Artificial Intelligence
Modes of Learning
There are two modes of learning to choose from: batch and stochastic. In stochastic learning, each propagation is followed immediately by a weight update. In batch learning many propagations occur before updating the weights, accumulating errors over the samples within a batch. Stochastic learning introduces "noise" into the gradient descent process, using the local gradient calculated from one data point. This reduces the chance of the network getting stuck in a local minima. Yet batch learning typically yields a faster, more stable descent to a local minima, since each update is performed in the direction of the average error of the batch samples. In modern applications a common compromise choice is to use "mini-batches", meaning batch learning but with a batch of small size and with stochastically selected samples.
Occam's razor
a problem-solving principle attributed to William of Ockham (c. 1287-1347), who was an English Franciscan friar and scholastic philosopher and theologian. The principle can be interpreted as stating Among competing hypotheses, the one with the fewest assumptions should be selected. law of parsimony: the scientific principle that things are usually connected or behave in the simplest or most economical way, especially with reference to alternative evolutionary pathways.
Long short term memory (LSTM) network
A deep learning RNN that unlike traditional RNNs doesn't have the vanishing gradient problem (compare the section on training algorithms below). LSTM is normally augmented by recurrent gates called forget gates. LSTM RNNs prevent backpropagated errors from vanishing or exploding. Instead errors can flow backwards through unlimited numbers of virtual layers in LSTM RNNs unfolded in space. That is, LSTM can learn "Very Deep Learning" tasks that require memories of events that happened thousands or even millions of discrete time steps ago. Problem-specific LSTM-like topologies can be evolved. LSTM works even when there are long delays, and it can handle signals that have a mix of low and high frequency components. Today, many applications use stacks of LSTM RNNs and train them by Connectionist Temporal Classification (CTC) to find an RNN weight matrix that maximizes the probability of the label sequences in a training set, given the corresponding input sequences. CTC achieves both alignment and recognition. Around 2007, LSTM started to revolutionise speech recognition, outperforming traditional models in certain speech applications. In 2009, CTC-trained LSTM was the first RNN to win pattern recognition contests, when it won several competitions in connected handwriting recognition. In 2014, the Chinese search giant Baidu used CTC-trained RNNs to break the Switchboard Hub5'00 speech recognition benchmark, without using any traditional speech processing methods. LSTM also improved large-vocabulary speech recognition, text-to-speech synthesis, also for Google Android, and photo-real talking heads. In 2015, Google's speech recognition reportedly experienced a dramatic performance jump of 49% through CTC-trained LSTM, which is now available through Google voice search to all smartphone users. LSTM has also become very popular in the field of Natural Language Processing. Unlike previous models based on HMMs and similar concepts, LSTM can learn to recognise context-sensitive languages. LSTM improved machine translation, Language Modeling and Multilingual Language Processing. LSTM combined with Convolutional Neural Networks (CNNs) also improved automatic image captioning and a plethora of other applications. Published by Hochreiter & Schmidhuber in 1997
Limitations of GD and BP
Gradient descent with backpropagation is not guaranteed to find the global minimum of the error function, but only a local minimum; also, it has trouble crossing plateaus in the error function landscape. This issue, caused by the non-convexity of error functions in neural networks, was long thought to be a major drawback, but in a 2015 review article, Yann LeCun et al. argue that in many practical problems, it isn't.[3] Backpropagation learning does not require normalization of input vectors; however, normalization could improve performance.[4]
Learning as an optimization problem
Before showing the mathematical derivation of the backpropagation algorithm, it helps to develop some intuitions about the relationship between the actual output of a neuron and the correct output for a particular training case. Consider a simple neural network with two input units, one output unit and no hidden units. Each neuron uses a linear output[note 1] that is the weighted sum of its input. Initially, before training, the weights will be set randomly. Then the neuron learns from training examples, which in this case consists of a set of tuples (x_{1}, x_{2}, t) where x_{1} and x_{2} are the inputs to the network and t is the correct output (the output the network should eventually produce given the identical inputs). The network given x_{1} and x_{2} will compute an output y which very likely differs from t (since the weights are initially random). A common method for measuring the discrepancy between the expected output t and the actual output y is using the squared error measure: E=(t-y)^2, where E is the discrepancy or error. As an example, consider the network on a single training case: (1, 1, 0), thus the input x_{1} and x_{2} are 1 and 1 respectively and the correct output, t is 0. Now if the actual output y is plotted on the x-axis against the error E on the y-axis, the result is a parabola. The minimum of the parabola corresponds to the output y which minimizes the error E. For a single training case, the minimum also touches the x-axis, which means the error will be zero and the network can produce an output y that exactly matches the expected output t. Therefore, the problem of mapping inputs to outputs can be reduced to an optimization problem of finding a function that will produce the minimal error. However, the output of a neuron depends on the weighted sum of all its inputs: y=x_1w_1 + x_2w_2, where w_{1} and w_{2} are the weights on the connection from the input units to the output unit. Therefore, the error also depends on the incoming weights to the neuron, which is ultimately what needs to be changed in the network to enable learning. If each weight is plotted on a separate horizontal axis and the error on the vertical axis, the result is a parabolic bowl (If a neuron has k weights, then the dimension of the error surface would be k+1, thus a k+1 dimensional equivalent of the 2D parabola). Error surface of a linear neuron with two input weights The backpropagation algorithm aims to find the set of weights that minimizes the error. There are several methods for finding the minima of a parabola or any function in any dimension. One way is analytically by solving systems of equations, however this relies on the network being a linear system, and the goal is to be able to also train multi-layer, non-linear networks (since a multi-layered linear network is equivalent to a single-layer network). The method used in backpropagation is gradient descent.
Multi-layer perceptron network
Class of networks that consists of multiple layers of computational units, usually interconnected in a feed-forward way. Each neuron in one layer has directed connections to the neurons of the subsequent layer. In many applications the units of these networks apply a sigmoid function as an activation function.
Issues with recurrent neural networks (RNN)
Most RNNs used to have scaling issues. In particular, RNNs could not be easily trained for large numbers of neuron units nor for large numbers of inputs units. Recent advances in training techniques have greatly increased the capabilities of recurrent neural networks[citation needed]. Successful training has been mostly in time series problems with few inputs and in chemical process control.
Training data collection
Online learning is used for dynamic environments that provide a continuous stream of new training data patterns. Offline learning makes use of a training set of static patterns.
Delta rule
Perceptrons can be trained by a simple learning algorithm that is usually called the delta rule. It calculates the errors between calculated output and sample output data, and uses this to create an adjustment to the weights, thus implementing a form of gradient descent.
RNN's related fields and models
RNNs may behave chaotically. In such cases, dynamical systems theory may be used for analysis. Recurrent neural networks are in fact recursive neural networks with a particular structure: that of a linear chain. Whereas recursive neural networks operate on any hierarchical structure, combining child representations into parent representations, recurrent neural networks operate on the linear progression of time, combining the previous time step and a hidden representation into the representation for the current time step. In particular, recurrent neural networks can appear as nonlinear versions of finite impulse response and infinite impulse response filters and also as a nonlinear autoregressive exogenous model (NARX).[63]
An analogy for understanding gradient descent
The basic intuition behind gradient descent can be illustrated by a hypothetical scenario. A person is stuck in the mountains and is trying to get down (i.e. trying to find the minima). There is heavy fog such that visibility is extremely low. Therefore, the path down the mountain is not visible, so he must use local information to find the minima. He can use the method of gradient descent, which involves looking at the steepness of the hill at his current position, then proceeding in the direction with the steepest descent (i.e. downhill). If he was trying to find the top of the mountain (i.e. the maxima), then he would proceed in the direction steepest ascent (i.e. uphill). Using this method, he would eventually find his way down the mountain. However, assume also that the steepness of the hill is not immediately obvious with simple observation, but rather it requires a sophisticated instrument to measure, which the person happens to have at the moment. It takes quite some time to measure the steepness of the hill with the instrument, thus he should minimize his use of the instrument if he wanted to get down the mountain before sunset. The difficulty then is choosing the frequency at which he should measure the steepness of the hill so not to go off track. In this analogy, the person represents the backpropagation algorithm, and the path taken down the mountain represents the sequence of parameter settings that the algorithm will explore. The steepness of the hill represents the slope of the error surface at that point. The instrument used to measure steepness is differentiation (the slope of the error surface can be calculated by taking the derivative of the squared error function at that point). The direction he chooses to travel in aligns with the gradient of the error surface at that point. The amount of time he travels before taking another measurement is the learning rate of the algorithm. See the limitation section for a discussion of the limitations of this type of "hill climbing" algorithm.
History of CNN's
The design of convolutional neural networks follows the discovery of visual mechanisms in living organisms. Early 1968 work[11] showed that the animal visual cortex contains complex arrangements of cells, responsible for detecting light in small, overlapping sub-regions of the visual field, called receptive fields. The paper identified two basic cell types: simple cells, which respond maximally to specific edge-like patterns within their receptive field, and complex cells, which have larger receptive fields and are locally invariant to the exact position of the pattern. These cells act as local filters over the input space. The neocognitron, a predecessor to convolutional networks,[12] was introduced in a 1980 paper.[9][13] The neocognitron differs from convolutional networks because it does not force units located at several positions to have the same trainable weights. This idea appears in 1986 in the book version of the original backpropagation paper[14] (Figure 14). They were developed in 1988 for temporal signals.[15] Their design was improved in 1998,[16] generalized in 2003,[17] and simplified in the same year.[18] The famous LeNet-5 network can classify digits successfully and is applied to recognising hand-written check (cheque) numbers. However, given more complex problems the breadth and depth of the network increases and becomes limited by computing resources thus hindering performance. A different convolution-based design was proposed in 1988[19] for application to decomposition of one-dimensional electromyography convolved signals via de-convolution. This design was modified in 1989 to other de-convolution-based designs.[20][21] Following the 2005 paper that established the value of GPGPU for machine learning,[22] several publications described more efficient ways to train convolutional neural networks using GPU computing.[23][24][25][26] In 2011, they were refined and implemented on a GPU, with impressive results.[7] In 2012, Ciresan et al. significantly improved on the best performance in the literature for multiple image databases, including the MNIST database, the NORB database, the HWDB1.0 dataset (Chinese characters), the CIFAR10 dataset (dataset of 60000 32x32 labeled RGB images),[9] and the ImageNet dataset.[27]
Motivation of backprogation
The goal of any supervised learning algorithm is to find a function that best maps a set of inputs to its correct output. An example would be a simple classification task, where the input is an image of an animal, and the correct output would be the name of the animal. The goal and motivation for developing the backpropagation algorithm was to find a way to train a multi-layered neural network such that it can learn the appropriate internal representations to allow it to learn any arbitrary mapping of input to output.[1]
Universal Approximation Theorem
The universal approximation theorem for neural networks states that every continuous function that maps intervals of real numbers to some output interval of real numbers can be approximated arbitrarily closely by a multi-layer perceptron with just one hidden layer. This result holds for a wide range of activation functions, e.g. for the sigmoidal functions. Multi-layer networks use a variety of learning techniques, the most popular being back-propagation.
Global optimization methods
Training the weights in a neural network can be modeled as a non-linear global optimization problem. A target function can be formed to evaluate the fitness or error of a particular weight vector as follows: First, the weights in the network are set according to the weight vector. Next, the network is evaluated against the training sequence. Typically, the sum-squared-difference between the predictions and the target values specified in the training sequence is used to represent the error of the current weight vector. Arbitrary global optimization techniques may then be used to minimize this target function. The most common global optimization method for training RNNs is genetic algorithms, especially in unstructured networks.[60][61][62] Initially, the genetic algorithm is encoded with the neural network weights in a predefined manner where one gene in the chromosome represents one weight link, henceforth; the whole network is represented as a single chromosome. The fitness function is evaluated as follows: 1) each weight encoded in the chromosome is assigned to the respective weight link of the network ; 2) the training set of examples is then presented to the network which propagates the input signals forward ; 3) the mean-squared-error is returned to the fitness function ; 4) this function will then drive the genetic selection process. There are many chromosomes that make up the population; therefore, many different neural networks are evolved until a stopping criterion is satisfied. A common stopping scheme is: 1) when the neural network has learnt a certain percentage of the training data or 2) when the minimum value of the mean-squared-error is satisfied or 3) when the maximum number of training generations has been reached. The stopping criterion is evaluated by the fitness function as it gets the reciprocal of the mean-squared-error from each neural network during training. Therefore, the goal of the genetic algorithm is to maximize the fitness function, hence, reduce the mean-squared-error. Other global (and/or evolutionary) optimization techniques may be used to seek a good set of weights such as simulated annealing or particle swarm optimization.
overfitting
Usually a learning algorithm is trained using some set of "training data": exemplary situations for which the desired output is known. The goal is that the algorithm will also perform well on predicting the output when fed "validation data" that was not encountered during its training. Overfitting is the use of models or procedures that violate Occam's razor, for example by including more adjustable parameters than are ultimately optimal, or by using a more complicated approach than is ultimately optimal. For an example where there are too many adjustable parameters, consider a dataset where training data for y can be adequately predicted by a linear function of two dependent variables. Such a function requires only three parameters (the intercept and two slopes). Replacing this simple function with a new, more complex quadratic function, or with a new, more complex linear function on more than two dependent variables, carries a risk: Occam's razor implies that any given complex function is a priori less probable than any given simple function. If the new, more complicated function is selected instead of the simple function, and if there was not a large enough gain in training-data fit to offset the complexity increase, then the new complex function "overfits" the data, and the complex overfitted function will likely perform worse than the simpler function on validation data outside the training dataset, even though the complex function performed as well, or perhaps even better, on the training dataset.[2] When comparing different types of models, complexity cannot be measured solely by counting how many parameters exist in each model; the expressivity of each parameter must be considered as well. For example, it is nontrivial to directly compare the complexity of a neural net (which can track curvilinear relationships) with m parameters to a regression model with n parameters.[2] Overfitting is especially likely in cases where learning was performed too long or where training examples are rare, causing the learner to adjust to very specific random features of the training data, that have no causal relation to the target function. In this process of overfitting, the performance on the training examples still increases while the performance on unseen data becomes worse. As a simple example, consider a database of retail purchases that includes the item bought, the purchaser, and the date and time of purchase. It's easy to construct a model that will fit the training set perfectly by using the date and time of purchase to predict the other attributes; but this model will not generalize at all to new data, because those past times will never occur again. Generally, a learning algorithm is said to overfit relative to a simpler one if it is more accurate in fitting known data (hindsight) but less accurate in predicting new data (foresight). One can intuitively understand overfitting from the fact that information from all past experience can be divided into two groups: information that is relevant for the future and irrelevant information ("noise"). Everything else being equal, the more difficult a criterion is to predict (i.e., the higher its uncertainty), the more noise exists in past information that needs to be ignored. The problem is determining which part to ignore. A learning algorithm that can reduce the chance of fitting noise is called robust. Consequences[edit] The most obvious consequence of overfitting is poor performance on the validation dataset. Other negative consequences include:[2] A function that is overfitted is likely to request more information about each item in the validation dataset than does the optimal function; gathering this additional unneeded data can be expensive or error-prone, especially if each individual piece of information must be gathered by human observation and manual data-entry. A more complex, overfitted function is likely to be less portable than a simple one. At one extreme, a one-variable linear regression is so portable that, if necessary, it could even be done by hand. At the other extreme are models that can be reproduced only by exactly duplicating the original modeler's entire setup, making reuse or scientific reproduction difficult.
Distinguishing features of CNN's
While traditional multilayer perceptron (MLP) models were successfully used for image recognition, due to the full connectivity between nodes they suffer from the curse of dimensionality and thus do not scale well to higher resolution images. For example, in CIFAR-10, images are only of size 32x32x3 (32 wide, 32 high, 3 color channels), so a single fully connected neuron in a first hidden layer of a regular neural network would have 32*32*3 = 3,072 weights. A 200x200 image, however, would lead to neurons that have 200*200*3 = 120,000 weights. Such network architecture does not take into account the spatial structure of data, treating input pixels which are far apart and close together on exactly the same footing. Clearly, the full connectivity of neurons is wasteful in the framework of image recognition, and the huge number of parameters quickly leads to overfitting. Convolutional neural networks are biologically inspired variants of multilayer perceptrons, designed to emulate the behaviour of a visual cortex. These models mitigate the challenges posed by the MLP architecture by exploiting the strong spatially local correlation present in natural images. As opposed to MLPs, CNN have the following distinguishing features: 3D volumes of neurons. The layers of a CNN have neurons arranged in 3 dimensions: width, height and depth. The neurons inside a layer are only connected to a small region of the layer before it, called a receptive field. Distinct types of layers, both locally and completely connected, are stacked to form a CNN architecture. Local connectivity: following the concept of receptive fields, CNNs exploit spatially local correlation by enforcing a local connectivity pattern between neurons of adjacent layers. The architecture thus ensures that the learnt "filters" produce the strongest response to a spatially local input pattern. Stacking many such layers leads to non-linear "filters" that become increasingly "global" (i.e. responsive to a larger region of pixel space). This allows the network to first create good representations of small parts of the input, then assemble representations of larger areas from them. Shared weights: In CNNs, each filter is replicated across the entire visual field. These replicated units share the same parameterization (weight vector and bias) and form a feature map. This means that all the neurons in a given convolutional layer detect exactly the same feature. Replicating units in this way allows for features to be detected regardless of their position in the visual field, thus constituting the property of translation invariance. Together, these properties allow convolutional neural networks to achieve better generalization on vision problems. Weight sharing also helps by dramatically reducing the number of free parameters being learnt, thus lowering the memory requirements for running the network. Decreasing the memory footprint allows the training of larger, more powerful networks.
Recurrent Neural Network (RNN)
a class of artificial neural network where connections between units form a directed cycle. This creates an internal state of the network which allows it to exhibit dynamic temporal behavior. Unlike feedforward neural networks, RNNs can use their internal memory to process arbitrary sequences of inputs. This makes them applicable to tasks such as unsegmented connected handwriting recognition or speech recognition.
Gradient descent
a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or of the approximate gradient) of the function at the current point. If instead one takes steps proportional to the positive of the gradient, one approaches a local maximum of that function; the procedure is then known as gradient ascent. Gradient descent is also known as steepest descent, or the method of steepest descent. Gradient descent should not be confused with the method of steepest descent for approximating integrals. To minimize total error, gradient descent can be used to change each weight in proportion to the derivative of the error with respect to that weight, provided the non-linear activation functions are differentiable. Various methods for doing so were developed in the 1980s and early 1990s by Paul Werbos, Ronald J. Williams, Tony Robinson, Jürgen Schmidhuber, Sepp Hochreiter, Barak Pearlmutter, and others. The standard method is called "backpropagation through time" or BPTT, and is a generalization of back-propagation for feed-forward networks,[50][51] and like that method, is an instance of Automatic differentiation in the reverse accumulation mode or Pontryagin's minimum principle. A more computationally expensive online variant is called "Real-Time Recurrent Learning" or RTRL,[52][53] which is an instance of Automatic differentiation in the forward accumulation mode with stacked tangent vectors. Unlike BPTT this algorithm is local in time but not local in space.[54][55] There also is an online hybrid between BPTT and RTRL with intermediate complexity,[56][57] and there are variants for continuous time.[58] A major problem with gradient descent for standard RNN architectures is that error gradients vanish exponentially quickly with the size of the time lag between important events.[12][59] The Long short term memory architecture together with a BPTT/RTRL hybrid learning method was introduced in an attempt to overcome these problems.[16] Gradient descent is based on the observation that if the multi-variable function F(x) is defined and differentiable in a neighborhood of a point a, then F(x) decreases fastest if one goes from a in the direction of the negative gradient of F at a, -{upside down delta}F(a).
automatic differentiation
a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program. not symbolic or numerical differentiation.
Convolutional neural network (CNN or ConvNet)
a type of feed-forward artificial neural network in which the connectivity pattern between its neurons is inspired by the organization of the animal visual cortex, whose individual neurons are arranged in such a way that they respond to overlapping regions tiling the visual field.[1] Convolutional networks were inspired by biological processes[2] and are variations of multilayer perceptrons designed to use minimal amounts of preprocessing.[3] They have wide applications in image and video recognition, recommender systems and natural language processing. When used for image recognition, convolutional neural networks (CNNs) consist of multiple layers of small neuron collections which process portions of the input image, called receptive fields. The outputs of these collections are then tiled so that their input regions overlap, to obtain a better representation of the original image; this is repeated for every such layer. Tiling allows CNNs to tolerate translation of the input image.[6] Convolutional networks may include local or global pooling layers, which combine the outputs of neuron clusters.[7][8] They also consist of various combinations of convolutional and fully connected layers, with pointwise nonlinearity applied at the end of or after each layer.[9] To reduce the number of free parameters and improve generalization, a convolution operation on small regions of input is introduced. One major advantage of convolutional networks is the use of shared weight in convolutional layers, which means that the same filter (weights bank) is used for each pixel in the layer; this both reduces memory footprint and improves performance.[3] Some time delay neural networks also use a very similar architecture to convolutional neural networks, especially those for image recognition and/or classification tasks, since the tiling of neuron outputs can be done in timed stages, in a manner useful for analysis of images.[10] Compared to other image classification algorithms, convolutional neural networks use relatively little pre-processing. This means that the network is responsible for learning the filters that in traditional algorithms were hand-engineered. The lack of dependence on prior knowledge and human effort in designing features is a major advantage for CNNs.
Backpropagation
an abbreviation for "backward propagation of errors", is a common method of training artificial neural networks used in conjunction with an optimization method such as gradient descent. The method calculates the gradient of a loss function with respect to all the weights in the network. The gradient is fed to the optimization method which in turn uses it to update the weights, in an attempt to minimize the loss function. Backpropagation requires a known, desired output for each input value in order to calculate the loss function gradient. It is therefore usually considered to be a supervised learning method, although it is also used in some unsupervised networks such as autoencoders. It is a generalization of the delta rule to multi-layered feedforward networks, made possible by using the chain rule to iteratively compute gradients for each layer. Backpropagation requires that the activation function used by the artificial neurons (or "nodes") be differentiable. The backpropagation learning algorithm can be divided into two phases: propagation and weight update. Phase 1: Propagation[edit] Each propagation involves the following steps: Forward propagation of a training pattern's input through the neural network in order to generate the propagation's output activations. Backward propagation of the propagation's output activations through the neural network using the training pattern target in order to generate the deltas (the difference between the targeted and actual output values) of all output and hidden neurons. Phase 2: Weight update[edit] For each weight-synapse follow the following steps: Multiply its output delta and input activation to get the gradient of the weight. Subtract a ratio (percentage) from the gradient of the weight. This ratio (percentage) influences the speed and quality of learning; it is called the learning rate. The greater the ratio, the faster the neuron trains; the lower the ratio, the more accurate the training is. The sign of the gradient of a weight indicates where the error is increasing, this is why the weight must be updated in the opposite direction. Repeat phase 1 and 2 until the performance of the network is satisfactory.
Recursive Neural Network
created by applying the same set of weights recursively over a differentiable graph-like structure, by traversing the structure in topological order. Such networks are typically also trained by the reverse mode of automatic differentiation. They were introduced to learn distributed representations of structure, such as logical terms. A special case of recursive neural networks is the RNN itself whose structure corresponds to a linear chain. Recursive neural networks have been applied to natural language processing. The Recursive Neural Tensor Network uses a tensor-based composition function for all nodes in the tree.
Single-layer perceptron network
simplest kind of neural network and FNN. consists of a single layer of output nodes; the inputs are fed directly to the outputs via a series of weights. The sum of the products of the weights and the inputs is calculated in each node, and if the value is above some threshold (typically 0) the neuron fires and takes the activated value (typically 1); otherwise it takes the deactivated value (typically -1). Neurons with this kind of activation function are also called artificial neurons or linear threshold units. In the literature the term perceptron often refers to networks consisting of just one of these units. A similar neuron was described by Warren McCulloch and Walter Pitts in the 1940s. Single-unit perceptrons are only capable of learning linearly separable patterns; in 1969 in a famous monograph entitled Perceptrons, Marvin Minsky and Seymour Papert showed that it was impossible for a single-layer perceptron network to learn an XOR function.
feedforward neural networks
the first and simplest type of artificial neural network devised. an artificial neural network wherein connections between the units do not form a cycle. This is different from recurrent neural networks. In this network, the information moves in only one direction, forward, from the input nodes, through the hidden nodes (if any) and to the output nodes. There are no cycles or loops in the network. 1) Single-layer perceptron 2) Multi-layer perceptron
