Lucy Xia

Ye Tian, Haolei Weng, Lucy Xia et al.

Unsupervised learning has been widely used in many real-world applications. One of the simplest and most important unsupervised learning models is the Gaussian mixture model (GMM). In this work, we study the multi-task learning problem on GMMs, which aims to leverage potentially similar GMM parameter structures among tasks to obtain improved learning performance compared to single-task learning. We propose a multi-task GMM learning procedure based on the EM algorithm that effectively utilizes unknown similarities between related tasks and is robust against a fraction of outlier tasks from arbitrary distributions. The proposed procedure is shown to achieve the minimax optimal rate of convergence for both parameter estimation error and the excess mis-clustering error, in a wide range of regimes. Moreover, we generalize our approach to tackle the problem of transfer learning for GMMs, where similar theoretical results are derived. Additionally, iterative unsupervised multi-task and transfer learning methods may suffer from an initialization alignment problem, and two alignment algorithms are proposed to resolve the issue. Finally, we demonstrate the effectiveness of our methods through simulations and real data examples. To the best of our knowledge, this is the first work studying multi-task and transfer learning on GMMs with theoretical guarantees.

Lucy Xia, Richard Zhao, Yanhui Wu et al.

This paper addresses the challenges in classifying textual data obtained from open online platforms, which are vulnerable to distortion. Most existing classification methods minimize the overall classification error and may yield an undesirably large type I error (relevant textual messages are classified as irrelevant), particularly when available data exhibit an asymmetry between relevant and irrelevant information. Data distortion exacerbates this situation and often leads to fallacious prediction. To deal with inestimable data distortion, we propose the use of the Neyman-Pearson (NP) classification paradigm, which minimizes type II error under a user-specified type I error constraint. Theoretically, we show that the NP oracle is unaffected by data distortion when the class conditional distributions remain the same. Empirically, we study a case of classifying posts about worker strikes obtained from a leading Chinese microblogging platform, which are frequently prone to extensive, unpredictable and inestimable censorship. We demonstrate that, even though the training and test data are susceptible to different distortion and therefore potentially follow different distributions, our proposed NP methods control the type I error on test data at the targeted level. The methods and implementation pipeline proposed in our case study are applicable to many other problems involving data distortion.

Xin Tong, Lucy Xia, Jiacheng Wang et al.

The Neyman-Pearson (NP) paradigm in binary classification seeks classifiers that achieve a minimal type II error while enforcing the prioritized type I error controlled under some user-specified level $α$. This paradigm serves naturally in applications such as severe disease diagnosis and spam detection, where people have clear priorities among the two error types. Recently, Tong, Feng and Li (2018) proposed a nonparametric umbrella algorithm that adapts all scoring-type classification methods (e.g., logistic regression, support vector machines, random forest) to respect the given type I error upper bound $α$ with high probability, without specific distributional assumptions on the features and the responses. Universal the umbrella algorithm is, it demands an explicit minimum sample size requirement on class $0$, which is often the more scarce class, such as in rare disease diagnosis applications. In this work, we employ the parametric linear discriminant analysis (LDA) model and propose a new parametric thresholding algorithm, which does not need the minimum sample size requirements on class $0$ observations and thus is suitable for small sample applications such as rare disease diagnosis. Leveraging both the existing nonparametric and the newly proposed parametric thresholding rules, we propose four LDA-based NP classifiers, for both low- and high-dimensional settings. On the theoretical front, we prove NP oracle inequalities for one proposed classifier, where the rate for excess type II error benefits from the explicit parametric model assumption. Furthermore, as NP classifiers involve a sample splitting step of class $0$ observations, we construct a new adaptive sample splitting scheme that can be applied universally to NP classifiers, and this adaptive strategy reduces the type II error of these classifiers.

Jianqing Fan, Yang Feng, Lucy Xia

Measuring conditional dependence is an important topic in statistics with broad applications including graphical models. Under a factor model setting, a new conditional dependence measure based on projection is proposed. The corresponding conditional independence test is developed with the asymptotic null distribution unveiled where the number of factors could be high-dimensional. It is also shown that the new test has control over the asymptotic significance level and can be calculated efficiently. A generic method for building dependency graphs without Gaussian assumption using the new test is elaborated. Numerical results and real data analysis show the superiority of the new method.

Dong Dai, Philippe Rigollet, Lucy Xia et al.

We consider the problem of aggregating a general collection of affine estimators for fixed design regression. Relevant examples include some commonly used statistical estimators such as least squares, ridge and robust least squares estimators. Dalalyan and Salmon (2012) have established that, for this problem, exponentially weighted (EW) model selection aggregation leads to sharp oracle inequalities in expectation, but similar bounds in deviation were not previously known. While results indicate that the same aggregation scheme may not satisfy sharp oracle inequalities with high probability, we prove that a weaker notion of oracle inequality for EW that holds with high probability. Moreover, using a generalization of the newly introduced $Q$-aggregation scheme we also prove sharp oracle inequalities that hold with high probability. Finally, we apply our results to universal aggregation and show that our proposed estimator leads simultaneously to all the best known bounds for aggregation, including $\ell_q$-aggregation, $q \in (0,1)$, with high probability.