Classification of Data Generated by Gaussian Mixture Models Using Deep ReLU Networks

• URL: http://arxiv.org/abs/2308.08030v1
• Date: Tue, 15 Aug 2023 20:40:42 GMT
• Title: Classification of Data Generated by Gaussian Mixture Models Using Deep ReLU Networks
• Authors: Tian-Yi Zhou, Xiaoming Huo
• Abstract summary: This paper studies the binary classification of data from $math RMs. generated under Gaussian Mixture networks. We obtain $d2013x neural analysis rates for the first time convergence rates. Results provide a theoretical verification of deep neural networks in practical classification problems.
• Score: 28.437011792990347
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• Out-Of-Domain Unlabeled Data Improves Generalization [0.7589678255312519]
  We propose a novel framework for incorporating unlabeled data into semi-supervised classification problems. We show that unlabeled samples can be harnessed to narrow the generalization gap. We validate our claims through experiments conducted on a variety of synthetic and real-world datasets.
  arXiv  Detail & Related papers  (2023-09-29T02:00:03Z)
• On Excess Risk Convergence Rates of Neural Network Classifiers [8.329456268842227]
  We study the performance of plug-in classifiers based on neural networks in a binary classification setting as measured by their excess risks. We analyze the estimation and approximation properties of neural networks to obtain a dimension-free, uniform rate of convergence.
  arXiv  Detail & Related papers  (2023-09-26T17:14:10Z)
• Benign Overfitting in Deep Neural Networks under Lazy Training [72.28294823115502]
  We show that when the data distribution is well-separated, DNNs can achieve Bayes-optimal test error for classification. Our results indicate that interpolating with smoother functions leads to better generalization.
  arXiv  Detail & Related papers  (2023-05-30T19:37:44Z)
• Learning Distributions by Generative Adversarial Networks: Approximation and Generalization [0.6768558752130311]
  We study how well generative adversarial networks learn from finite samples by analyzing the convergence rates of these models. Our analysis is based on a new inequality oracle that decomposes the estimation error of GAN into the discriminator and generator approximation errors. For generator approximation error, we show that neural network can approximately transform a low-dimensional source distribution to a high-dimensional target distribution.
  arXiv  Detail & Related papers  (2022-05-25T09:26:17Z)
• On the Effective Number of Linear Regions in Shallow Univariate ReLU Networks: Convergence Guarantees and Implicit Bias [50.84569563188485]
  We show that gradient flow converges in direction when labels are determined by the sign of a target network with $r$ neurons. Our result may already hold for mild over- parameterization, where the width is $tildemathcalO(r)$ and independent of the sample size.
  arXiv  Detail & Related papers  (2022-05-18T16:57:10Z)
• Predicting Unreliable Predictions by Shattering a Neural Network [145.3823991041987]
  Piecewise linear neural networks can be split into subfunctions. Subfunctions have their own activation pattern, domain, and empirical error. Empirical error for the full network can be written as an expectation over subfunctions.
  arXiv  Detail & Related papers  (2021-06-15T18:34:41Z)
• Learning Gaussian Mixtures with Generalised Linear Models: Precise Asymptotics in High-dimensions [79.35722941720734]
  Generalised linear models for multi-class classification problems are one of the fundamental building blocks of modern machine learning tasks. We prove exacts characterising the estimator in high-dimensions via empirical risk minimisation. We discuss how our theory can be applied beyond the scope of synthetic data.
  arXiv  Detail & Related papers  (2021-06-07T16:53:56Z)
• Towards an Understanding of Benign Overfitting in Neural Networks [104.2956323934544]
  Modern machine learning models often employ a huge number of parameters and are typically optimized to have zero training loss. We examine how these benign overfitting phenomena occur in a two-layer neural network setting. We show that it is possible for the two-layer ReLU network interpolator to achieve a near minimax-optimal learning rate.
  arXiv  Detail & Related papers  (2021-06-06T19:08:53Z)
• Non-asymptotic Excess Risk Bounds for Classification with Deep Convolutional Neural Networks [6.051520664893158]
  We consider the problem of binary classification with a class of general deep convolutional neural networks. We define the prefactors of the risk bounds in terms of the input data dimension and other model parameters. We show that the classification methods with CNNs can circumvent the curse of dimensionality.
  arXiv  Detail & Related papers  (2021-05-01T15:55:04Z)
• Binary Classification of Gaussian Mixtures: Abundance of Support Vectors, Benign Overfitting and Regularization [39.35822033674126]
  We study binary linear classification under a generative Gaussian mixture model. We derive novel non-asymptotic bounds on the classification error of the latter. Our results extend to a noisy model with constant probability noise flips.
  arXiv  Detail & Related papers  (2020-11-18T07:59:55Z)
• Good Classifiers are Abundant in the Interpolating Regime [64.72044662855612]
  We develop a methodology to compute precisely the full distribution of test errors among interpolating classifiers. We find that test errors tend to concentrate around a small typical value $varepsilon*$, which deviates substantially from the test error of worst-case interpolating model. Our results show that the usual style of analysis in statistical learning theory may not be fine-grained enough to capture the good generalization performance observed in practice.
  arXiv  Detail & Related papers  (2020-06-22T21:12:31Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.