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 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 A Geometric Interpretation of Stochastic Gradient Descent Using Diffusion Metrics

Entropy ◽  
2020 ◽  
Vol 22 (1) ◽  
pp. 101
Author(s):  
Rita Fioresi ◽  
Pratik Chaudhari ◽  
Stefano Soatto
Keyword(s):  
Neural Networks ◽  
General Relativity ◽  
Gradient Descent ◽  
Deep Neural Networks ◽  
Deterministic Model ◽  
Geometric Interpretation ◽  
Stochastic Gradient ◽  
Stochastic Gradient Descent ◽  
Diffusion Matrix

This paper is a step towards developing a geometric understanding of a popular algorithm for training deep neural networks named stochastic gradient descent (SGD). We built upon a recent result which observed that the noise in SGD while training typical networks is highly non-isotropic. That motivated a deterministic model in which the trajectories of our dynamical systems are described via geodesics of a family of metrics arising from a certain diffusion matrix; namely, the covariance of the stochastic gradients in SGD. Our model is analogous to models in general relativity: the role of the electromagnetic field in the latter is played by the gradient of the loss function of a deep network in the former.

scholarly journals MonolithNet: Training monolithic deep neural networks via a partitioned training strategy

2018 ◽  
Vol 4 (1) ◽  
pp. 3
Author(s):  
Rene Bidart ◽  
Alexander Wong
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Gradient Descent ◽  
Deep Neural Network ◽  
Deep Neural Networks ◽  
Stochastic Gradient ◽  
Stochastic Gradient Descent ◽  
Neural Net ◽  
Training Strategy ◽  
Effective Manner

In this study, we explore the training of monolithic deep neural net-works in an effective manner. One of the biggest challenges withtraining such networks to the desired level of accuracy is the dif-ficulty in converging to a good solution using iterative optimizationmethods such as stochastic gradient descent due to the enormousnumber of parameters that need to be learned. To achieve this,we introduce a partitioned training strategy, where proxy layersare connected to different partitions of a deep neural network toenable isolated training of a much smaller number of parametersto convergence. To illustrate the efficacy of this training strategy,we introduce MonolithNet, a massive residual deep neural networkconsisting of 437 million parameters. The trained MonolithNet wasable to achieve a top-1 accuracy of 97% on the CIFAR10 imageclassification dataset, which demonstrates the feasibility of the pro-posed training strategy for training monolithic deep neural networksto high accuracies.

scholarly journals Optical Recognition of Handwritten Logic Formulas Using Neural Networks

Electronics ◽  
2021 ◽  
Vol 10 (22) ◽  
pp. 2761
Author(s):  
Vaios Ampelakiotis ◽  
Isidoros Perikos ◽  
Ioannis Hatzilygeroudis ◽  
George Tsihrintzis
Keyword(s):  
Neural Networks ◽  
Character Recognition ◽  
Gradient Descent ◽  
Feedforward Neural Networks ◽  
Stochastic Gradient ◽  
Stochastic Gradient Descent ◽  
Training Algorithms ◽  
Gradient Descent Algorithm ◽  
Two Stages ◽  
And Training

In this paper, we present a handwritten character recognition (HCR) system that aims to recognize first-order logic handwritten formulas and create editable text files of the recognized formulas. Dense feedforward neural networks (NNs) are utilized, and their performance is examined under various training conditions and methods. More specifically, after three training algorithms (backpropagation, resilient propagation and stochastic gradient descent) had been tested, we created and trained an NN with the stochastic gradient descent algorithm, optimized by the Adam update rule, which was proved to be the best, using a trainset of 16,750 handwritten image samples of 28 × 28 each and a testset of 7947 samples. The final accuracy achieved is 90.13%. The general methodology followed consists of two stages: the image processing and the NN design and training. Finally, an application has been created that implements the methodology and automatically recognizes handwritten logic formulas. An interesting feature of the application is that it allows for creating new, user-oriented training sets and parameter settings, and thus new NN models.

scholarly journals On the Convergence of (Stochastic) Gradient Descent with Extrapolation for Non-Convex Minimization

2019 ◽  
Author(s):  
Yi Xu ◽  
Zhuoning Yuan ◽  
Sen Yang ◽  
Rong Jin ◽  
Tianbao Yang
Keyword(s):  
Convex Optimization ◽  
Gradient Descent ◽  
Optimization Problems ◽  
Upper Bounds ◽  
Convex Minimization ◽  
Stochastic Gradient ◽  
Stochastic Gradient Descent ◽  
First Order ◽  
Convex Optimization Problems ◽  
Learning Tasks

Extrapolation is a well-known technique for solving convex optimization and variational inequalities and recently attracts some attention for non-convex optimization. Several recent works have empirically shown its success in some machine learning tasks. However, it has not been analyzed for non-convex minimization and there still remains a gap between the theory and the practice. In this paper, we analyze gradient descent  and stochastic gradient descent with extrapolation for finding an approximate first-order stationary point in smooth non-convex optimization problems. Our convergence upper bounds show that the algorithms with extrapolation can be accelerated than without extrapolation.

scholarly journals Deep Convolutional Spiking Neural Networks for Image Classification

2021 ◽  
Author(s):  
Ruthvik Vaila
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Artificial Neural Networks ◽  
Gradient Descent ◽  
Stochastic Gradient ◽  
Spiking Neural Networks ◽  
Stochastic Gradient Descent ◽  
Data Set ◽  
Learning Capabilities ◽  
Artificial Neural

Spiking neural networks are biologically plausible counterparts of artificial neural networks. Artificial neural networks are usually trained with stochastic gradient descent (SGD) and spiking neural networks are trained with bioinspired spike timing dependent plasticity (STDP). Spiking networks could potentially help in reducing power usage owing to their binary activations. In this work, we use unsupervised STDP in the feature extraction layers of a neural network with instantaneous neurons to extract meaningful features. The extracted binary feature vectors are then classified using classification layers containing neurons with binary activations. Gradient descent (backpropagation) is used only on the output layer to perform training for classification. Surrogate gradients are proposed to perform backpropagation with binary gradients. The accuracies obtained for MNIST and the balanced EMNIST data set compare favorably with other approaches. The effect of the stochastic gradient descent (SGD) approximations on learning capabilities of our network are also explored. We also studied catastrophic forgetting and its effect on spiking neural networks (SNNs). For the experiments regarding catastrophic forgetting, in the classification sections of the network we use a modified synaptic intelligence that we refer to as cost per synapse metric as a regularizer to immunize the network against catastrophic forgetting in a Single-Incremental-Task scenario (SIT). In catastrophic forgetting experiments, we use MNIST and EMNIST handwritten digits datasets that were divided into five and ten incremental subtasks respectively. We also examine behavior of the spiking neural network and empirically study the effect of various hyperparameters on its learning capabilities using the software tool SPYKEFLOW that we developed. We employ MNIST, EMNIST and NMNIST data sets to produce our results.

scholarly journals THE USE OF CONTROL THEORY METHODS IN TRAINING NEURAL NETWORKS ON THE EXAMPLE OF TEETH RECOGNITION ON PANORAMIC X-RAY IMAGES

2021 ◽  
Vol 13 (2) ◽  
pp. 36-40
Author(s):  
A. Smorodin
Keyword(s):  
Neural Networks ◽  
Control Theory ◽  
Gradient Descent ◽  
Deep Neural Networks ◽  
Discrete Dynamical System ◽  
Stochastic Gradient Descent ◽  
Network Training ◽  
Panoramic Images ◽  
Important Field ◽  
New Algorithms

The article investigated a modification of stochastic gradient descent (SGD), based on the previously developed stabilization theory of discrete dynamical system cycles. Relation between stabilization of cycles in discrete dynamical systems and finding extremum points allowed us to apply new control methods to accelerate gradient descent when approaching local minima. Gradient descent is often used in training deep neural networks on a par with other iterative methods.  Two gradient SGD and Adam were experimented, and we conducted comparative experiments.  All experiments were conducted during solving a practical problem of teeth recognition on 2-D panoramic images. Network training showed that the new method outperforms the SGD in its capabilities and as for parameters chosen it approaches the capabilities of Adam, which is a “state of the art” method. Thus, practical utility of using control theory in the training of deep neural networks and possibility of expanding its applicability in the process of creating new algorithms in this important field are shown.

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