scholarly journals Interval universal approximation for neural networks

2022 ◽  
Vol 6 (POPL) ◽  
pp. 1-29
Author(s):  
Zi Wang ◽  
Aws Albarghouthi ◽  
Gautam Prakriya ◽  
Somesh Jha
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Approximation Problem ◽  
Activation Function ◽  
Constructive Proof ◽  
Universal Approximation ◽  
Activation Functions ◽  
Complete Problems ◽  
Robust Network ◽  
Interval Approximation

To verify safety and robustness of neural networks, researchers have successfully applied abstract interpretation , primarily using the interval abstract domain. In this paper, we study the theoretical power and limits of the interval domain for neural-network verification. First, we introduce the interval universal approximation (IUA) theorem. IUA shows that neural networks not only can approximate any continuous function f (universal approximation) as we have known for decades, but we can find a neural network, using any well-behaved activation function, whose interval bounds are an arbitrarily close approximation of the set semantics of f (the result of applying f to a set of inputs). We call this notion of approximation interval approximation . Our theorem generalizes the recent result of Baader et al. from ReLUs to a rich class of activation functions that we call squashable functions . Additionally, the IUA theorem implies that we can always construct provably robust neural networks under ℓ ∞ -norm using almost any practical activation function. Second, we study the computational complexity of constructing neural networks that are amenable to precise interval analysis. This is a crucial question, as our constructive proof of IUA is exponential in the size of the approximation domain. We boil this question down to the problem of approximating the range of a neural network with squashable activation functions. We show that the range approximation problem (RA) is a Δ 2 -intermediate problem, which is strictly harder than NP -complete problems, assuming coNP ⊄ NP . As a result, IUA is an inherently hard problem : No matter what abstract domain or computational tools we consider to achieve interval approximation, there is no efficient construction of such a universal approximator. This implies that it is hard to construct a provably robust network, even if we have a robust network to start with.

Deep neural network based on generalized neo-fuzzy neurons and its learning based on backpropagation

2021 ◽  
Vol 26 (jai2021.26(1)) ◽  
pp. 32-41
Author(s):  
Bodyanskiy Y ◽  
◽  
Antonenko T ◽  
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Learning Process ◽  
Activation Function ◽  
Point Of View ◽  
Basic Unit ◽  
Theoretical Point ◽  
Activation Functions ◽  
Approximation Properties ◽  
Network Training

Modern approaches in deep neural networks have a number of issues related to the learning process and computational costs. This article considers the architecture grounded on an alternative approach to the basic unit of the neural network. This approach achieves optimization in the calculations and gives rise to an alternative way to solve the problems of the vanishing and exploding gradient. The main issue of the article is the usage of the deep stacked neo-fuzzy system, which uses a generalized neo-fuzzy neuron to optimize the learning process. This approach is non-standard from a theoretical point of view, so the paper presents the necessary mathematical calculations and describes all the intricacies of using this architecture from a practical point of view. From a theoretical point, the network learning process is fully disclosed. Derived all necessary calculations for the use of the backpropagation algorithm for network training. A feature of the network is the rapid calculation of the derivative for the activation functions of neurons. This is achieved through the use of fuzzy membership functions. The paper shows that the derivative of such function is a constant, and this is a reason for the statement of increasing in the optimization rate in comparison with neural networks which use neurons with more common activation functions (ReLU, sigmoid). The paper highlights the main points that can be improved in further theoretical developments on this topic. In general, these issues are related to the calculation of the activation function. The proposed methods cope with these points and allow approximation using the network, but the authors already have theoretical justifications for improving the speed and approximation properties of the network. The results of the comparison of the proposed network with standard neural network architectures are shown

scholarly journals On Transformative Adaptive Activation Functions in Neural Networks for Gene Expression Inference

2019 ◽  
Author(s):  
Vladimír Kunc ◽  
Jiří Kléma
Keyword(s):  
Neural Network ◽  
Gene Expression ◽  
Neural Networks ◽  
Expression Profiling ◽  
Mean Absolute Error ◽  
Absolute Error ◽  
Activation Function ◽  
Activation Functions ◽  
Hyperbolic Tangent ◽  
Sigmoid Activation Function

AbstractMotivationGene expression profiling was made cheaper by the NIH LINCS program that profiles only ~1, 000 selected landmark genes and uses them to reconstruct the whole profile. The D–GEX method employs neural networks to infer the whole profile. However, the original D–GEX can be further significantly improved.ResultsWe have analyzed the D–GEX method and determined that the inference can be improved using a logistic sigmoid activation function instead of the hyperbolic tangent. Moreover, we propose a novel transformative adaptive activation function that improves the gene expression inference even further and which generalizes several existing adaptive activation functions. Our improved neural network achieves average mean absolute error of 0.1340 which is a significant improvement over our reimplementation of the original D–GEX which achieves average mean absolute error 0.1637

scholarly journals Overview of Configuring Adaptive Activation Functions for Deep Neural Networks - A Comparative Study

2021 ◽  
Vol 3 (1) ◽  
pp. 10-22
Author(s):  
Wang Haoxiang ◽  
Smys S
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Learning Process ◽  
Deep Neural Networks ◽  
Research Work ◽  
Nonlinear Function ◽  
Activation Function ◽  
Classification Error ◽  
Activation Functions ◽  
Research Article

Recently, the deep neural networks (DNN) have demonstrated many performances in the pattern recognition paradigm. The research studies on DNN include depth layer networks, filters, training and testing datasets. Deep neural network is providing many solutions for nonlinear partial differential equations (PDE). This research article comprises of many activation functions for each neuron. Besides, these activation networks are allowing many neurons within the neuron networks. In this network, the multitude of the functions will be selected between node by node to minimize the classification error. This is the reason for selecting the adaptive activation function for deep neural networks. Therefore, the activation functions are adapted with every neuron on the network, which is used to reduce the classification error during the process. This research article discusses the scaling factor for activation function that provides better optimization for the process in the dynamic changes of procedure. The proposed adaptive activation function has better learning capability than fixed activation function in any neural network. The research articles compare the convergence rate, early training function, and accuracy between existing methods. Besides, this research work provides improvements in debt ideas of the learning process of various neural networks. This learning process works and tests the solution available in the domain of various frequency bands. In addition to that, both forward and inverse problems of the parameters in the overriding equation will be identified. The proposed method is very simple architecture and efficiency, robustness, and accuracy will be high when considering the nonlinear function. The overall classification performance will be improved in the resulting networks, which have been trained with common datasets. The proposed work is compared with the recent findings in neuroscience research and proved better performance.

scholarly journals Efficient Quantization for Neural Networks with Binary Weights and Low Bitwidth Activations

2019 ◽  
Vol 33 ◽  
pp. 3854-3861
Author(s):  
Kun Huang ◽  
Bingbing Ni ◽  
Xiaokang Yang
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Deep Neural Network ◽  
Activation Function ◽  
Limited Resources ◽  
Portable Devices ◽  
Training Procedure ◽  
Activation Functions ◽  
First Time ◽  
And Training

Quantization has shown stunning efficiency on deep neural network, especially for portable devices with limited resources. Most existing works uncritically extend weight quantization methods to activations. However, we take the view that best performance can be obtained by applying different quantization methods to weights and activations respectively. In this paper, we design a new activation function dubbed CReLU from the quantization perspective and further complement this design with appropriate initialization method and training procedure. Moreover, we develop a specific quantization strategy in which we formulate the forward and backward approximation of weights with binary values and quantize the activations to low bitwdth using linear or logarithmic quantizer. We show, for the first time, our final quantized model with binary weights and ultra low bitwidth activations outperforms the previous best models by large margins on ImageNet as well as achieving nearly a 10.85× theoretical speedup with ResNet-18. Furthermore, ablation experiments and theoretical analysis demonstrate the effectiveness and robustness of CReLU in comparison with other activation functions.

scholarly journals Universal Approximation Theorem for Tessarine-Valued Neural Networks

2021 ◽  
Author(s):  
Rafael A. F. Carniello ◽  
Wington L. Vital ◽  
Marcos Eduardo Valle
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Approximation Theorem ◽  
Superior Performance ◽  
Universal Approximation ◽  
Activation Functions ◽  
Hidden Layer ◽  
Arbitrary Precision ◽  
Vector Valued Function ◽  
Vector Valued

The universal approximation theorem ensures that any continuous real-valued function defined on a compact subset can be approximated with arbitrary precision by a single hidden layer neural network. In this paper, we show that the universal approximation theorem also holds for tessarine-valued neural networks. Precisely, any continuous tessarine-valued function can be approximated with arbitrary precision by a single hidden layer tessarine-valued neural network with split activation functions in the hidden layer. A simple numerical example, confirming the theoretical result and revealing the superior performance of a tessarine-valued neural network over a real-valued model for interpolating a vector-valued function, is presented in the paper.

scholarly journals Stacked Heterogeneous Neural Networks for Time Series Forecasting

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Florin Leon ◽  
Mihai Horia Zaharia
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Time Series ◽  
Case Studies ◽  
Multilayer Perceptron ◽  
Neural Model ◽  
Activation Function ◽  
Time Series Forecasting ◽  
The Other ◽  
Activation Functions

A hybrid model for time series forecasting is proposed. It is a stacked neural network, containing one normal multilayer perceptron with bipolar sigmoid activation functions, and the other with an exponential activation function in the output layer. As shown by the case studies, the proposed stacked hybrid neural model performs well on a variety of benchmark time series. The combination of weights of the two stack components that leads to optimal performance is also studied.

scholarly journals Two novel finite time convergent recurrent neural networks for tackling complex-valued systems of linear equation

Filomat ◽  
2020 ◽  
Vol 34 (15) ◽  
pp. 5009-5018
Author(s):  
Lei Ding ◽  
Lin Xiao ◽  
Kaiqing Zhou ◽  
Yonghong Lan ◽  
Yongsheng Zhang
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Linear Equation ◽  
Recurrent Neural Networks ◽  
Activation Function ◽  
Convergence Speed ◽  
The Other ◽  
Activation Functions ◽  
Complex Valued ◽  
Nonlinear Activation Function

Compared to the linear activation function, a suitable nonlinear activation function can accelerate the convergence speed. Based on this finding, we propose two modified Zhang neural network (ZNN) models using different nonlinear activation functions to tackle the complex-valued systems of linear equation (CVSLE) problems in this paper. To fulfill this goal, we first propose a novel neural network called NRNN-SBP model by introducing the sign-bi-power activation function. Then, we propose another novel neural network called NRNN-IRN model by introducing the tunable activation function. Finally, simulative results demonstrate that the convergence speed of NRNN-SBP and the NRNN-IRN is faster than that of the FTRNN model. On the other hand, these results also reveal that different nonlinear activation function will have a different effect on the convergence rate for different CVSLE problems.

SCORING MODELING BASED ON NEURAL NETWORKS FOR DETERMINING A BANK BORROWER'S RATING

2020 ◽  
Vol 2020 (10) ◽  
pp. 54-62
Author(s):  
Oleksii VASYLIEV ◽  
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Network Architecture ◽  
Statistical Data ◽  
Activation Function ◽  
Decision Making Process ◽  
Neural Network Architecture ◽  
Acceptable Accuracy ◽  
The Neural Network ◽  
Sigmoid Activation Function

The problem of applying neural networks to calculate ratings used in banking in the decision-making process on granting or not granting loans to borrowers is considered. The task is to determine the rating function of the borrower based on a set of statistical data on the effectiveness of loans provided by the bank. When constructing a regression model to calculate the rating function, it is necessary to know its general form. If so, the task is to calculate the parameters that are included in the expression for the rating function. In contrast to this approach, in the case of using neural networks, there is no need to specify the general form for the rating function. Instead, certain neural network architecture is chosen and parameters are calculated for it on the basis of statistical data. Importantly, the same neural network architecture can be used to process different sets of statistical data. The disadvantages of using neural networks include the need to calculate a large number of parameters. There is also no universal algorithm that would determine the optimal neural network architecture. As an example of the use of neural networks to determine the borrower's rating, a model system is considered, in which the borrower's rating is determined by a known non-analytical rating function. A neural network with two inner layers, which contain, respectively, three and two neurons and have a sigmoid activation function, is used for modeling. It is shown that the use of the neural network allows restoring the borrower's rating function with quite acceptable accuracy.

Analysis of Non-Linear Activation Functions for Classification Tasks Using Convolutional Neural Networks

2019 ◽  
Vol 12 (3) ◽  
pp. 156-161 ◽  
Author(s):  
Aman Dureja ◽  
Payal Pahwa
Keyword(s):  
Neural Networks ◽  
Deep Neural Networks ◽  
Activation Function ◽  
Primary Objective ◽  
Experimental Comparison ◽  
Activation Functions ◽  
Practical Applications ◽  
Network Activation ◽  
Non Linear ◽  
Hidden Layer

Background: In making the deep neural network, activation functions play an important role. But the choice of activation functions also affects the network in term of optimization and to retrieve the better results. Several activation functions have been introduced in machine learning for many practical applications. But which activation function should use at hidden layer of deep neural networks was not identified. Objective: The primary objective of this analysis was to describe which activation function must be used at hidden layers for deep neural networks to solve complex non-linear problems. Methods: The configuration for this comparative model was used by using the datasets of 2 classes (Cat/Dog). The number of Convolutional layer used in this network was 3 and the pooling layer was also introduced after each layer of CNN layer. The total of the dataset was divided into the two parts. The first 8000 images were mainly used for training the network and the next 2000 images were used for testing the network. Results: The experimental comparison was done by analyzing the network by taking different activation functions on each layer of CNN network. The validation error and accuracy on Cat/Dog dataset were analyzed using activation functions (ReLU, Tanh, Selu, PRelu, Elu) at number of hidden layers. Overall the Relu gave best performance with the validation loss at 25th Epoch 0.3912 and validation accuracy at 25th Epoch 0.8320. Conclusion: It is found that a CNN model with ReLU hidden layers (3 hidden layers here) gives best results and improve overall performance better in term of accuracy and speed. These advantages of ReLU in CNN at number of hidden layers are helpful to effectively and fast retrieval of images from the databases.

scholarly journals Hardware implementation of radial-basis neural networks with Gaussian activation functions on FPGA

2021 ◽  
Author(s):  
Volodymyr Shymkovych ◽  
Sergii Telenyk ◽  
Petro Kravets
Keyword(s):  
Neural Networks ◽  
Hardware Implementation ◽  
Gaussian Function ◽  
Activation Function ◽  
Rbf Neural Networks ◽  
Activation Functions ◽  
Rbf Network ◽  
Combination Scheme ◽  
Radial Basis ◽  
Hidden Layer

AbstractThis article introduces a method for realizing the Gaussian activation function of radial-basis (RBF) neural networks with their hardware implementation on field-programmable gaits area (FPGAs). The results of modeling of the Gaussian function on FPGA chips of different families have been presented. RBF neural networks of various topologies have been synthesized and investigated. The hardware component implemented by this algorithm is an RBF neural network with four neurons of the latent layer and one neuron with a sigmoid activation function on an FPGA using 16-bit numbers with a fixed point, which took 1193 logic matrix gate (LUTs—LookUpTable). Each hidden layer neuron of the RBF network is designed on an FPGA as a separate computing unit. The speed as a total delay of the combination scheme of the block RBF network was 101.579 ns. The implementation of the Gaussian activation functions of the hidden layer of the RBF network occupies 106 LUTs, and the speed of the Gaussian activation functions is 29.33 ns. The absolute error is ± 0.005. The Spartan 3 family of chips for modeling has been used to get these results. Modeling on chips of other series has been also introduced in the article. RBF neural networks of various topologies have been synthesized and investigated. Hardware implementation of RBF neural networks with such speed allows them to be used in real-time control systems for high-speed objects.

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