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1. Robust estimation via generalized quasi-gradients

   https://doi.org/10.1093/imaiai/iaab018  

   Zhu, Banghua ; Jiao, Jiantao ; Steinhardt, Jacob ( August 2021 , Information and Inference: A Journal of the IMA)

   Abstract We explore why many recently proposed robust estimation problems are efficiently solvable, even though the underlying optimization problems are non-convex. We study the loss landscape of these robust estimation problems, and identify the existence of ’generalized quasi-gradients’. Whenever these quasi-gradients exist, a large family of no-regret algorithms are guaranteed to approximate the global minimum; this includes the commonly used filtering algorithm. For robust mean estimation of distributions under bounded covariance, we show that any first-order stationary point of the associated optimization problem is an approximate global minimum if and only if the corruption level $\epsilon < 1/3$. Consequently, any optimization algorithm that approaches a stationary point yields an efficient robust estimator with breakdown point $1/3$. With carefully designed initialization and step size, we improve this to $1/2$, which is optimal. For other tasks, including linear regression and joint mean and covariance estimation, the loss landscape is more rugged: there are stationary points arbitrarily far from the global minimum. Nevertheless, we show that generalized quasi-gradients exist and construct efficient algorithms. These algorithms are simpler than previous ones in the literature, and for linear regression we improve the estimation error from $O(\sqrt{\epsilon })$ to the optimal rate of $O(\epsilon )$ formore »small $\epsilon $ assuming certified hypercontractivity. For mean estimation with near-identity covariance, we show that a simple gradient descent algorithm achieves breakdown point $1/3$ and iteration complexity $\tilde{O}(d/\epsilon ^2)$.« less
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, wemore »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.« less
3. Fast Algorithms for Geometric Consensuses

   Har-Peled, Sariel ; Jones, Mitchell ( June 2020 , 36th International Symposium on Computational Geometry, SoCG 2020)

   Cabello, Sergio ; Chen, Danny (Ed.)

   Let P be a set of n points in ℝ^d in general position. A median hyperplane (roughly) splits the point set P in half. The yolk of P is the ball of smallest radius intersecting all median hyperplanes of P. The egg of P is the ball of smallest radius intersecting all hyperplanes which contain exactly d points of P. We present exact algorithms for computing the yolk and the egg of a point set, both running in expected time O(n^(d-1) log n). The running time of the new algorithm is a polynomial time improvement over existing algorithms. We also present algorithms for several related problems, such as computing the Tukey and center balls of a point set, among others.
4. Nonparametric, Tuning-Free Estimation of S-Shaped Functions

   https://doi.org/10.1111/rssb.12481  

   Feng, Oliver Y. ; Chen, Yining ; Han, Qiyang ; Carroll, Raymond J. ; Samworth, Richard J. ( April 2022 , Journal of the Royal Statistical Society Series B: Statistical Methodology)

   We consider the nonparametric estimation of an S-shaped regression function. The least squares estimator provides a very natural, tuning-free approach, but results in a non-convex optimization problem, since the inflection point is unknown. We show that the estimator may nevertheless be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal-dual bases algorithm for its efficient, sequential computation. After developing a projection framework that demonstrates the consistency and robustness to misspecification of the estimator, our main theoretical results provide sharp oracle inequalities that yield worst-case and adaptive risk bounds for the estimation of the regression function, as well as a rate of convergence for the estimation of the inflection point. These results reveal not only that the estimator achieves the minimax optimal rate of convergence for both the estimation of the regression function and its inflection point (up to a logarithmic factor in the latter case), but also that it is able to achieve an almost-parametric rate when the true regression function is piecewise affine with not too many affine pieces. Simulations and a real data application to air pollution modelling also confirm the desirable finite-sample properties of the estimator,more »and our algorithm is implemented in the R package Sshaped.

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5. Estimating location parameters in sample-heterogeneous distributions

   https://doi.org/10.1093/imaiai/iaab013  

   Pensia, Ankit ; Jog, Varun ; Loh, Po-Ling ( June 2021 , Information and Inference: A Journal of the IMA)

   Abstract Estimating the mean of a probability distribution using i.i.d. samples is a classical problem in statistics, wherein finite-sample optimal estimators are sought under various distributional assumptions. In this paper, we consider the problem of mean estimation when independent samples are drawn from $d$-dimensional non-identical distributions possessing a common mean. When the distributions are radially symmetric and unimodal, we propose a novel estimator, which is a hybrid of the modal interval, shorth and median estimators and whose performance adapts to the level of heterogeneity in the data. We show that our estimator is near optimal when data are i.i.d. and when the fraction of ‘low-noise’ distributions is as small as $\varOmega \left (\frac{d \log n}{n}\right )$, where $n$ is the number of samples. We also derive minimax lower bounds on the expected error of any estimator that is agnostic to the scales of individual data points. Finally, we extend our theory to linear regression. In both the mean estimation and regression settings, we present computationally feasible versions of our estimators that run in time polynomial in the number of data points.