mhjensen
537 lines
21 KiB
HTML
537 lines
21 KiB
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'sections': [('What is Machine Learning?', 2, None, '___sec0'),
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('Types of Machine Learning', 2, None, '___sec1'),
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('Artificial neurons', 2, None, '___sec2'),
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('Neural network types', 2, None, '___sec3'),
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('Feed-forward neural networks', 2, None, '___sec4'),
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('Recurrent neural networks', 2, None, '___sec5'),
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('Other types of networks', 2, None, '___sec6'),
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<!-- ------------------- main content ---------------------- -->
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<center><h1>Data Analysis and Machine Learning: Elements of machine learning</h1></center> <!-- document title -->
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<p>
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<!-- author(s): Morten Hjorth-Jensen -->
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<center>
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<b>Morten Hjorth-Jensen</b> [1, 2]
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</center>
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<p>
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<!-- institution(s) -->
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<center>[1] <b>Department of Physics, University of Oslo</b></center>
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<br>
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<p>
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<center><h4>Nov 26, 2017</h4></center> <!-- date -->
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<br>
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec0">What is Machine Learning? </h2>
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<p>
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Machine learning is the science of giving computers the ability to learn without being explicitly programmed.
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The idea is that there exist generic algorithms which can be used to find patterns in a broad class of data sets without
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having to write code specifically for each problem. The algorithm will build its own logic based on the data.
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<p>
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Machine learning is a subfield of computer science, and is closely related to computational statistics.
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It evolved from the study of pattern recognition in artificial intelligence (AI) research, and has made contributions to
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AI tasks like computer vision, natural language processing
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and speech recognition. It has also, especially in later years,
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found applications in a wide variety of other areas, including bioinformatics, economy, physics, finance and marketing.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec1">Types of Machine Learning </h2>
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<p>
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The approaches to machine learning are many, but are often split into two main categories.
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In <em>supervised learning</em> we know the answer to a problem,
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and let the computer deduce the logic behind it. On the other hand, <em>unsupervised learning</em>
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is a method for finding patterns and relationship in data sets without any prior knowledge of the system.
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Some authours also operate with a third category, namely <em>reinforcement learning</em>. This is a paradigm
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of learning inspired by behavioural psychology, where learning is achieved by trial-and-error,
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solely from rewards and punishment.
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<p>
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Another way to categorize machine learning tasks is to consider the desired output of a system.
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Some of the most common tasks are:
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<ul>
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<li> Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.</li>
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<li> Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.</li>
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<li> Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.</li>
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</ul>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec2">Artificial neurons </h2>
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The field of artificial neural networks has a long history of development, and is closely connected with
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the advancement of computer science and computers in general. A model of artificial neurons
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was first developed by McCulloch and Pitts in 1943 to study signal processing in the brain and
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has later been refined by others. The general idea is to mimic neural networks in the human brain, which
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is composed of billions of neurons that communicate with each other by sending electrical signals.
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Each neuron accumulates its incoming signals,
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which must exceed an activation threshold to yield an output. If the threshold is not overcome, the neuron
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remains inactive, i.e. has zero output.
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<p>
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This behaviour has inspired a simple mathematical model for an artificial neuron.
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$$
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\begin{equation}
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y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
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\label{artificialNeuron}
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\end{equation}
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$$
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Here, the output \( y \) of the neuron is the value of its activation function, which have as input
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a weighted sum of signals \( x_i, \dots ,x_n \) received by \( n \) other neurons.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec3">Neural network types </h2>
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<p>
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An artificial neural network (NN), is a computational model that consists of layers of connected neurons, or <em>nodes</em>.
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It is supposed to mimic a biological nervous system by letting each neuron interact with other neurons
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by sending signals in the form of mathematical functions between layers.
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A wide variety of different NNs have
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been developed, but most of them consist of an input layer, an output layer and eventual layers in-between, called
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<em>hidden layers</em>. All layers can contain an arbitrary number of nodes, and each connection between two nodes
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is associated with a weight variable.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec4">Feed-forward neural networks </h2>
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The feed-forward neural network (FFNN) was the first and simplest type of NN devised. In this network,
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the information moves in only one direction: forward through the layers.
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<p>
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Nodes are represented by circles, while the arrows display the connections between the nodes, including the
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direction of information flow. Additionally, each arrow corresponds to a weight variable, not displayed here.
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We observe that each node in a layer is connected to <em>all</em> nodes in the subsequent layer,
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making this a so-called <em>fully-connected</em> FFNN.
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<p>
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A different variant of FFNNs are <em>convolutional neural networks</em> (CNNs), which have a connectivity pattern
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inspired by the animal visual cortex. Individual neurons in the visual cortex only respond to stimuli from
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small sub-regions of the visual field, called a receptive field. This makes the neurons well-suited to exploit the strong
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spatially local correlation present in natural images. The response of each neuron can be approximated mathematically
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as a convolution operation.
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<p>
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CNNs emulate the behaviour of neurons in the visual cortex by enforcing a <em>local</em> connectivity pattern
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between nodes of adjacent layers: Each node
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in a convolutional layer is connected only to a subset of the nodes in the previous layer,
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in contrast to the fully-connected FFNN.
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Often, CNNs
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consist of several convolutional layers that learn local features of the input, with a fully-connected layer at the end,
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which gathers all the local data and produces the outputs. They have wide applications in image and video recognition
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec5">Recurrent neural networks </h2>
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<p>
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So far we have only mentioned NNs where information flows in one direction: forward. <em>Recurrent neural networks</em> on
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the other hand, have connections between nodes that form directed <em>cycles</em>. This creates a form of
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internal memory which are able to capture information on what has been calculated before; the output is dependent
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on the previous computations. Recurrent NNs make use of sequential information by performing the same task for
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every element in a sequence, where each element depends on previous elements. An example of such information is
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sentences, making recurrent NNs especially well-suited for handwriting and speech recognition.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec6">Other types of networks </h2>
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<p>
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There are many other kinds of NNs that have been developed. One type that is specifically designed for interpolation
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in multidimensional space is the radial basis function (RBF) network. RBFs are typically made up of three layers:
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an input layer, a hidden layer with non-linear radial symmetric activation functions and a linear output layer (''linear'' here
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means that each node in the output layer has a linear activation function). The layers are normally fully-connected and
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there are no cycles, thus RBFs can be viewed as a type of fully-connected FFNN. They are however usually treated as
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a separate type of NN due the unusual activation functions.
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<p>
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Other types of NNs could also be mentioned, but are outside the scope of this work. We will now move on to a detailed description
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of how a fully-connected FFNN works, and how it can be used to interpolate data sets.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec7">Multilayer perceptrons </h2>
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<p>
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One use often so-called fully-connected feed-forward neural networks with three
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or more layers (an input layer, one or more hidden layers and an output layer)
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consisting of neurons that have non-linear activation functions.
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<p>
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Such networks are often called <em>multilayer perceptrons</em> (MLPs)
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec8">Why multilayer perceptrons? </h2>
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<p>
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According to the <em>Universal approximation theorem</em>, a feed-forward neural network with just a single hidden layer containing
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a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy,
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assuming the activation function for the hidden layer is a <b>non-constant, bounded and monotonically-increasing continuous function</b>.
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Note that the requirements on the activation function only applies to the hidden layer, the output nodes are always
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assumed to be linear, so as to not restrict the range of output values.
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<p>
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We note that this theorem is only applicable to a NN with <em>one</em> hidden layer.
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Therefore, we can easily construct an NN
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that employs activation functions which do not satisfy the above requirements, as long as we have at least one layer
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with activation functions that <em>do</em>. Furthermore, although the universal approximation theorem
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lays the theoretical foundation for regression with neural networks, it does not say anything about how things work in practice:
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A neural network can still be able to approximate a given function reasonably well without having the flexibility to fit <em>all other</em>
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functions.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec9">Mathematical model </h2>
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$$
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\begin{equation}
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y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(u)
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\label{artificialNeuron2}
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\end{equation}
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$$
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In an FFNN of such neurons, the <em>inputs</em> \( x_i \)
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are the <em>outputs</em> of the neurons in the preceding layer. Furthermore, an MLP is fully-connected,
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which means that each neuron receives a weighted sum of the outputs of <em>all</em> neurons in the previous layer.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec10">Mathematical model </h2>
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<p>
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First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( u_i^1 \) of the input coordinates \( x_j \),
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$$
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\begin{equation}
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u_i^1 = \sum_{j=1}^2 w_{ij}^1 x_j + b_i^1
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\label{_auto1}
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\end{equation}
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$$
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This value is the argument to the activation function \( f_1 \) of each neuron \( i \),
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producing the output \( y_i^1 \) of all neurons in layer 1,
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$$
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\begin{equation}
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y_i^1 = f_1(u_i^1) = f_1\left(\sum_{j=1}^2 w_{ij}^1 x_j + b_i^1\right)
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\label{outputLayer1}
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\end{equation}
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$$
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where we assume that all nodes in the same layer have identical activation functions, hence the notation \( f_l \)
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$$
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\begin{equation}
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y_i^l = f_l(u_i^l) = f_l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
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\label{generalLayer}
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\end{equation}
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$$
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where \( N_l \) is the number of nodes in layer \( l \). When the output of all the nodes in the first hidden layer are computed,
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the values of the subsequent layer can be calculated and so forth until the output is obtained.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec11">Mathematical model </h2>
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<p>
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The output of neuron \( i \) in layer 2 is thus,
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$$
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\begin{align}
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y_i^2 &= f_2\left(\sum_{j=1}^3 w_{ij}^2 y_j^1 + b_i^2\right)
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\label{_auto2}\\
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&= f_2\left[\sum_{j=1}^3 w_{ij}^2f_1\left(\sum_{k=1}^2 w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
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\label{outputLayer2}
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\end{align}
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$$
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where we have substituted \( y_m^1 \) with. Finally, the NN output yields,
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$$
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\begin{align}
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y_1^3 &= f_3\left(\sum_{j=1}^3 w_{1m}^3 y_j^2 + b_1^3\right)
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\label{_auto3}\\
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&= f_3\left[\sum_{j=1}^3 w_{1j}^3 f_2\left(\sum_{k=1}^3 w_{jk}^2 f_1\left(\sum_{m=1}^2 w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
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+ b_1^3\right]
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\label{_auto4}
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\end{align}
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$$
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec12">Mathematical model </h2>
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<p>
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We can generalize this expression to an MLP with \( l \) hidden layers. The complete functional form
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is,
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$$
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\begin{align}
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&y^{l+1}_1\! = \!f_{l+1}\!\left[\!\sum_{j=1}^{N_l}\! w_{1j}^3 f_l\!\left(\!\sum_{k=1}^{N_{l-1}}\! w_{jk}^2 f_{l-1}\!\left(\!
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\dots \!f_1\!\left(\!\sum_{n=1}^{N_0} \!w_{mn}^1 x_n\! + \!b_m^1\!\right)
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\!\dots \!\right) \!+ \!b_k^2\!\right)
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\!+ \!b_1^3\!\right] &&
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\label{completeNN}
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\end{align}
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$$
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which illustrates a basic property of MLPs: The only independent variables are the input values \( x_n \).
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec13">Mathematical model </h2>
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<p>
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This confirms that an MLP,
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despite its quite convoluted mathematical form, is nothing more than an analytic function, specifically a
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mapping of real-valued vectors \( \vec{x} \in \mathbb{R}^n \rightarrow \vec{y} \in \mathbb{R}^m \).
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In our example, \( n=2 \) and \( m=1 \). Consequentially,
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the number of input and output values of the function we want to fit must be equal to the number of inputs and outputs of our MLP.
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<p>
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Furthermore, the flexibility and universality of a MLP can be illustrated by realizing that
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the expression is essentially a nested sum of scaled activation functions of the form
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$$
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\begin{equation}
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h(x) = c_1 f(c_2 x + c_3) + c_4
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\label{_auto5}
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\end{equation}
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$$
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where the parameters \( c_i \) are weights and biases. By adjusting these parameters, the activation functions
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can be shifted up and down or left and right, change slope or be rescaled
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which is the key to the flexibility of a neural network.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h3 id="___sec14">Matrix-vector notation </h3>
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We can introduce a more convenient notation for the activations in a NN.
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<p>
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Additionally, we can represent the biases and activations
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as layer-wise column vectors \( \vec{b}_l \) and \( \vec{y}_l \), so that the \( i \)-th element of each vector
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is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively.
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<p>
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We have that \( \mathrm{W}_l \) is a \( N_{l-1} \times N_l \) matrix, while \( \vec{b}_l \) and \( \vec{y}_l \) are \( N_l \times 1 \) column vectors.
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With this notation, the sum in becomes a matrix-vector multiplication, and we can write
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the equation for the activations of hidden layer 2 in
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$$
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\begin{equation}
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\vec{y}_2 = f_2(\mathrm{W}_2 \vec{y}_{1} + \vec{b}_{2}) =
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f_2\left(\left[\begin{array}{ccc}
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w^2_{11} &w^2_{12} &w^2_{13} \\
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w^2_{21} &w^2_{22} &w^2_{23} \\
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w^2_{31} &w^2_{32} &w^2_{33} \\
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\end{array} \right] \cdot
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\left[\begin{array}{c}
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y^1_1 \\
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y^1_2 \\
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y^1_3 \\
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\end{array}\right] +
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\left[\begin{array}{c}
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b^2_1 \\
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b^2_2 \\
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b^2_3 \\
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\end{array}\right]\right).
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\label{_auto6}
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\end{equation}
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$$
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h3 id="___sec15">Matrix-vector notation and activation </h3>
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The activation of node \( i \) in layer 2 is
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$$
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\begin{equation}
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y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) =
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f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
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\label{_auto7}
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\end{equation}
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$$
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This is not just a convenient and compact notation, but also
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a useful and intuitive way to think about MLPs: The output is calculated by a series of matrix-vector multiplications
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and vector additions that are used as input to the activation functions. For each operation
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\( \mathrm{W}_l \vec{y}_{l-1} \) we move forward one layer.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h3 id="___sec16">Activation functions </h3>
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<p>
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A property that characterizes a neural network, other than its connectivity, is the choice of activation function(s).
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As described in, the following restrictions are imposed on an activation function for a FFNN
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to fulfill the universal approximation theorem
|
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|
<ul>
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<li> Non-constant</li>
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<li> Bounded</li>
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<li> Monotonically-increasing</li>
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<li> Continuous</li>
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</ul>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h3 id="___sec17">Activation functions, Logistic and Hyperbolic ones </h3>
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<p>
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The second requirement excludes all linear functions. Furthermore, in a MLP with only linear activation functions, each
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layer simply performs a linear transformation of its inputs.
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|
<p>
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Regardless of the number of layers,
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the output of the NN will be nothing but a linear function of the inputs. Thus we need to introduce some kind of
|
|
non-linearity to the NN to be able to fit non-linear functions
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Typical examples are the logistic <em>Sigmoid</em>
|
|
$$
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|
\begin{equation}
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f(x) = \frac{1}{1 + e^{-x}},
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\label{sigmoidActivationFunction}
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\end{equation}
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$$
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|
and the <em>hyperbolic tangent</em> function
|
|
$$
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\begin{equation}
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f(x) = \tanh(x)
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\label{tanhActivationFunction}
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\end{equation}
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|
$$
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h3 id="___sec18">Relevance </h3>
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The <em>sigmoid</em> function are more biologically plausible because
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|
the output of inactive neurons are zero. Such activation function are called <em>one-sided</em>. However,
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|
it has been shown that the hyperbolic tangent
|
|
performs better than the sigmoid for training MLPs.
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|
has become the most popular for <em>deep neural networks</em>
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<p>
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<!-- ------------------- end of main content --------------- -->
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