Wei Qian

Wei Qian, Chenxu Zhao, Yangyi Li et al.

Despite the recent progress in deep neural networks (DNNs), it remains challenging to explain the predictions made by DNNs. Existing explanation methods for DNNs mainly focus on post-hoc explanations where another explanatory model is employed to provide explanations. The fact that post-hoc methods can fail to reveal the actual original reasoning process of DNNs raises the need to build DNNs with built-in interpretability. Motivated by this, many self-explaining neural networks have been proposed to generate not only accurate predictions but also clear and intuitive insights into why a particular decision was made. However, existing self-explaining networks are limited in providing distribution-free uncertainty quantification for the two simultaneously generated prediction outcomes (i.e., a sample's final prediction and its corresponding explanations for interpreting that prediction). Importantly, they also fail to establish a connection between the confidence values assigned to the generated explanations in the interpretation layer and those allocated to the final predictions in the ultimate prediction layer. To tackle the aforementioned challenges, in this paper, we design a novel uncertainty modeling framework for self-explaining networks, which not only demonstrates strong distribution-free uncertainty modeling performance for the generated explanations in the interpretation layer but also excels in producing efficient and effective prediction sets for the final predictions based on the informative high-level basis explanations. We perform the theoretical analysis for the proposed framework. Extensive experimental evaluation demonstrates the effectiveness of the proposed uncertainty framework.

Wei Qian, Yuqian Zhang, Yudong Chen

$k$-means clustering is a fundamental problem in unsupervised learning. The problem concerns finding a partition of the data points into $k$ clusters such that the within-cluster variation is minimized. Despite its importance and wide applicability, a theoretical understanding of the $k$-means problem has not been completely satisfactory. Existing algorithms with theoretical performance guarantees often rely on sophisticated (sometimes artificial) algorithmic techniques and restricted assumptions on the data. The main challenge lies in the non-convex nature of the problem; in particular, there exist additional local solutions other than the global optimum. Moreover, the simplest and most popular algorithm for $k$-means, namely Lloyd's algorithm, generally converges to such spurious local solutions both in theory and in practice. In this paper, we approach the $k$-means problem from a new perspective, by investigating the structures of these spurious local solutions under a probabilistic generative model with $k$ ground truth clusters. As soon as $k=3$, spurious local minima provably exist, even for well-separated and balanced clusters. One such local minimum puts two centers at one true cluster, and the third center in the middle of the other two true clusters. For general $k$, one local minimum puts multiple centers at a true cluster, and one center in the middle of multiple true clusters. Perhaps surprisingly, we prove that this is essentially the only type of spurious local minima under a separation condition. Our results pertain to the $k$-means formulation for mixtures of Gaussians or bounded distributions. Our theoretical results corroborate existing empirical observations and provide justification for several improved algorithms for $k$-means clustering.

Wei Qian, Yuqian Zhang, Yudong Chen

This work studies the location estimation problem for a mixture of two rotation invariant log-concave densities. We demonstrate that Least Squares EM, a variant of the EM algorithm, converges to the true location parameter from a randomly initialized point. We establish the explicit convergence rates and sample complexity bounds, revealing their dependence on the signal-to-noise ratio and the tail property of the log-concave distribution. Moreover, we show that this global convergence property is robust under model mis-specification. Our analysis generalizes previous techniques for proving the convergence results for Gaussian mixtures. In particular, we make use of an angle-decreasing property for establishing global convergence of Least Squares EM beyond Gaussian settings, as $\ell_2$ distance contraction no longer holds globally for general log-concave mixtures.

Jeongyeol Kwon, Wei Qian, Constantine Caramanis et al.

The Expectation-Maximization algorithm is perhaps the most broadly used algorithm for inference of latent variable problems. A theoretical understanding of its performance, however, largely remains lacking. Recent results established that EM enjoys global convergence for Gaussian Mixture Models. For Mixed Linear Regression, however, only local convergence results have been established, and those only for the high SNR regime. We show here that EM converges for mixed linear regression with two components (it is known that it may fail to converge for three or more), and moreover that this convergence holds for random initialization. Our analysis reveals that EM exhibits very different behavior in Mixed Linear Regression from its behavior in Gaussian Mixture Models, and hence our proofs require the development of several new ideas.