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1. Sample-Optimal Identity Testing with High Probability

   Diakonikolas, Ilias ; Gouleakis, Themis ; Peebles, John ; Price, Eric ( July 2018 , 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. 

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2. Testing Mixtures of Discrete Distributions

   Aliakbarpour, M. ; Kumar, R. ; Rubinfeld, R ( January 2019 , Proceedings of the Thirty-Second Conference on Learning Theory (COLT 2019))

   There has been significant study on the sample complexity of testing properties of distributions over large domains. For many properties, it is known that the sample complexity can be substantially smaller than the domain size. For example, over a domain of size n, distinguishing the uniform distribution from distributions that are far from uniform in ℓ1-distance uses only O(n−−√) samples. However, the picture is very different in the presence of arbitrary noise, even when the amount of noise is quite small. In this case, one must distinguish if samples are coming from a distribution that is ϵ-close to uniform from the case where the distribution is (1−ϵ)-far from uniform. The latter task requires nearly linear in n samples (Valiant, 2008; Valiant and Valiant, 2017a). In this work, we present a noise model that on one hand is more tractable for the testing problem, and on the other hand represents a rich class of noise families. In our model, the noisy distribution is a mixture of the original distribution and noise, where the latter is known to the tester either explicitly or via sample access; the form of the noise is also known \emph{a priori}. Focusing on the identity and closeness testing problems leads to the following mixture testing question: Given samples of distributions p,q1,q2, can we test if p is a mixture of q1 and q2? We consider this general question in various scenarios that differ in terms of how the tester can access the distributions, and show that indeed this problem is more tractable. Our results show that the sample complexity of our testers are exactly the same as for the classical non-mixture case. 

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3. Testing Mixtures of Discrete Distributions

   Aliakbarpour, Maryam ; Kumar, Ravi ; Rubinfeld, Ronitt ( January 2019 , Proceedings of the Thirty-Second Conference on Learning Theory (PMLR))

   There has been significant study on the sample complexity of testing properties of distributions over large domains. For many properties, it is known that the sample complexity can be substantially smaller than the domain size. For example, over a domain of size n, distinguishing the uniform distribution from distributions that are far from uniform in ℓ1-distance uses only O(n−−√) samples. However, the picture is very different in the presence of arbitrary noise, even when the amount of noise is quite small. In this case, one must distinguish if samples are coming from a distribution that is ϵ-close to uniform from the case where the distribution is (1−ϵ)-far from uniform. The latter task requires nearly linear in n samples (Valiant, 2008; Valiant and Valiant, 2017a). In this work, we present a noise model that on one hand is more tractable for the testing problem, and on the other hand represents a rich class of noise families. In our model, the noisy distribution is a mixture of the original distribution and noise, where the latter is known to the tester either explicitly or via sample access; the form of the noise is also known \emph{a priori}. Focusing on the identity and closeness testing problems leads to the following mixture testing question: Given samples of distributions p,q1,q2, can we test if p is a mixture of q1 and q2? We consider this general question in various scenarios that differ in terms of how the tester can access the distributions, and show that indeed this problem is more tractable. Our results show that the sample complexity of our testers are exactly the same as for the classical non-mixture case. 

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4. Optimal Stopping Rules for Sequential Hypothesis Testing

   Daskalakis, Constantinos ; Kawase, Yasushi ( January 2017 , 25th Annual European Symposium on Algorithms (ESA))

   Suppose that we are given sample access to an unknown distribution p over n elements and an explicit distribution q over the same n elements. We would like to reject the null hypothesis“p=q” after seeing as few samples as possible, whenp6=q, while we never want to reject the null, when p=q. Well-known results show thatΘ(√n/2)samples are necessary and sufficient for distinguishing whether p equals q versus p is-far from q in total variation distance. However,this requires the distinguishing radiusto be fixed prior to deciding how many samples to request.Our goal is instead to design sequential hypothesis testers, i.e. online algorithms that request i.i.d.samples from p and stop as soon as they can confidently reject the hypothesis p=q, without being given a lower bound on the distance between p and q, whenp6=q. In particular, we want to minimize the number of samples requested by our tests as a function of the distance between p and q, and if p=q we want the algorithm, with high probability, to never reject the null. Our work is motivated by and addresses the practical challenge of sequential A/B testing in Statistics.We show that, when n= 2, any sequential hypothesis test must seeΩ(1dtv(p,q)2log log1dtv(p,q))samples, with high (constant) probability, before it rejects p=q, where dtv(p,q) is the—unknown to the tester—total variation distance between p and q. We match the dependence of this lower bound ondtv(p,q)by proposing a sequential tester that rejects p=q from at most O(√ndtv(p,q)2log log1dtv(p,q))samples with high (constant) probability. TheΩ(√n)dependence on the support size n is also known to be necessary. We similarly provide two-sample sequential hypothesis testers, when sample access is given to both p and q, and discuss applications to sequential A/B testing. 

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5. Testing Distributional Assumptions of Learning Algorithms

   Rubinfeld, Ronitt ; Vasilyan, Arsen ( June 2023 , 55th Annual ACM Symposium on Theory of Computing, STOC 2023)

   There are many important high dimensional function classes that have fast agnostic learning algorithms when strong assumptions on the distribution of examples can be made, such as Gaussianity or uniformity over the domain. But how can one be sufficiently confident that the data indeed satisfies the distributional assumption, so that one can trust in the output quality of the agnostic learning algorithm? We propose a model by which to systematically study the design of tester-learner pairs (A,T), such that if the distribution on examples in the data passes the tester T then one can safely trust the output of the agnostic learner A on the data. To demonstrate the power of the model, we apply it to the classical problem of agnostically learning halfspaces under the standard Gaussian distribution and present a tester-learner pair with a combined run-time of nÕ(1/є4). This qualitatively matches that of the best known ordinary agnostic learning algorithms for this task. In contrast, finite sample Gaussian distribution testers do not exist for the L1 and EMD distance measures. Previously it was known that half-spaces are well-approximated with low-degree polynomials relative to the Gaussian distribution. A key step in our analysis is showing that this is the case even relative to distributions whose low-degree moments approximately match those of a Gaussian. We also go beyond spherically-symmetric distributions, and give a tester-learner pair for halfspaces under the uniform distribution on {0,1}n with combined run-time of nÕ(1/є4). This is achieved using polynomial approximation theory and critical index machinery of [Diakonikolas, Gopalan, Jaiswal, Servedio, and Viola 2009]. Can one design agnostic learning algorithms under distributional assumptions and count on future technical work to produce, as a matter of course, tester-learner pairs with similar run-time? Our answer is a resounding no, as we show there exist some well-studied settings for which 2Õ(√n) run-time agnostic learning algorithms are available, yet the combined run-times of tester-learner pairs must be as high as 2Ω(n). On that account, the design of tester-learner pairs is a research direction in its own right independent of standard agnostic learning. To be specific, our lower bounds apply to the problems of agnostically learning convex sets under the Gaussian distribution and for monotone Boolean functions under the uniform distribution over {0,1}n. 

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