A Bernstein-von Mises theorem in the nonparametric right-censoring model

In the recent Bayesian nonparametric literature, many examples have been reported in which Bayesian estimators and posterior distributions do not achieve the optimal convergence rate, indicating that the Bernstein-von Mises theorem does not hold. In this article, we give a positive result in this direction by showing that the Bernstein-von Mises theorem holds in survival models for a large class of prior processes neutral to the right. We also show that, for an arbitrarily given convergence rate n^{-\alpha} with 0<\alpha \leq 1/2, a prior process neutral to the right can be chosen so that its posterior distribution achieves the convergence rate n^{-\alpha}.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Statistics (http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000052

Calibrating nonconvex penalized regression in ultra-high dimension

We investigate high-dimensional nonconvex penalized regression, where the number of covariates may grow at an exponential rate. Although recent asymptotic theory established that there exists a local minimum possessing the oracle property under general conditions, it is still largely an open problem how to identify the oracle estimator among potentially multiple local minima. There are two main obstacles: (1) due to the presence of multiple minima, the solution path is nonunique and is not guaranteed to contain the oracle estimator; (2) even if a solution path is known to contain the oracle estimator, the optimal tuning parameter depends on many unknown factors and is hard to estimate. To address these two challenging issues, we first prove that an easy-to-calculate calibrated CCCP algorithm produces a consistent solution path which contains the oracle estimator with probability approaching one. Furthermore, we propose a high-dimensional BIC criterion and show that it can be applied to the solution path to select the optimal tuning parameter which asymptotically identifies the oracle estimator. The theory for a general class of nonconvex penalties in the ultra-high dimensional setup is established when the random errors follow the sub-Gaussian distribution. Monte Carlo studies confirm that the calibrated CCCP algorithm combined with the proposed high-dimensional BIC has desirable performance in identifying the underlying sparsity pattern for high-dimensional data analysis.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1159 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Within-group fairness: A guidance for more sound between-group fairness

As they have a vital effect on social decision-making, AI algorithms not only should be accurate and but also should not pose unfairness against certain sensitive groups (e.g., non-white, women). Various specially designed AI algorithms to ensure trained AI models to be fair between sensitive groups have been developed. In this paper, we raise a new issue that between-group fair AI models could treat individuals in a same sensitive group unfairly. We introduce a new concept of fairness so-called within-group fairness which requires that AI models should be fair for those in a same sensitive group as well as those in different sensitive groups. We materialize the concept of within-group fairness by proposing corresponding mathematical definitions and developing learning algorithms to control within-group fairness and between-group fairness simultaneously. Numerical studies show that the proposed learning algorithms improve within-group fairness without sacrificing accuracy as well as between-group fairness

Smooth function approximation by deep neural networks with general activation functions

There has been a growing interest in expressivity of deep neural networks. However, most of the existing work about this topic focuses only on the specific activation function such as ReLU or sigmoid. In this paper, we investigate the approximation ability of deep neural networks with a broad class of activation functions. This class of activation functions includes most of frequently used activation functions. We derive the required depth, width and sparsity of a deep neural network to approximate any H\"older smooth function upto a given approximation error for the large class of activation functions. Based on our approximation error analysis, we derive the minimax optimality of the deep neural network estimators with the general activation functions in both regression and classification problems.Comment: 24 page

Improving Performance of Semi-Supervised Learning by Adversarial Attacks

Semi-supervised learning (SSL) algorithm is a setup built upon a realistic assumption that access to a large amount of labeled data is tough. In this study, we present a generalized framework, named SCAR, standing for Selecting Clean samples with Adversarial Robustness, for improving the performance of recent SSL algorithms. By adversarially attacking pre-trained models with semi-supervision, our framework shows substantial advances in classifying images. We introduce how adversarial attacks successfully select high-confident unlabeled data to be labeled with current predictions. On CIFAR10, three recent SSL algorithms with SCAR result in significantly improved image classification.Comment: 4 page