The Robustness of Estimator Composition

Neural Information Processing Systems

A composite estimator successively applies two (or more) estimators: on data decomposed into disjoint parts, it applies the first estimator on each part, then the second estimator on the outputs of the first estimator. And so on, if the composition is of more than two estimators. Informally, the breakdown point is the minimum fraction of data points which if significantly modified will also significantly modify the output of the estimator, so it is typically desirable to have a large breakdown point. Our main result shows that, under mild conditions on the individual estimators, the breakdown point of the composite estimator is the product of the breakdown points of the individual estimators. We also demonstrate several scenarios, ranging from regression to statistical testing, where this analysis is easy to apply, useful in understanding worst case robustness, and sheds powerful insights onto the associated data analysis.

The Nearest Neighbor Information Estimator is Adaptively Near Minimax Rate-Optimal

Neural Information Processing Systems

We analyze the Kozachenko–Leonenko (KL) fixed k-nearest neighbor estimator for the differential entropy. We obtain the first uniform upper bound on its performance for any fixed k over H\"{o}lder balls on a torus without assuming any conditions on how close the density could be from zero. Accompanying a recent minimax lower bound over the H\"{o}lder ball, we show that the KL estimator for any fixed k is achieving the minimax rates up to logarithmic factors without cognizance of the smoothness parameter s of the H\"{o}lder ball for $s \in (0,2]$ and arbitrary dimension d, rendering it the first estimator that provably satisfies this property.

Analysis of KNN Information Estimators for Smooth Distributions Machine Learning

KSG mutual information estimator, which is based on the distances of each sample to its k-th nearest neighbor, is widely used to estimate mutual information between two continuous random variables. Existing work has analyzed the convergence rate of this estimator for random variables whose densities are bounded away from zero in its support. In practice, however, KSG estimator also performs well for a much broader class of distributions, including not only those with bounded support and densities bounded away from zero, but also those with bounded support but densities approaching zero, and those with unbounded support. In this paper, we analyze the convergence rate of the error of KSG estimator for smooth distributions, whose support of density can be both bounded and unbounded. As KSG mutual information estimator can be viewed as an adaptive recombination of KL entropy estimators, in our analysis, we also provide convergence analysis of KL entropy estimator for a broad class of distributions.

Learning Additive Exponential Family Graphical Models via \ell_{2,1}-norm Regularized M-Estimation

Neural Information Processing Systems

We investigate a subclass of exponential family graphical models of which the sufficient statistics are defined by arbitrary additive forms. We propose two $\ell_{2,1}$-norm regularized maximum likelihood estimators to learn the model parameters from i.i.d. The first one is a joint MLE estimator which estimates all the parameters simultaneously. The second one is a node-wise conditional MLE estimator which estimates the parameters for each node individually. For both estimators, statistical analysis shows that under mild conditions the extra flexibility gained by the additive exponential family models comes at almost no cost of statistical efficiency.

Distributed Inference for Linear Support Vector Machine Machine Learning

The growing size of modern data brings many new challenges to existing statistical inference methodologies and theories, and calls for the development of distributed inferential approaches. This paper studies distributed inference for linear support vector machine (SVM) for the binary classification task. Despite a vast literature on SVM, much less is known about the inferential properties of SVM, especially in a distributed setting. In this paper, we propose a multi-round distributed linear-type (MDL) estimator for conducting inference for linear SVM. The proposed estimator is computationally efficient. In particular, it only requires an initial SVM estimator and then successively refines the estimator by solving simple weighted least squares problem. Theoretically, we establish the Bahadur representation of the estimator. Based on the representation, the asymptotic normality is further derived, which shows that the MDL estimator achieves the optimal statistical efficiency, i.e., the same efficiency as the classical linear SVM applying to the entire dataset in a single machine setup. Moreover, our asymptotic result avoids the condition on the number of machines or data batches, which is commonly assumed in distributed estimation literature, and allows the case of diverging dimension. We provide simulation studies to demonstrate the performance of the proposed MDL estimator.