Neural network (machine learning)

Overview

In machine learning, a neural network (NN) or artificial neural network (ANN) is a computational model inspired by the structure and functions of biological neural networks. A neural network consists of connected units or nodes called artificial neurons, which loosely model the neurons in the brain. These are connected by edges, which model the synapses in the brain. Each artificial neuron receives signals from connected neurons, then processes them and sends a signal to other connected neurons. The "signal" is a real number, and the output of each neuron is computed by some non-linear function of the totality of its inputs, called the activation function. The strength of the signal at each connection is determined by a weight, which adjusts as part of the training process. Groups of neurons are aggregated into layers. Each layer performs a transformation on its inputs. Signals travel from the first layer (the input layer) to the last layer (the output layer), typically passing through multiple intermediate layers (hidden layers). A network is typically called a deep neural network if it has at least two hidden layers. Deep neural networks are capable of learning sophisticated hierarchical representations. Training neural networks is a compute-intensive process, accelerated by the use of graphics processing units (GPUs), and large datasets.

In reality, such textures and outlines would not be represented by single nodes, but rather by associated weight patterns of multiple nodes. Architectural innovations such as convolutional neural networks (CNNs) significantly improved performance in computer vision tasks, while recurrent neural networks (RNNs) enabled modeling of sequential data such as speech and time-series information. Transformer architectures introduced attention mechanisms that allow neural networks to model long-range dependencies in data and have been the basis of large language models. Artificial neural networks are used for myriad tasks including chatbots, large-scale text, image, and video generation, and robotics.

Mathematical foundations

Deep neural networks are based on statistics developed over 200 years ago. The simplest kind of feedforward neural network (FNN) is a linear network, which consists of a single layer of output nodes with linear activation functions; the inputs are fed directly to the outputs via weights. The sum of the products of the weights and the inputs is calculated at each node. The mean squared errors between these calculated outputs and the given target values are minimized by adjusting to the weights. This technique is the method of least squares or linear regression. It was used to find a rough linear fit to a set of points by Legendre (1805) and Gauss (1795) for the prediction of planetary movement.

Perceptrons

Computers are based on John von Neumann's model. They execute explicit lists of instructions with access to memory to record their changing state. Neural networks instead originated from efforts to model information processing in biological systems via connectionism. Unlike the von Neumann model, connectionist computing does not separate memory and processing. Warren McCulloch and Walter Pitts (1943) considered a non-learning computational model for neural networks. This model paved the way for research to split into one branch focused on biological processes and another focused on artificial intelligence. McCulloch and Pitts also developed mathematical models of artificial neurons capable of representing logical functions. In the late 1940s, D. O. Hebb proposed a learning hypothesis based on neural plasticity that became known as Hebbian learning. It was used in many early neural network experiments, such as Rosenblatt's perceptron and the Hopfield network. Farley and Clark (1954) used computational machines to simulate a Hebbian network. Other neural networks computational machines were created by Rochester, Holland, Habit and Duda (1956). In 1958, psychologist Frank Rosenblatt described the perceptron, one of the first implemented neural networks, funded by the United States Office of Naval Research. R. D. Joseph (1960) mentioned an earlier perceptron-like device by B. G. Farley and W. A. Clark of the MIT Lincoln Laboratory; however, according to Joseph, "they dropped the subject." The first perceptrons did not have adaptive hidden units. However, Joseph (1960) discussed multilayer that did. Rosenblatt (1962) cited and adopted these ideas, crediting work by H. D. Block and B. W. Knight. However, these early efforts did not lead to a working learning algorithm for hidden units, i.e., deep learning. The perceptron raised public excitement in neural networks, causing the US government to drastically increase funding. This contributed to "the Golden Age of AI", fueled by the optimistic claims made by computer scientists regarding the ability of perceptrons to emulate human intelligence.

Historical foundations and the Dartmouth proposal

Artificial neural networks were identified as a promising direction for artificial intelligence research in the 1955 proposal for the Dartmouth Summer Research Project on Artificial Intelligence. Neural network models initially faced major limitations. Hardware constraints limited network size and training efficiency, while theoretical understanding of learning algorithms remained incomplete. Many models used single-layer perceptrons, which were restricted to solving linearly separable problems. These limitations were highlighted in the book Perceptrons by Marvin Minsky and Seymour Papert, which deflated interest during the late 1960s and 1970s.

1960s and 1970s

Fundamental research was conducted on NNs in the 1960s and 1970s. The first working deep learning algorithm was the group method of data handling, a method to train arbitrarily deep neural networks, published by Alexey Ivakhnenko and Valentin Lapa in the Soviet Union (1965). They regarded it as a form of polynomial regression, generalizing Rosenblatt's perceptron. A 1971 paper described a deep network with eight layers trained by this method, training layer by layer via regression analysis. Superfluous hidden units were pruned using a separate validation set. The activation functions of the nodes were Kolmogorov-Gabor polynomials, the first deep networks with multiplicative units or "gates". The first deep learning multilayer perceptron (MLP) trained by stochastic gradient descent was published in 1967 by Shun'ichi Amari. In computer experiments conducted by Amari's student S. Saito, a five layer MLP with two modifiable layers learned internal representations to classify non-linearily separable pattern classes. Subsequent developments in hardware and hyperparameter tuning made end-to-end stochastic gradient descent the dominant technique for reducing loss (error). In 1969, Kunihiko Fukushima introduced the ReLU (rectified linear unit) activation function. ReLU is the most common activation function. Nevertheless, research stagnated in the United States after Minsky and Papert (1969), who emphasized that perceptrons were incapable of processing the exclusive-or circuit. However, this insight was irrelevant for the deep networks of Ivakhnenko (1965) and Amari (1967). In 1976, transfer learning was introduced.

Backpropagation

Interest in neural networks revived during the 1980s because of the novel backpropagation algorithm, which allowed multi-layer neural networks to be trained efficiently by propagating error gradients backward (from output back to input) through network layers. Backpropagation is an efficient application of the chain rule derived by Gottfried Wilhelm Leibniz in 1673 to networks of differentiable nodes. The terminology "back-propagating errors" was introduced in 1962 by Rosenblatt, but he did not describe how to implement this. Henry J. Kelley developed a precursor of backpropagation in 1960 in the context of control theory. In 1970, Seppo Linnainmaa published the modern form of backpropagation in his master's thesis (1970). G.M. Ostrovski et al. republished it in 1971. Paul Werbos applied backpropagation to neural networks in 1982 (his 1974 PhD thesis, reprinted in a 1994 book, did not describe the algorithm). In 1986, David E. Rumelhart, et al., popularized backpropagation but did not cite the original work.

Convolutional neural networks

Deep learning architectures for convolutional neural networks (CNNs) with convolutional layers and downsampling layers and weight replication began with the neocognitron introduced by Kunihiko Fukushima in 1979. Fukushima's CNN architecture also introduced max pooling, a popular downsampling procedure for CNNs. CNNs have become an essential tool for computer vision. The time delay neural network (TDNN) was introduced in 1987 by Alex Waibel to apply CNNs to phoneme recognition. It used convolutions, weight sharing, and backpropagation. In 1988, Wei Zhang applied a backpropagation-trained CNN to recognizing individual letters. In 1989, Yann LeCun et al. created a CNN called LeNet for recognizing handwritten ZIP codes on mail. Training required 3 days. In 1990, Wei Zhang implemented a CNN on optical computing hardware. In 1991, a CNN was applied to medical image object segmentation and breast cancer detection in mammograms. LeNet-5 (1998), a 7-level CNN by Yann LeCun et al. that classifies hand-written digits, was applied by banks to recognize numbers on checks digitized in 32×32 pixel images. From 1988 onward, the use of neural networks transformed the field of protein structure prediction, in particular when the first cascading networks were trained on profiles (matrices) produced by multiple sequence alignments.

Recurrent neural networks

One source of RNN was statistical mechanics. In 1972, Shun'ichi Amari proposed to modify the weights of an Ising model by Hebbian learning rule as a model of associative memory, adding in learning. This was popularized as the Hopfield network by John Hopfield (1982). Another origin of RNN was neuroscience. The word "recurrent" is used to describe loop-like structures in anatomy. In 1901, Cajal observed "recurrent semicircles" in the cerebellar cortex. Hebb considered "reverberating circuit" as an explanation for short-term memory. The McCulloch and Pitts paper (1943) considered neural networks that contain cycles, and noted that the current activity of such networks can be affected by activity indefinitely far in the past. In 1982 Crossbar Adaptive Array, a recurrent neural network with an array architecture (rather than a multilayer perceptron architecture), used direct recurrent connections from the output to the supervisor (teaching) inputs. In addition to computing actions (decisions), it computed internal state evaluations (emotions) of consequence situations. Eliminating the external supervisor, it introduced the self-learning method in neural networks. In cognitive psychology, the American Psychologist journal in the early 1980s debated the relation between cognition and emotion. Social psychologist Robert Zajonc in 1980 stated that emotion is computed first and is independent from cognition, while Richard Lazarus in 1982 stated that cognition is computed first and is inseparable from emotion. It was an example of a debate where an RNN contributed to an issue and also addressed cognitive psychology. The Jordan network (1986) and the Elman network (1990) applied RNN to study cognitive psychology. A distinct computationally efficient recurrent neural model, the Random neural network was introduced by Erol Gelenbe (1989) and used over several decades in numerous applications including tumor detection from MR images, video compression, network routing, and cyberattack detection. In the 1980s, backpropagation did not work well for deep RNNs. In 1991, Jürgen Schmidhuber proposed the "neural sequence chunker" or "neural history compressor" which introduced self-supervised pre-training (the "P" in ChatGPT) and neural knowledge distillation. In 1993, a neural history compressor system solved a "Very Deep Learning" task that required more than 1000 layers in an RNN unfolded in time. In 1991, Sepp Hochreiter's diploma thesis identified and analyzed the vanishing gradient problem and proposed recurrent residual connections to solve it. He and Schmidhuber introduced long short-term memory (LSTM), which set accuracy records in multiple application domains. This was not the ultimate version of LSTM, which required the forget gate, and was introduced in 1999. It became the default choice for RNN architecture. During 1985–1995, inspired by statistical mechanics, several architectures and methods were developed by Terry Sejnowski, Peter Dayan, Geoffrey Hinton, and others, including the Boltzmann machine, restricted Boltzmann machine, Helmholtz machine, and the wake-sleep algorithm. These were designed for unsupervised learning of deep generative models.

Modern deep learning

Between 2009 and 2012, NNs began winning prizes in image recognition contests, approaching human performance on various tasks, initially in pattern recognition and handwriting recognition. In 2011, DanNet, a CNN by Dan Ciresan, Ueli Meier, Jonathan Masci, Luca Maria Gambardella, and Schmidhuber achieved for the first time superhuman performance in a visual pattern recognition contest, outperforming traditional methods by a factor of 3. It then won more contests. They also showed how max-pooling CNNs on GPU improved performance. In October 2012, AlexNet by Alex Krizhevsky, Ilya Sutskever, and Hinton won the large-scale ImageNet competition by a significant margin over shallow machine learning methods. Further incremental improvements included the VGG-16 network by Karen Simonyan and Andrew Zisserman and Google's Inceptionv3. In 2012, Ng and Dean created a network that learned to recognize higher-level concepts, such as cats, from training only on unlabeled images. Unsupervised pre-training and increased computing power from GPUs and distributed computing allowed the use of larger networks, particularly in image and visual recognition problems, which became known as "deep learning". Radial basis function and wavelet networks were introduced in 2013. These can be shown to offer best approximation properties and have been applied in nonlinear system identification and classification applications. Generative adversarial networks (GANs) (Ian Goodfellow et al., 2014) became state of the art in generative modeling from 2014–2018. GAN was originally published in 1991 by Schmidhuber, who called it "artificial curiosity": two neural networks contest with each other in the form of a zero-sum game, where one network's gain is the other network's loss. The first network is a generative model that models a probability distribution over output patterns. The second network learns by gradient descent to predict the reactions of the environment to these patterns. Excellent image quality was achieved by Nvidia's StyleGAN (2018) based on the Progressive GAN by Tero Karras et al. Here, the GAN generator is grown from small to large scale in a pyramidal fashion. Image generation by GAN reached popular success, and provoked discussions concerning deepfakes. Diffusion models (2015) eclipsed GANs in generative modeling thereafter, with systems such as DALL·E 2 (2022) and Stable Diffusion (2022). In 2014, the state of the art was training "[a] very deep neural network" with 20 to 30 layers. Stacking too many layers led to a steep reduction in training accuracy, known as the "degradation" problem. In 2015, training very deep networks advanced with the highway network, published in May, and the residual neural network (ResNet) in December. ResNet behaves like an open-gated Highway Net.

Transformers

During the 2010s, the seq2seq model was developed, and attention mechanisms were added. This led to the modern transformer architecture in 2017 in "Attention Is All You Need". It requires computation time that is quadratic in the size of the context window. Schmidhuber's fast weight controller (1992) scales linearly and was later shown to be equivalent to the unnormalized linear transformer. Transformers have increasingly become the model of choice for natural language processing. Many modern large language models such as GPT, Gemini, Grok, DeepSeek, and Qwen use this architecture.

Elements

NNs began as an attempt to replicate the architecture of the brain in the digital realm. It immediately showed promise for handling non-linear relationships, but encountered obstacles. Models soon reoriented to applying mathematical insights to improve empirical results, at the expense of biological fidelity.

Neuron

NNs are composed of digital neurons, conceptually derived from biological neurons. Each neuron has one or more numerical inputs and produces a single numerical output. An input (e.g., an image) is typically parceled across the set of input neurons (each getting a piece of the image). Each neuron's inputs, weighted by the weights of the connections from each of those inputs are summed. A bias term is added to this sum. The result is then passed through a nonlinear activation function to produce the neuron's output. Small enough outputs may be zeroed out (ignored).