The standard method of generating random weights and biases in feedforward neural networks with random hidden nodes, selects them both from the uniform distribution over the same fixed interval. In this work, we show the drawbacks of this approach and propose a new method of generating random parameters. This method ensures the most nonlinear fragments of sigmoids, which are most useful in modeling target function nonlinearity, are kept in the input hypercube. In addition, we show how to generate activation functions with uniformly distributed slope angles.
Neural networks with random hidden nodes have gained increasing interest from researchers and practical applications. This is due to their unique features such as very fast training and universal approximation property. In these networks the weights and biases of hidden nodes determining the nonlinear feature mapping are set randomly and are not learned. Appropriate selection of the intervals from which weights and biases are selected is extremely important. This topic has not yet been sufficiently explored in the literature. In this work a method of generating random weights and biases is pro…
In this work, a method of random parameters generation for randomized learning of a single-hidden-layer feedforward neural network is proposed. The method firstly, randomly selects the slope angles of the hidden neurons activation functions from an interval adjusted to the target function, then randomly rotates the activation functions, and finally distributes them across the input space. For complex target functions the proposed method gives better results than the approach commonly used in practice, where the random parameters are selected from the fixed interval. This is because it introduc…
We prove a negative result for the approximation of functions defined on compact subsets of $\mathbb{R}^d$ (where $d \geq 2$) using feedforward neural networks with one hidden layer and arbitrary continuous activation function. In a nutshell, this result claims the existence of target functions that are as difficult to approximate using these neural networks as one may want. We also demonstrate an analogous result (for general $d \in \mathbb{N}$) for neural networks with an \emph{arbitrary} number of hidden layers, for activation functions that are either rational functions or continuous splin…
Aalto University（阿尔托大学）；Université de Montréal（蒙特利尔大学）
Stochastic binary hidden units in a multi-layer perceptron (MLP) network give at least three potential benefits when compared to deterministic MLP networks. (1) They allow to learn one-to-many type of mappings. (2) They can be used in structured prediction problems, where modeling the internal structure of the output is important. (3) Stochasticity has been shown to be an excellent regularizer, which makes generalization performance potentially better in general. However, training stochastic networks is considerably more difficult. We study training using M samples of hidden activations per in…
This work contributes to the development of a new data-driven method (D-DM) of feedforward neural networks (FNNs) learning. This method was proposed recently as a way of improving randomized learning of FNNs by adjusting the network parameters to the target function fluctuations. The method employs logistic sigmoid activation functions for hidden nodes. In this study, we introduce other activation functions, such as bipolar sigmoid, sine function, saturating linear functions, reLU, and softplus. We derive formulas for their parameters, i.e. weights and biases. In the simulation study, we evalu…
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；National University of Singapore（新加坡国立大学）
The back-propagation (BP) algorithm has been considered the de-facto method for training deep neural networks. It back-propagates errors from the output layer to the hidden layers in an exact manner using the transpose of the feedforward weights. However, it has been argued that this is not biologically plausible because back-propagating error signals with the exact incoming weights are not considered possible in biological neural systems. In this work, we propose a biologically plausible paradigm of neural architecture based on related literature in neuroscience and asymmetric BP-like methods…
A general Boltzmann machine with continuous visible and discrete integer valued hidden states is introduced. Under mild assumptions about the connection matrices, the probability density function of the visible units can be solved for analytically, yielding a novel parametric density function involving a ratio of Riemann-Theta functions. The conditional expectation of a hidden state for given visible states can also be calculated analytically, yielding a derivative of the logarithmic Riemann-Theta function. The conditional expectation can be used as activation function in a feedforward neural …
In this paper, we present a local geometric analysis to interpret how deep feedforward neural networks extract low-dimensional features from high-dimensional data. Our study shows that, in a local geometric region, the optimal weight in one layer of the neural network and the optimal feature generated by the previous layer comprise a low-rank approximation of a matrix that is determined by the Bayes action of this layer. This result holds (i) for analyzing both the output layer and the hidden layers of the neural network, and (ii) for neuron activation functions with non-vanishing gradients. W…
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）
We establish upper bounds for the minimal number of hidden units for which a binary stochastic feedforward network with sigmoid activation probabilities and a single hidden layer is a universal approximator of Markov kernels. We show that each possible probabilistic assignment of the states of $n$ output units, given the states of $k\geq1$ input units, can be approximated arbitrarily well by a network with $2^{k-1}(2^{n-1}-1)$ hidden units.
It is well known that Artificial Neural Networks are universal approximators. The classical result proves that, given a continuous function on a compact set on an n-dimensional space, then there exists a one-hidden-layer feedforward network which approximates the function. Such result proves the existence, but it does not provide a method for finding it. In this paper, a constructive approach to the proof of this property is given for the case of two-hidden-layer feedforward networks. This approach is based on an approximation of continuous functions by simplicial maps. Once a triangulation of…
Feedforward neural networks are widely used as universal predictive models to fit data distribution. Common gradient-based learning, however, suffers from many drawbacks making the training process ineffective and time-consuming. Alternative randomized learning does not use gradients but selects hidden node parameters randomly. This makes the training process extremely fast. However, the problem in randomized learning is how to determine the random parameters. A recently proposed method uses autoencoders for unsupervised parameter learning. This method showed superior performance on classifica…
Recurrent neural networks (RNNs) have been drawing much attention with great success in many applications like speech recognition and neural machine translation. Long short-term memory (LSTM) is one of the most popular RNN units in deep learning applications. LSTM transforms the input and the previous hidden states to the next states with the affine transformation, multiplication operations and a nonlinear activation function, which makes a good data representation for a given task. The affine transformation includes rotation and reflection, which change the semantic or syntactic information o…
Politecnico di Torino（都灵理工大学）；IEIIT-CNR Torino（意大利国家研究委员会电子、电信与信息工程研究所都灵分部）；INRIA（法国国家信息与自动化研究所）；Ecole polytechnique（巴黎综合理工学院）
We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation function in the output node (LSE networks) is a smooth universal approximator of continuous functions over convex, compact sets. By using a logarithmic transform, this class of networks maps to a family of subtraction-free ratios of generalized posynomials, which we also show to be universal approximators of positive functions over log-convex, compact subsets of the positive orthant. The main adva…
University of Pennsylvania（宾夕法尼亚大学）；Universität Leipzig（莱比锡大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）；Duke University（杜克大学）
We state concentration inequalities for the output of the hidden layers of a stochastic deep neural network (SDNN), as well as for the output of the whole SDNN. These results allow us to introduce an expected classifier (EC), and to give probabilistic upper bound for the classification error of the EC. We also state the optimal number of layers for the SDNN via an optimal stopping procedure. We apply our analysis to a stochastic version of a feedforward neural network with ReLU activation function.
Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院工业与系统工程学院）
In this paper, we explore some basic questions on the complexity of training neural networks with ReLU activation function. We show that it is NP-hard to train a two-hidden layer feedforward ReLU neural network. If dimension of the input data and the network topology is fixed, then we show that there exists a polynomial time algorithm for the same training problem. We also show that if sufficient over-parameterization is provided in the first hidden layer of ReLU neural network, then there is a polynomial time algorithm which finds weights such that output of the over-parameterized ReLU neural…
Saarland University（萨尔兰大学）；University of Tübingen（蒂宾根大学）
In the recent literature the important role of depth in deep learning has been emphasized. In this paper we argue that sufficient width of a feedforward network is equally important by answering the simple question under which conditions the decision regions of a neural network are connected. It turns out that for a class of activation functions including leaky ReLU, neural networks having a pyramidal structure, that is no layer has more hidden units than the input dimension, produce necessarily connected decision regions. This implies that a sufficiently wide hidden layer is necessary to guar…
Hebei University of Technology（河北工业大学）；Southwest Petroleum University（西南石油大学）
The three-way decisions strategy has been employed to construct network topology in a single hidden layer feedforward neural network (SFNN). However, this model has a general performance, and does not consider the process costs, since it has fixed threshold parameters. Inspired by the sequential three-way decisions (STWD), this paper proposes STWD with an SFNN (STWD-SFNN) to enhance the performance of networks on structured datasets. STWD-SFNN adopts multi-granularity levels to dynamically learn the number of hidden layer nodes from coarse to fine, and set the sequential threshold parameters. …
In this paper, we describe the hyper-parameter search problem in the field of machine learning and present a heuristic approach in an attempt to tackle it. In most learning algorithms, a set of hyper-parameters must be determined before training commences. The choice of hyper-parameters can affect the final model's performance significantly, but yet determining a good choice of hyper-parameters is in most cases complex and consumes large amount of computing resources. In this paper, we show the differences between an exhaustive search of hyper-parameters and a heuristic search, and show that t…
University of Torontoland（多伦多兰大学）；University of Edenborrow（伊甸伯勒大学）
We describe a novel family of models of multi- layer feedforward neural networks in which the activation functions are encoded via penalties in the training problem. Our approach is based on representing a non-decreasing activation function as the argmin of an appropriate convex optimiza- tion problem. The new framework allows for algo- rithms such as block-coordinate descent methods to be applied, in which each step is composed of a simple (no hidden layer) supervised learning problem that is parallelizable across data points and/or layers. Experiments indicate that the pro- posed models prov…
Singapore University of Technology and Design（新加坡科技设计大学）
The universal approximation theorem, in one of its most general versions, says that if we consider only continuous activation functions $σ$, then a standard feedforward neural network with one hidden layer is able to approximate any continuous multivariate function $f$ to any given approximation threshold $\varepsilon$, if and only if $σ$ is non-polynomial. In this paper, we give a direct algebraic proof of the theorem. Furthermore we shall explicitly quantify the number of hidden units required for approximation. Specifically, if $X\subseteq \mathbb{R}^n$ is compact, then a neural network wit…
Beta Basis Function Neural Network (BBFNN) is a special kind of kernel basis neural networks. It is a feedforward network typified by the use of beta function as a hidden activation function. Beta is a flexible transfer function representing richer forms than the common existing functions. As in every network, the architecture setting as well as the learning method are two main gauntlets faced by BBFNN. In this paper, new architecture and training algorithm are proposed for the BBFNN. An Extreme Learning Machine (ELM) is used as a training approach of BBFNN with the aim of quickening the train…
The University of Hong Kong（香港大学）；Wells Fargo（富国银行）
Network initialization is the first and critical step for training neural networks. In this paper, we propose a novel network initialization scheme based on the celebrated Stein's identity. By viewing multi-layer feedforward neural networks as cascades of multi-index models, the projection weights to the first hidden layer are initialized using eigenvectors of the cross-moment matrix between the input's second-order score function and the response. The input data is then forward propagated to the next layer and such a procedure can be repeated until all the hidden layers are initialized. Final…
In this paper, we first introduce batch normalization to the edRVFL network. This re-normalization method can help the network avoid divergence of the hidden features. Then we propose novel variants of Ensemble Deep Random Vector Functional Link (edRVFL). Weighted edRVFL (WedRVFL) uses weighting methods to give training samples different weights in different layers according to how the samples were classified confidently in the previous layer thereby increasing the ensemble's diversity and accuracy. Furthermore, a pruning-based edRVFL (PedRVFL) has also been proposed. We prune some inferior ne…
Simons Center for Systems Biology（西蒙斯系统生物学中心）；Institute for Advanced Study（高等研究院）；Princeton Neuroscience Institute（普林斯顿神经科学研究所）；Princeton University（普林斯顿大学）
A model of associative memory is studied, which stores and reliably retrieves many more patterns than the number of neurons in the network. We propose a simple duality between this dense associative memory and neural networks commonly used in deep learning. On the associative memory side of this duality, a family of models that smoothly interpolates between two limiting cases can be constructed. One limit is referred to as the feature-matching mode of pattern recognition, and the other one as the prototype regime. On the deep learning side of the duality, this family corresponds to feedforward…
University of Science and Technology of China（中国科学技术大学）；York University（约克大学）；IFLYTEK Research（科大讯飞研究院）
We introduce a new structure for memory neural networks, called feedforward sequential memory networks (FSMN), which can learn long-term dependency without using recurrent feedback. The proposed FSMN is a standard feedforward neural networks equipped with learnable sequential memory blocks in the hidden layers. In this work, we have applied FSMN to several language modeling (LM) tasks. Experimental results have shown that the memory blocks in FSMN can learn effective representations of long history. Experiments have shown that FSMN based language models can significantly outperform not only fe…
A particular case of Recurrent Neural Network (RNN) was introduced at the beginning of the 2000s under the name of Echo State Networks (ESNs). The ESN model overcomes the limitations during the training of the RNNs while introducing no significant disadvantages. Although the model presents some well-identified drawbacks when the parameters are not well initialised. The performance of an ESN is highly dependent on its internal parameters and pattern of connectivity of the hidden-hidden weights Often, the tuning of the network parameters can be hard and can impact in the accuracy of the models. …
Université de Montréal（蒙特利尔大学）；University of Southern California（南加州大学）
Recurrent Neural Networks (RNNs) are powerful models for sequential data that have the potential to learn long-term dependencies. However, they are computationally expensive to train and difficult to parallelize. Recent work has shown that normalizing intermediate representations of neural networks can significantly improve convergence rates in feedforward neural networks . In particular, batch normalization, which uses mini-batch statistics to standardize features, was shown to significantly reduce training time. In this paper, we show that applying batch normalization to the hidden-to-hidden…
Université de Montréal（蒙特利尔大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）
This paper explores the complexity of deep feedforward networks with linear pre-synaptic couplings and rectified linear activations. This is a contribution to the growing body of work contrasting the representational power of deep and shallow network architectures. In particular, we offer a framework for comparing deep and shallow models that belong to the family of piecewise linear functions based on computational geometry. We look at a deep rectifier multi-layer perceptron (MLP) with linear outputs units and compare it with a single layer version of the model. In the asymptotic regime, when …
University of Queensland（昆士兰大学）；International Computer Science Institute（国际计算机科学研究所）
An interesting approach to analyzing neural networks that has received renewed attention is to examine the equivalent kernel of the neural network. This is based on the fact that a fully connected feedforward network with one hidden layer, a certain weight distribution, an activation function, and an infinite number of neurons can be viewed as a mapping into a Hilbert space. We derive the equivalent kernels of MLPs with ReLU or Leaky ReLU activations for all rotationally-invariant weight distributions, generalizing a previous result that required Gaussian weight distributions. Additionally, th…
Feedforward neural networks with random hidden nodes suffer from a problem with the generation of random weights and biases as these are difficult to set optimally to obtain a good projection space. Typically, random parameters are drawn from an interval which is fixed before or adapted during the learning process. Due to the different functions of the weights and biases, selecting them both from the same interval is not a good idea. Recently more sophisticated methods of random parameters generation have been developed, such as the data-driven method proposed in \cite{Anon19}, where the sigmo…
Université catholique de Louvain（鲁汶天主教大学）；Institute of Neuroinformatics（神经信息学研究所）；University of Zürich（苏黎世大学）；ETH Zürich（苏黎世联邦理工学院）
While the backpropagation of error algorithm enables deep neural network training, it implies (i) bidirectional synaptic weight transport and (ii) update locking until the forward and backward passes are completed. Not only do these constraints preclude biological plausibility, but they also hinder the development of low-cost adaptive smart sensors at the edge, as they severely constrain memory accesses and entail buffering overhead. In this work, we show that the one-hot-encoded labels provided in supervised classification problems, denoted as targets, can be viewed as a proxy for the error s…
Single layer feedforward networks with random weights are successful in a variety of classification and regression problems. These networks are known for their non-iterative and fast training algorithms. A major drawback of these networks is that they require a large number of hidden units. In this paper, we propose the use of multi-activation hidden units. Such units increase the number of tunable parameters and enable formation of complex decision surfaces, without increasing the number of hidden units. We experimentally show that multi-activation hidden units can be used either to improve t…
Scalability properties of deep neural networks raise key research questions, particularly as the problems considered become larger and more challenging. This paper expands on the idea of conditional computation introduced by Bengio, et. al., where the nodes of a deep network are augmented by a set of gating units that determine when a node should be calculated. By factorizing the weight matrix into a low-rank approximation, an estimation of the sign of the pre-nonlinearity activation can be efficiently obtained. For networks using rectified-linear hidden units, this implies that the computatio…
Single layer feedforward networks with random weights are known for their non-iterative and fast training algorithms and are successful in a variety of classification and regression problems. A major drawback of these networks is that they require a large number of hidden units. In this paper, we propose a technique to reduce the number of hidden units substantially without affecting the accuracy of the networks significantly. We introduce the concept of primary and secondary hidden units. The weights for the primary hidden units are chosen randomly while the secondary hidden units are derived…
George Cybenko's landmark 1989 paper showed that there exists a feedforward neural network, with exactly one hidden layer (and a finite number of neurons), that can arbitrarily approximate a given continuous function $f$ on the unit hypercube. The paper did not address how to find the weight/parameters of such a network, or if finding them would be computationally feasible. This paper outlines an algorithm for a neural network with exactly one hidden layer to reconstruct any continuous scalar or vector valued continuous function.
Tokyo Institute of Technology（东京工业大学）；Denso IT Laboratory（电装信息技术研究所）
Target Propagation (TP) is a biologically more plausible algorithm than the error backpropagation (BP) to train deep networks, and improving practicality of TP is an open issue. TP methods require the feedforward and feedback networks to form layer-wise autoencoders for propagating the target values generated at the output layer. However, this causes certain drawbacks; e.g., careful hyperparameter tuning is required to synchronize the feedforward and feedback training, and frequent updates of the feedback path are usually required than that of the feedforward path. Learning of the feedforward …
Regional, Lattice and Logical Representations of Neural Networks
Center for Mathematics, Computing and Cognition（数学、计算与认知中心）；Federal University of ABC, Brazil（巴西阿布特大学）；Institute of Mathematics and Statistics（数学与统计研究所）；University of São Paulo, Brazil（巴西圣保罗大学）
The Hong Kong University of Science and Technology（香港科技大学）
Sparse connectivity is an important factor behind the success of convolutional neural networks and recurrent neural networks. In this paper, we consider the problem of learning sparse connectivity for feedforward neural networks (FNNs). The key idea is that a unit should be connected to a small number of units at the next level below that are strongly correlated. We use Chow-Liu's algorithm to learn a tree-structured probabilistic model for the units at the current level, use the tree to identify subsets of units that are strongly correlated, and introduce a new unit with receptive field over …
Changsha University of Science & Technology（长沙理工大学）；Honam University（湖南大学）
Recurrent Neural Networks (RNNs) have been widely applied to deal with temporal problems, such as flood forecasting and financial data processing. On the one hand, traditional RNNs models amplify the gradient issue due to the strict time serial dependency, making it difficult to realize a long-term memory function. On the other hand, RNNs cells are highly complex, which will significantly increase computational complexity and cause waste of computational resources during model training. In this paper, an improved Time Feedforward Connections Recurrent Neural Networks (TFC-RNNs) model was first…
We seek to impose linear, equality constraints in feedforward neural networks. As top layer predictors are usually nonlinear, this is a difficult task if we seek to deploy standard convex optimization methods and strong duality. To overcome this, we introduce a new saddle-point Lagrangian with auxiliary predictor variables on which constraints are imposed. Elimination of the auxiliary variables leads to a dual minimization problem on the Lagrange multipliers introduced to satisfy the linear constraints. This minimization problem is combined with the standard learning problem on the weight matr…
Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；Beijing Forestry University（北京林业大学）；Griffith University（格里菲斯大学）
There is a wide gap between symbolic reasoning and deep learning. In this research, we explore the possibility of using deep learning to improve symbolic reasoning. Briefly, in a reasoning system, a deep feedforward neural network is used to guide rewriting processes after learning from algebraic reasoning examples produced by humans. To enable the neural network to recognise patterns of algebraic expressions with non-deterministic sizes, reduced partial trees are used to represent the expressions. Also, to represent both top-down and bottom-up information of the expressions, a centralisation …
Université de Montréal（蒙特利尔大学）；Aalto University School of Science（阿尔托大学理学院）
In this paper, we explore different ways to extend a recurrent neural network (RNN) to a \textit{deep} RNN. We start by arguing that the concept of depth in an RNN is not as clear as it is in feedforward neural networks. By carefully analyzing and understanding the architecture of an RNN, however, we find three points of an RNN which may be made deeper; (1) input-to-hidden function, (2) hidden-to-hidden transition and (3) hidden-to-output function. Based on this observation, we propose two novel architectures of a deep RNN which are orthogonal to an earlier attempt of stacking multiple recurre…
Baltic Institute of Advanced Technology (BPTI)（波罗的海高级技术研究所（BPTI））
In this work we systematically analyze general properties of differential equations used as machine learning models. We demonstrate that the gradient of the loss function with respect to to the hidden state can be considered as a generalized momentum conjugate to the hidden state, allowing application of the tools of classical mechanics. In addition, we show that not only residual networks, but also feedforward neural networks with small nonlinearities and the weights matrices deviating only slightly from identity matrices can be related to the differential equations. We propose a differential…
This paper is concerned with the sparsification of the input-hidden weights of ELM (Extreme Learning Machine). For ordinary feedforward neural networks, the sparsification is usually done by introducing certain regularization technique into the learning process of the network. But this strategy can not be applied for ELM, since the input-hidden weights of ELM are supposed to be randomly chosen rather than to be learned. To this end, we propose a modified ELM, called ELM-LC (ELM with local connections), which is designed for the sparsification of the input-hidden weights as follows: The hidden …
The Ohio State University（俄亥俄州立大学）；Carnegie Mellon University（卡内基梅隆大学）
We study model recovery for data classification, where the training labels are generated from a one-hidden-layer neural network with sigmoid activations, also known as a single-layer feedforward network, and the goal is to recover the weights of the neural network. We consider two network models, the fully-connected network (FCN) and the non-overlapping convolutional neural network (CNN). We prove that with Gaussian inputs, the empirical risk based on cross entropy exhibits strong convexity and smoothness {\em uniformly} in a local neighborhood of the ground truth, as soon as the sample comple…
The performance of Feedforward neural network (FNN) fully de-pends upon the selection of architecture and training algorithm. FNN architecture can be tweaked using several parameters, such as the number of hidden layers, number of hidden neurons at each hidden layer and number of connections between layers. There may be exponential combinations for these architectural attributes which may be unmanageable manually, so it requires an algorithm which can automatically design an optimal architecture with high generalization ability. Numerous optimization algorithms have been utilized for FNN archi…
We establish, for the first time, connections between feedforward neural networks with ReLU activation and tropical geometry --- we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden layer can be characterized by zonotopes, which serve as building blocks for deeper networks; we relate decision boundaries of such neural networks to tropical hypersurfaces, a major object of study in tropical geometry; and we prove that linear regions of such neural networks corresp…
We propose a novel family of connectionist models based on kernel machines and consider the problem of learning layer-by-layer a compositional hypothesis class, i.e., a feedforward, multilayer architecture, in a supervised setting. In terms of the models, we present a principled method to "kernelize" (partly or completely) any neural network (NN). With this method, we obtain a counterpart of any given NN that is powered by kernel machines instead of neurons. In terms of learning, when learning a feedforward deep architecture in a supervised setting, one needs to train all the components simult…
International University of Business Agriculture and Technology（国际商业农业与技术大学）；Mississippi State University（密西西比州立大学）；Rajshahi University of Engineering and Technology（拉杰沙希工程技术大学）
Nowadays, deep learning can be employed to a wide ranges of fields including medicine, engineering, etc. In deep learning, Convolutional Neural Network (CNN) is extensively used in the pattern and sequence recognition, video analysis, natural language processing, spam detection, topic categorization, regression analysis, speech recognition, image classification, object detection, segmentation, face recognition, robotics, and control. The benefits associated with its near human level accuracies in large applications lead to the growing acceptance of CNN in recent years. The primary contribution…
