Tian-Yi Zhou

Tian-Yi Zhou, Xiaoming Huo

This paper studies the binary classification of unbounded data from ${\mathbb R}^d$ generated under Gaussian Mixture Models (GMMs) using deep ReLU neural networks. We obtain $\unicode{x2013}$ for the first time $\unicode{x2013}$ non-asymptotic upper bounds and convergence rates of the excess risk (excess misclassification error) for the classification without restrictions on model parameters. The convergence rates we derive do not depend on dimension $d$, demonstrating that deep ReLU networks can overcome the curse of dimensionality in classification. While the majority of existing generalization analysis of classification algorithms relies on a bounded domain, we consider an unbounded domain by leveraging the analyticity and fast decay of Gaussian distributions. To facilitate our analysis, we give a novel approximation error bound for general analytic functions using ReLU networks, which may be of independent interest. Gaussian distributions can be adopted nicely to model data arising in applications, e.g., speeches, images, and texts; our results provide a theoretical verification of the observed efficiency of deep neural networks in practical classification problems.

Tian-Yi Zhou, Matthew Lau, Jizhou Chen et al. · gatech

Anomaly detection is an important problem in many application areas, such as network security. Many deep learning methods for unsupervised anomaly detection produce good empirical performance but lack theoretical guarantees. By casting anomaly detection into a binary classification problem, we establish non-asymptotic upper bounds and a convergence rate on the excess risk on rectified linear unit (ReLU) neural networks trained on synthetic anomalies. Our convergence rate on the excess risk matches the minimax optimal rate in the literature. Furthermore, we provide lower and upper bounds on the number of synthetic anomalies that can attain this optimality. For practical implementation, we relax some conditions to improve the search for the empirical risk minimizer, which leads to competitive performance to other classification-based methods for anomaly detection. Overall, our work provides the first theoretical guarantees of unsupervised neural network-based anomaly detectors and empirical insights on how to design them well.

Matthew Lau, Tian-Yi Zhou, Xiangchi Yuan et al. · gatech

Anomaly detection (AD) is a critical task across domains such as cybersecurity and healthcare. In the unsupervised setting, an effective and theoretically-grounded principle is to train classifiers to distinguish normal data from (synthetic) anomalies. We extend this principle to semi-supervised AD, where training data also include a limited labeled subset of anomalies possibly present in test time. We propose a theoretically-grounded and empirically effective framework for semi-supervised AD that combines known and synthetic anomalies during training. To analyze semi-supervised AD, we introduce the first mathematical formulation of semi-supervised AD, which generalizes unsupervised AD. Here, we show that synthetic anomalies enable (i) better anomaly modeling in low-density regions and (ii) optimal convergence guarantees for neural network classifiers -- the first theoretical result for semi-supervised AD. We empirically validate our framework on five diverse benchmarks, observing consistent performance gains. These improvements also extend beyond our theoretical framework to other classification-based AD methods, validating the generalizability of the synthetic anomaly principle in AD.