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1. 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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2. 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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3. Sharp Constants in Uniformity Testing via the Huber Statistic

   Gupta, S ; Price, Eric ( January 2022 , Conference on Learning Theory)

   Uniformity testing is one of the most well-studied problems in property testing, with many known test statistics, including ones based on counting collisions, singletons, and the empirical TV distance. It is known that the optimal sample complexity to distinguish the uniform distribution on m elements from any ϵ-far distribution with 1−δ probability is n=Θ(mlog(1/δ)√ϵ2+log(1/δ)ϵ2), which is achieved by the empirical TV tester. Yet in simulation, these theoretical analyses are misleading: in many cases, they do not correctly rank order the performance of existing testers, even in an asymptotic regime of all parameters tending to 0 or ∞. We explain this discrepancy by studying the \emph{constant factors} required by the algorithms. We show that the collisions tester achieves a sharp maximal constant in the number of standard deviations of separation between uniform and non-uniform inputs. We then introduce a new tester based on the Huber loss, and show that it not only matches this separation, but also has tails corresponding to a Gaussian with this separation. This leads to a sample complexity of (1+o(1))mlog(1/δ)√ϵ2 in the regime where this term is dominant, unlike all other existing testers. 

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4. Tester-Learners for Halfspaces: Universal Algorithms

   Gollakota, Aravind ; Klivans, Adam R. ; Stavropoulos, Konstantinos ; Vasilyan, Arsen ( December 2023 , 37th Conference on Neural Information Processing Systems, NeurIPS 2023)

   We give the first tester-learner for halfspaces that succeeds universally over a wide class of structured distributions. Our universal tester-learner runs in fully polynomial time and has the following guarantee: the learner achieves error O(opt)+ϵ on any labeled distribution that the tester accepts, and moreover, the tester accepts whenever the marginal is any distribution that satisfies a Poincare inequality. In contrast to prior work on testable learning, our tester is not tailored to any single target distribution but rather succeeds for an entire target class of distributions. The class of Poincare distributions includes all strongly log-concave distributions, and, assuming the Kannan--Lovasz--Simonovits (KLS) conjecture, includes all log-concave distributions. In the special case where the label noise is known to be Massart, our tester-learner achieves error opt+ϵ while accepting all log-concave distributions unconditionally (without assuming KLS).Our tests rely on checking hypercontractivity of the unknown distribution using a sum-of-squares (SOS) program, and crucially make use of the fact that Poincare distributions are certifiably hypercontractive in the SOS framework. 

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5. Lifting uniform learners via distributional decomposition

   https://doi.org/10.1145/3564246.3585212  

   Blanc, Guy ; Lange, Jane ; Malik, Ali ; Tan, Li-Yang ( January 2023 , Proceedings of the 55th Annual ACM Symposium on Theory of Computing (STOC 2023))

   We show how any PAC learning algorithm that works under the uniform distribution can be transformed, in a blackbox fashion, into one that works under an arbitrary and unknown distribution ‍D. The efficiency of our transformation scales with the inherent complexity of ‍D, running in (n, (md)d) time for distributions over n whose pmfs are computed by depth-d decision trees, where m is the sample complexity of the original algorithm. For monotone distributions our transformation uses only samples from ‍D, and for general ones it uses subcube conditioning samples. A key technical ingredient is an algorithm which, given the aforementioned access to D, produces an optimal decision tree decomposition of D: an approximation of D as a mixture of uniform distributions over disjoint subcubes. With this decomposition in hand, we run the uniform-distribution learner on each subcube and combine the hypotheses using the decision tree. This algorithmic decomposition lemma also yields new algorithms for learning decision tree distributions with runtimes that exponentially improve on the prior state of the art—results of independent interest in distribution learning. 

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