicitly given distribution q or ε-far from q in ℓ_1-distance, and (3) Closeness Testing: Testing whether all the p_i’s are equal to a distribution q which we have sample access to, or ε-far from q in ℓ_1-distance. By assuming an additional natural condition about the source distributions, we provide sample optimal testers for all of these problems.

Gatmiry, Khashayar; Aliakbarpour, Maryam; Jegelka, Stefanie(
, Annual Conference on Neural Information Processing Systems (NeurIPS 2020))

null
(Ed.)

Determinantal point processes (DPPs) are popular probabilistic models of diversity. In this paper, we investigate DPPs from a new perspective: property testing of distributions. Given sample access to an unknown distribution q over the subsets of a ground set, we aim to distinguish whether q is a DPP distribution or ϵ-far from all DPP distributions in ℓ1-distance. In this work, we propose the first algorithm for testing DPPs. Furthermore, we establish a matching lower bound on the sample complexity of DPP testing. This lower bound also extends to showing a new hardness result for the problem of testing the more general class of log-submodular distributions

Bhattacharyya, Arnab; Gayen, Sutanu; Kandasamy, Saravanan; Vinodchandran, N. V.(
, International Conference on Algorithmic Learning Theory)

null
(Ed.)

We study the problems of identity and closeness testing of n-dimensional product distributions. Prior works of Canonne et al. (2017) and Daskalakis and Pan (2017) have established tight sample complexity bounds for non-tolerant testing over a binary alphabet: given two product distributions P and Q over a binary alphabet, distinguish between the cases P = Q and dTV(P;Q) > epsilon . We build on this prior work to give a more comprehensive map of the complexity of testing of product distributions by investigating tolerant testing with respect to several natural distance measures and over an arbitrary alphabet. Our study gives a fine-grained understanding of how the sample complexity of tolerant testing varies with the distance measures for product distributions. In addition,
we also extend one of our upper bounds on product distributions to bounded-degree Bayes nets.

Diakonikolas, Ilias; Gouleakis, Themis; Peebles, John; Price, Eric(
, 45th International Colloquium on Automata, Languages, and Automata)

We study the problem of testing identity against a given distribution with a focus on the high confidence regime. More precisely, given samples from an unknown distribution p over n elements, an explicitly given distribution q, and parameters 0< epsilon, delta < 1, we wish to distinguish, with probability at least 1-delta, whether the distributions are identical versus epsilon-far in total variation distance. Most prior work focused on the case that delta = Omega(1), for which the sample complexity of identity testing is known to be Theta(sqrt{n}/epsilon^2). Given such an algorithm, one can achieve arbitrarily small values of delta via black-box amplification, which multiplies the required number of samples by Theta(log(1/delta)). We show that black-box amplification is suboptimal for any delta = o(1), and give a new identity tester that achieves the optimal sample complexity. Our new upper and lower bounds show that the optimal sample complexity of identity testing is Theta((1/epsilon^2) (sqrt{n log(1/delta)} + log(1/delta))) for any n, epsilon, and delta. For the special case of uniformity testing, where the given distribution is the uniform distribution U_n over the domain, our new tester is surprisingly simple: to test whether p = U_n versus d_{TV} (p, U_n) >= epsilon, we simply threshold d_{TV}({p^}, U_n), where {p^} is the empirical probability distribution. The fact that this simple "plug-in" estimator is sample-optimal is surprising, even in the constant delta case. Indeed, it was believed that such a tester would not attain sublinear sample complexity even for constant values of epsilon and delta. An important contribution of this work lies in the analysis techniques that we introduce in this context. First, we exploit an underlying strong convexity property to bound from below the expectation gap in the completeness and soundness cases. Second, we give a new, fast method for obtaining provably correct empirical estimates of the true worst-case failure probability for a broad class of uniformity testing statistics over all possible input distributions - including all previously studied statistics for this problem. We believe that our novel analysis techniques will be useful for other distribution testing problems as well.

Daskalakis, Constantinos, Dikkala, Nishanth, and Kamath, Gautam. Testing Ising Models. Retrieved from https://par.nsf.gov/biblio/10079726. Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms .

Daskalakis, Constantinos, Dikkala, Nishanth, and Kamath, Gautam.
"Testing Ising Models". Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms (). Country unknown/Code not available. https://par.nsf.gov/biblio/10079726.

@article{osti_10079726,
place = {Country unknown/Code not available},
title = {Testing Ising Models},
url = {https://par.nsf.gov/biblio/10079726},
abstractNote = {Given samples from an unknown multivariate distribution p, is it possible to distinguish whether p is the product of its marginals versus p being far from every product distribution? Similarly, is it possible to distinguish whether p equals a given distribution q versus p and q being far from each other? These problems of testing independence and goodness-of-fit have received enormous attention in statistics, information theory, and theoretical computer science, with sample-optimal algorithms known in several interesting regimes of parameters [BFF+01, Pan08, VV17, ADK15, DK16]. Unfortunately, it has also been understood that these problems become intractable in large dimensions, necessitating exponential sample complexity. Motivated by the exponential lower bounds for general distributions as well as the ubiquity of Markov Random Fields (MRFs) in the modeling of high-dimensional distributions, we initiate the study of distribution testing on structured multivariate distributions, and in particular the prototypical example of MRFs: the Ising Model. We demonstrate that, in this structured setting, we can avoid the curse of dimensionality, obtaining sample and time efficient testers for independence and goodness-of-fit. One of the key technical challenges we face along the way is bounding the variance of functions of the Ising model.},
journal = {Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms},
author = {Daskalakis, Constantinos and Dikkala, Nishanth and Kamath, Gautam},
}

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