scholarly journals Generalization Error Bounds of Gradient Descent for Learning Over-Parameterized Deep ReLU Networks

2020 ◽  
Vol 34 (04) ◽  
pp. 3349-3356
Author(s):  
Yuan Cao ◽  
Quanquan Gu
Keyword(s):  
Neural Networks ◽  
Error Bounds ◽  
Gradient Descent ◽  
Deep Neural Networks ◽  
Empirical Studies ◽  
Training Data ◽  
Generalization Error ◽  
Generalization Performance ◽  
Gradient Based ◽  
Random Initialization

Empirical studies show that gradient-based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. Very recently, a line of work explains in theory that with over-parameterization and proper random initialization, gradient-based methods can find the global minima of the training loss for DNNs. However, existing generalization error bounds are unable to explain the good generalization performance of over-parameterized DNNs. The major limitation of most existing generalization bounds is that they are based on uniform convergence and are independent of the training algorithm. In this work, we derive an algorithm-dependent generalization error bound for deep ReLU networks, and show that under certain assumptions on the data distribution, gradient descent (GD) with proper random initialization is able to train a sufficiently over-parameterized DNN to achieve arbitrarily small generalization error. Our work sheds light on explaining the good generalization performance of over-parameterized deep neural networks.

scholarly journals VECA: A Method for Detecting Overfitting in Neural Networks (Student Abstract)

2020 ◽  
Vol 34 (10) ◽  
pp. 13791-13792
Author(s):  
Liangzhu Ge ◽  
Yuexian Hou ◽  
Yaju Jiang ◽  
Shuai Yao ◽  
Chao Yang
Keyword(s):  
Neural Networks ◽  
Strong Correlation ◽  
Good Predictor ◽  
Deep Neural Networks ◽  
Training Data ◽  
Training Process ◽  
Generalization Performance ◽  
Validation Set ◽  
Fully Connected ◽  
Fully Connected Networks

Despite their widespread applications, deep neural networks often tend to overfit the training data. Here, we propose a measure called VECA (Variance of Eigenvalues of Covariance matrix of Activation matrix) and demonstrate that VECA is a good predictor of networks' generalization performance during the training process. Experiments performed on fully-connected networks and convolutional neural networks trained on benchmark image datasets show a strong correlation between test loss and VECA, which suggest that we can calculate the VECA to estimate generalization performance without sacrificing training data to be used as a validation set.

scholarly journals Mutual Information Based Learning Rate Decay for Stochastic Gradient Descent Training of Deep Neural Networks

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 560
Author(s):  
Shrihari Vasudevan
Keyword(s):  
Neural Networks ◽  
Mutual Information ◽  
Gradient Descent ◽  
Deep Neural Networks ◽  
Learning Rate ◽  
Stochastic Gradient ◽  
Stochastic Gradient Descent ◽  
Novel Approach ◽  
The Neural Network ◽  
Gradient Based

This paper demonstrates a novel approach to training deep neural networks using a Mutual Information (MI)-driven, decaying Learning Rate (LR), Stochastic Gradient Descent (SGD) algorithm. MI between the output of the neural network and true outcomes is used to adaptively set the LR for the network, in every epoch of the training cycle. This idea is extended to layer-wise setting of LR, as MI naturally provides a layer-wise performance metric. A LR range test determining the operating LR range is also proposed. Experiments compared this approach with popular alternatives such as gradient-based adaptive LR algorithms like Adam, RMSprop, and LARS. Competitive to better accuracy outcomes obtained in competitive to better time, demonstrate the feasibility of the metric and approach.

scholarly journals Stronger convergence results for deep residual networks: network width scales linearly with training data size

2020 ◽  
Author(s):  
Talha Cihad Gulcu
Keyword(s):  
Neural Networks ◽  
Gradient Descent ◽  
Deep Neural Networks ◽  
Full Rank ◽  
Network Size ◽  
Training Data ◽  
Quadratic Loss ◽  
First Order ◽  
Convergence Results ◽  
Rank Gradient

Abstract Deep neural networks are highly expressive machine learning models with the ability to interpolate arbitrary datasets. Deep nets are typically optimized via first-order methods, and the optimization process crucially depends on the characteristics of the network as well as the dataset. This work sheds light on the relation between the network size and the properties of the dataset with an emphasis on deep residual networks (ResNets). Our contribution is that if the network Jacobian is full rank, gradient descent for the quadratic loss and smooth activation converges to the global minima even if the network width $m$ of the ResNet scales linearly with the sample size $n$ and logarithmically with the network depth $H$. Consequently, our work is able to provide a theoretical guarantee for the convergence of deep neural networks in the $m=\varOmega (n,\log H)$ regime.

scholarly journals Evaluation of Power Insulator Detection Efficiency with the Use of Limited Training Dataset

2020 ◽  
Vol 10 (6) ◽  
pp. 2104
Author(s):  
Michał Tomaszewski ◽  
Paweł Michalski ◽  
Jakub Osuchowski
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Object Detection ◽  
Convolutional Neural Network ◽  
Deep Neural Networks ◽  
Detection Efficiency ◽  
Training Data ◽  
Training Dataset ◽  
Training Set ◽  
Convolutional Network

This article presents an analysis of the effectiveness of object detection in digital images with the application of a limited quantity of input. The possibility of using a limited set of learning data was achieved by developing a detailed scenario of the task, which strictly defined the conditions of detector operation in the considered case of a convolutional neural network. The described solution utilizes known architectures of deep neural networks in the process of learning and object detection. The article presents comparisons of results from detecting the most popular deep neural networks while maintaining a limited training set composed of a specific number of selected images from diagnostic video. The analyzed input material was recorded during an inspection flight conducted along high-voltage lines. The object detector was built for a power insulator. The main contribution of the presented papier is the evidence that a limited training set (in our case, just 60 training frames) could be used for object detection, assuming an outdoor scenario with low variability of environmental conditions. The decision of which network will generate the best result for such a limited training set is not a trivial task. Conducted research suggests that the deep neural networks will achieve different levels of effectiveness depending on the amount of training data. The most beneficial results were obtained for two convolutional neural networks: the faster region-convolutional neural network (faster R-CNN) and the region-based fully convolutional network (R-FCN). Faster R-CNN reached the highest AP (average precision) at a level of 0.8 for 60 frames. The R-FCN model gained a worse AP result; however, it can be noted that the relationship between the number of input samples and the obtained results has a significantly lower influence than in the case of other CNN models, which, in the authors’ assessment, is a desired feature in the case of a limited training set.

scholarly journals A Diffusion Approximation Theory of Momentum Stochastic Gradient Descent in Nonconvex Optimization

2021 ◽  
Author(s):  
Tianyi Liu ◽  
Zhehui Chen ◽  
Enlu Zhou ◽  
Tuo Zhao
Keyword(s):  
Neural Networks ◽  
Nonconvex Optimization ◽  
Gradient Descent ◽  
Deep Neural Networks ◽  
Optimization Problems ◽  
Saddle Points ◽  
Stochastic Gradient ◽  
Stochastic Gradient Descent ◽  
Nonconvex Optimization Problems ◽  
Empirical Success

Momentum stochastic gradient descent (MSGD) algorithm has been widely applied to many nonconvex optimization problems in machine learning (e.g., training deep neural networks, variational Bayesian inference, etc.). Despite its empirical success, there is still a lack of theoretical understanding of convergence properties of MSGD. To fill this gap, we propose to analyze the algorithmic behavior of MSGD by diffusion approximations for nonconvex optimization problems with strict saddle points and isolated local optima. Our study shows that the momentum helps escape from saddle points but hurts the convergence within the neighborhood of optima (if without the step size annealing or momentum annealing). Our theoretical discovery partially corroborates the empirical success of MSGD in training deep neural networks.

scholarly journals Domain Generalization Using a Mixture of Multiple Latent Domains

2020 ◽  
Vol 34 (07) ◽  
pp. 11749-11756 ◽  
Author(s):  
Toshihiko Matsuura ◽  
Tatsuya Harada
Keyword(s):  
Neural Networks ◽  
Deep Neural Networks ◽  
Multiple Source ◽  
Web Crawling ◽  
Not Given ◽  
Adversarial Learning ◽  
Generalized Model ◽  
Generalization Performance ◽  
Invariant Feature ◽  
Feature Extractor

When domains, which represent underlying data distributions, vary during training and testing processes, deep neural networks suffer a drop in their performance. Domain generalization allows improvements in the generalization performance for unseen target domains by using multiple source domains. Conventional methods assume that the domain to which each sample belongs is known in training. However, many datasets, such as those collected via web crawling, contain a mixture of multiple latent domains, in which the domain of each sample is unknown. This paper introduces domain generalization using a mixture of multiple latent domains as a novel and more realistic scenario, where we try to train a domain-generalized model without using domain labels. To address this scenario, we propose a method that iteratively divides samples into latent domains via clustering, and which trains the domain-invariant feature extractor shared among the divided latent domains via adversarial learning. We assume that the latent domain of images is reflected in their style, and thus, utilize style features for clustering. By using these features, our proposed method successfully discovers latent domains and achieves domain generalization even if the domain labels are not given. Experiments show that our proposed method can train a domain-generalized model without using domain labels. Moreover, it outperforms conventional domain generalization methods, including those that utilize domain labels.

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