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We provide a computationally and statistically efficient estimator for the classical problem of trun-cated linear regression, where the dependent variabley=wTx+εand its corresponding vector ofcovariatesx∈Rkare only revealed if the dependent variable falls in some subsetS⊆R; otherwisethe existence of the pair(x,y)is hidden. This problem has remained a challenge since the earlyworks of Tobin (1958); Amemiya (1973); Hausman and Wise (1977); Breen et al. (1996), its appli-cations are abundant, and its history dates back even further to the work of Galton, Pearson, Lee,and Fisher Galton (1897); Pearson and Lee (1908); Lee (1914); Fisher (1931). While consistent es-timators of the regression coefficients have been identified, the error rates are not well-understood,especially in high-dimensional settings.Under a “thickness assumption” about the covariance matrix of the covariates in the revealed sample, we provide a computationally efficient estimator for the coefficient vectorwfromnre-vealed samples that attains`2errorO(√k/n), recovering the guarantees of least squares in thestandard (untruncated) linear regression setting. Our estimator uses Projected Stochastic Gradi-ent Descent (PSGD) on the negative log-likelihood of the truncated sample, and only needs ora-cle access to the setS, which may otherwise be arbitrary, and in particular may be non-convex.PSGD must be restricted to an appropriately defined convex cone to guarantee that the negativelog-likelihood is strongly convex, which in turn is established using concentration of matrices onvariables with sub-exponential tails. We perform experiments on simulated data to illustrate the accuracy of our estimator.As a corollary of our work, we show that SGD provably learns the parameters of single-layerneural networks with noisy Relu activation functions Nair and Hinton (2010); Bengio et al. (2013);Gulcehre et al. (2016), given linearly many, in the number of network parameters, input-outputpairs in the realizable setting. 

In this work, we consider the sample complexity required for testing the monotonicity of distributions over partial orders. A distribution p over a poset is {\em monotone} if, for any pair of domain elements x and y such that x⪯y, p(x)≤p(y). To understand the sample complexity of this problem, we introduce a new property called \emph{bigness} over a finite domain, where the distribution is T-big if the minimum probability for any domain element is at least T. We establish a lower bound of Ω(n/logn) for testing bigness of distributions on domains of size n. We then build on these lower bounds to give Ω(n/logn) lower bounds for testing monotonicity over a matching poset of size n and significantly improved lower bounds over the hypercube poset. We give sublinear sample complexity bounds for testing bigness and for testing monotonicity over the matching poset. We then give a number of tools for analyzing upper bounds on the sample complexity of the monotonicity testing problem. 

A wide range of learning tasks require human input in labeling massive data. The collected data though are usually low quality and contain inaccuracies and errors. As a result, modern science and business face the problem of learning from unreliable data sets. In this work, we provide a generic approach that is based on \textit{verification} of only few records of the data set to guarantee high quality learning outcomes for various optimization objectives. Our method, identifies small sets of critical records and verifies their validity. We show that many problems only need poly(1/ε) verifications, to ensure that the output of the computation is at most a factor of (1±ε) away from the truth. For any given instance, we provide an \textit{instance optimal} solution that verifies the minimum possible number of records to approximately certify correctness. Then using this instance optimal formulation of the problem we prove our main result: “every function that satisfies some Lipschitz continuity condition can be certified with a small number of verifications”. We show that the required Lipschitz continuity condition is satisfied even by some NP-complete problems, which illustrates the generality and importance of this theorem. In case this certification step fails, an invalid record will be identified. Removing these records and repeating until success, guarantees that the result will be accurate and will depend only on the verified records. Surprisingly, as we show, for several computation tasks more efficient methods are possible. These methods always guarantee that the produced result is not affected by the invalid records, since any invalid record that affects the output will be detected and verified. 

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. 

We present O(log logn)-round algorithms in the Massively Parallel Computation (MPC) model, with ˜O(n) memory per machine, that compute a maximal independent set, a 1 + ε approximation of maximum matching, and a 2 + ε approximation of minimum vertex cover, for any n-vertex graph and any constant ε > 0. These improve the state of the art as follows: • Our MIS algorithm leads to a simple O(log log Δ)-round MIS algorithm in the CONGESTED-CLIQUE model of distributed computing, which improves on the ˜O (plog Δ)-round algorithm of Ghaffari [PODC’17]. • OurO(log logn)-round (1+ε)-approximate maximum matching algorithm simplifies or improves on the following prior work: O(log2 logn)-round (1 + ε)-approximation algorithm of Czumaj et al. [STOC’18] and O(log logn)-round (1 + ε)- approximation algorithm of Assadi et al. [arXiv’17]. • Our O(log logn)-round (2+ε)-approximate minimum vertex cover algorithm improves on an O(log logn)-round O(1)- approximation of Assadi et al. [arXiv’17]. 

We provide an efficient algorithm for the classical problem, going back to Galton, Pearson, and Fisher, of estimating, with arbitrary accuracy the parameters of a multivariate normal distribution from truncated samples. Truncated samples from a d-variate normal (μ,Σ) means a samples is only revealed if it falls in some subset S⊆ℝd; otherwise the samples are hidden and their count in proportion to the revealed samples is also hidden. We show that the mean μ and covariance matrix Σ can be estimated with arbitrary accuracy in polynomial-time, as long as we have oracle access to S, and S has non-trivial measure under the unknown d-variate normal distribution. Additionally we show that without oracle access to S, any non-trivial estimation is impossible. 

We provide an efficient algorithm for the classical problem, going back to Galton, Pearson, and Fisher, of estimating, with arbitrary accuracy the parameters of a multivariate normal distribution from truncated samples. Truncated samples from a d-variate normal (μ,Σ) means a samples is only revealed if it falls in some subset S⊆ℝd; otherwise the samples are hidden and their count in proportion to the revealed samples is also hidden. We show that the mean μ and covariance matrix Σ can be estimated with arbitrary accuracy in polynomial-time, as long as we have oracle access to S, and S has non-trivial measure under the unknown d-variate normal distribution. Additionally we show that without oracle access to S, any non-trivial estimation is impossible.