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  Medical imaging

  Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to reveal internal structures hidden by the skin and bones, as well as to diagnose and treat disease. Medical imaging also establishes a database of normal anatomy and physiology to make it possible to identify abnormalities. Although imaging of removed organs and tissues can be performed for medical reasons, such procedures are usually considered part of pathology instead of medical imaging. Measurement and recording techniques that are not primarily designed to produce images, such as electroencephalography (EEG), magnetoencephalography (MEG), electrocardiography (ECG), and others, represent other technologies that produce data susceptible to representation as a parameter graph versus time or maps that contain data about the measurement locations. In a limited comparison, these technologies can be considered forms of medical imaging in another discipline of medical instrumentation. As of 2010, 5 billion medical imaging studies had been conducted worldwide. Radiation exposure from medical imaging in 2006 made up about 50% of total ionizing radiation exposure in the United States. Medical imaging equipment is manufactured using technology from the semiconductor industry, including CMOS integrated circuit chips, power semiconductor devices, sensors such as image sensors (particularly CMOS sensors) and biosensors, and processors such as microcontrollers, microprocessors, digital signal processors, media processors and system-on-chip devices. As of 2015, annual shipments of medical imaging chips amount to 46 million units and $1.1 billion. The term "noninvasive" is used to denote a procedure where no instrument is introduced into a patient's body, which is the case for most imaging techniques used. == History == In 1972, engineer Godfrey Hounsfield from the British company EMI invented the X-ray computed tomography device for head diagnosis, which is commonly referred to as computed tomography (CT). The CT nucleus method is based on the projecting X-rays through a section of the human head, which are then processed by computer to reconstruct the cross-sectional image, known as image reconstruction. In 1975, EMI successfully developed a CT device for the entire body, enabling the clear acquisition of tomographic images of various parts of the human body. This revolutionary diagnostic technique earned Hounsfield and physicist Allan Cormack the Nobel Prize in Physiology or Medicine in 1979. Digital image processing technology for medical applications was inducted into the Space Foundation's Space Technology Hall of Fame in 1994. By 2010, over 5 billion medical imaging studies had been conducted worldwide. Radiation exposure from medical imaging in 2006 accounted for about 50% of total ionizing radiation exposure in the United States. Medical imaging equipment is manufactured using technology from the semiconductor industry, including CMOS integrated circuit chips, power semiconductor devices, sensors such as image sensors (particularly CMOS sensors) and biosensors, as well as processors like microcontrollers, microprocessors, digital signal processors, media processors and system-on-chip devices. As of 2015, annual shipments of medical imaging chips reached 46 million units, generating a market value of $1.1 billion. == Types == In the clinical context, "invisible light" medical imaging is generally equated to radiology or "clinical imaging". "Visible light" medical imaging involves digital video or still pictures that can be seen without special equipment. Dermatology and wound care are two modalities that use visible light imagery. Interpretation of medical images is generally undertaken by a physician specialising in radiology known as a radiologist; however, this may be undertaken by any healthcare professional who is trained and certified in radiological clinical evaluation. Increasingly interpretation is being undertaken by non-physicians, for example radiographers frequently train in interpretation as part of expanded practice. Diagnostic radiography designates the technical aspects of medical imaging and in particular the acquisition of medical images. The radiographer (also known as a radiologic technologist) is usually responsible for acquiring medical images of diagnostic quality; although other professionals may train in this area, notably some radiological interventions performed by radiologists are done so without a radiographer. As a field of scientific investigation, medical imaging constitutes a sub-discipline of biomedical engineering, medical physics or medicine depending on the context: Research and development in the area of instrumentation, image acquisition (e.g., radiography), modeling and quantification are usually the preserve of biomedical engineering, medical physics, and computer science; Research into the application and interpretation of medical images is usually the preserve of radiology and the medical sub-discipline relevant to medical condition or area of medical science (neuroscience, cardiology, psychiatry, psychology, etc.) under investigation. Many of the techniques developed for medical imaging also have scientific and industrial applications. === Radiography === Two forms of radiographic images are in use in medical imaging. Projection radiography and fluoroscopy, with the latter being useful for catheter guidance. These 2D techniques are still in wide use despite the advance of 3D tomography due to the low cost, high resolution, and depending on the application, lower radiation dosages with 2D technique. This imaging modality uses a wide beam of X-rays for image acquisition and is the first imaging technique available in modern medicine. Fluoroscopy produces real-time images of internal structures of the body in a similar fashion to radiography, but employs a constant input of X-rays, at a lower dose rate. Contrast media, such as barium, iodine, and air are used to visualize internal organs as they work. Fluoroscopy is also used in image-guided procedures when constant feedback during a procedure is required. An image receptor is required to convert the radiation into an image after it has passed through the area of interest. Early on, this was a fluorescing screen, which gave way to an Image Amplifier (IA) which was a large vacuum tube that had the receiving end coated with cesium iodide, and a mirror at the opposite end. Eventually the mirror was replaced with a TV camera. Projectional radiographs, more commonly known as X-rays, are often used to determine the type and extent of a fracture as well as for detecting pathological changes in the lungs. With the use of radio-opaque contrast media, such as barium, they can also be used to visualize the structure of the stomach and intestines – this can help diagnose ulcers or certain types of colon cancer. === Magnetic resonance imaging === A magnetic resonance imaging instrument (MRI scanner), or "nuclear magnetic resonance (NMR) imaging" scanner as it was originally known, uses powerful magnets to polarize and excite hydrogen nuclei (i.e., single protons) of water molecules in human tissue, producing a detectable signal that is spatially encoded, resulting in images of the body. The MRI machine emits a radio frequency (RF) pulse at the resonant frequency of the hydrogen atoms on water molecules. Radio frequency antennas ("RF coils") send the pulse to the area of the body to be examined. The RF pulse is absorbed by protons, causing their direction with respect to the primary magnetic field to change. When the RF pulse is turned off, the protons "relax" back to alignment with the primary magnet and emit radio waves in the process. This radio-frequency emission from the hydrogen atoms on water is what is detected and reconstructed into an image. The resonant frequency of a spinning magnetic dipole (of which protons are one example) is called the Larmor frequency and is determined by the strength of the main magnetic field and the chemical environment of the nuclei of interest. MRI uses three electromagnetic fields: a very strong (typically 1.5 to 3 teslas) static magnetic field to polarize the hydrogen nuclei, called the primary field; gradient fields that can be modified to vary in space and time (on the order of 1 kHz) for spatial encoding, often simply called gradients; and a spatially homogeneous radio-frequency (RF) field for manipulation of the hydrogen nuclei to produce measurable signals, collected through an RF antenna. Like CT, MRI traditionally creates a two-dimensional image of a thin "slice" of the body and is therefore considered a tomographic imaging technique. Modern MRI instruments are capable of producing images in the form of 3D blocks, which may be considered a generalization of the single-slice

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  Fitness function

  A fitness function is a particular type of objective or cost function that is used to summarize, as a single figure of merit, how close a given candidate solution is to achieving the set aims. It is an important component of evolutionary algorithms (EA), such as genetic programming, evolution strategies or genetic algorithms. An EA is a metaheuristic that reproduces the basic principles of biological evolution as a computer algorithm in order to solve challenging optimization or planning tasks, at least approximately. For this purpose, many candidate solutions are generated, which are evaluated using a fitness function in order to guide the evolutionary development towards the desired goal. Similar quality functions are also used in other metaheuristics, such as ant colony optimization or particle swarm optimization. In the field of EAs, each candidate solution, also called an individual, is commonly represented as a string of numbers (referred to as a chromosome). After each round of testing or simulation the idea is to delete the n worst individuals, and to breed n new ones from the best solutions. Each individual must therefore to be assigned a quality number indicating how close it has come to the overall specification, and this is generated by applying the fitness function to the test or simulation results obtained from that candidate solution. Two main classes of fitness functions exist: one where the fitness function does not change, as in optimizing a fixed function or testing with a fixed set of test cases; and one where the fitness function is mutable, as in niche differentiation or co-evolving the set of test cases. Another way of looking at fitness functions is in terms of a fitness landscape, which shows the fitness for each possible chromosome. In the following, it is assumed that the fitness is determined based on an evaluation that remains unchanged during an optimization run. A fitness function does not necessarily have to be able to calculate an absolute value, as it is sometimes sufficient to compare candidates in order to select the better one. A relative indication of fitness (candidate a is better than b) is sufficient in some cases, such as tournament selection or Pareto optimization. == Requirements of evaluation and fitness function == The quality of the evaluation and calculation of a fitness function is fundamental to the success of an EA optimisation. It implements Darwin's principle of "survival of the fittest". Without fitness-based selection mechanisms for mate selection and offspring acceptance, EA search would be blind and hardly distinguishable from the Monte Carlo method. When setting up a fitness function, one must always be aware that it is about more than just describing the desired target state. Rather, the evolutionary search on the way to the optimum should also be supported as much as possible (see also section on auxiliary objectives), if and insofar as this is not already done by the fitness function alone. If the fitness function is designed badly, the algorithm will either converge on an inappropriate solution, or will have difficulty converging at all. Definition of the fitness function is not straightforward in many cases and often is performed iteratively if the fittest solutions produced by an EA is not what is desired. Interactive genetic algorithms address this difficulty by outsourcing evaluation to external agents which are normally humans. == Computational efficiency == The fitness function should not only closely align with the designer's goal, but also be computationally efficient. Execution speed is crucial, as a typical evolutionary algorithm must be iterated many times in order to produce a usable result for a non-trivial problem. Fitness approximation may be appropriate, especially in the following cases: Fitness computation time of a single solution is extremely high Precise model for fitness computation is missing The fitness function is uncertain or noisy. Alternatively or also in addition to the fitness approximation, the fitness calculations can also be distributed to a parallel computer in order to reduce the execution times. Depending on the population model of the EA used, both the EA itself and the fitness calculations of all offspring of one generation can be executed in parallel. == Multi-objective optimization == Practical applications usually aim at optimizing multiple and at least partially conflicting objectives. Two fundamentally different approaches are often used for this purpose, Pareto optimization and optimization based on fitness calculated using the weighted sum. === Weighted sum and penalty functions === When optimizing with the weighted sum, the single values of the O {\displaystyle O} objectives are first normalized so that they can be compared. This can be done with the help of costs or by specifying target values and determining the current value as the degree of fulfillment. Costs or degrees of fulfillment can then be compared with each other and, if required, can also be mapped to a uniform fitness scale. Without loss of generality, fitness is assumed to represent a value to be maximized. Each objective o i {\displaystyle o_{i}} is assigned a weight w i {\displaystyle w_{i}} in the form of a percentage value so that the overall raw fitness f r a w {\displaystyle f_{raw}} can be calculated as a weighted sum: f r a w = ∑ i = 1 O o i ⋅ w i w i t h ∑ i = 1 O w i = 1 {\displaystyle f_{raw}=\sum _{i=1}^{O}{o_{i}\cdot w_{i}}\quad {\mathsf {with}}\quad \sum _{i=1}^{O}{w_{i}}=1} A violation of R {\displaystyle R} restrictions r j {\displaystyle r_{j}} can be included in the fitness determined in this way in the form of penalty functions. For this purpose, a function p f j ( r j ) {\displaystyle pf_{j}(r_{j})} can be defined for each restriction which returns a value between 0 {\displaystyle 0} and 1 {\displaystyle 1} depending on the degree of violation, with the result being 1 {\displaystyle 1} if there is no violation. The previously determined raw fitness is multiplied by the penalty function(s) and the result is then the final fitness f f i n a l {\displaystyle f_{final}} : f f i n a l = f r a w ⋅ ∏ j = 1 R p f j ( r j ) = ∑ i = 1 O ( o i ⋅ w i ) ⋅ ∏ j = 1 R p f j ( r j ) {\displaystyle f_{final}=f_{raw}\cdot \prod _{j=1}^{R}{pf_{j}(r_{j})}=\sum _{i=1}^{O}{(o_{i}\cdot w_{i})}\cdot \prod _{j=1}^{R}{pf_{j}(r_{j})}} This approach is simple and has the advantage of being able to combine any number of objectives and restrictions. The disadvantage is that different objectives can compensate each other and that the weights have to be defined before the optimization. This means that the compromise lines must be defined before optimization, which is why optimization with the weighted sum is also referred to as the a priori method. In addition, certain solutions may not be obtained, see the section on the comparison of both types of optimization. === Pareto optimization === A solution is called Pareto-optimal if the improvement of one objective is only possible with a deterioration of at least one other objective. The set of all Pareto-optimal solutions, also called Pareto set, represents the set of all optimal compromises between the objectives. The figure below on the right shows an example of the Pareto set of two objectives f 1 {\displaystyle f_{1}} and f 2 {\displaystyle f_{2}} to be maximized. The elements of the set form the Pareto front (green line). From this set, a human decision maker must subsequently select the desired compromise solution. Constraints are included in Pareto optimization in that solutions without constraint violations are per se better than those with violations. If two solutions to be compared each have constraint violations, the respective extent of the violations decides. It was recognized early on that EAs with their simultaneously considered solution set are well suited to finding solutions in one run that cover the Pareto front sufficiently well. They are therefore well suited as a-posteriori methods for multi-objective optimization, in which the final decision is made by a human decision maker after optimization and determination of the Pareto front. Besides the SPEA2, the NSGA-II and NSGA-III have established themselves as standard methods. The advantage of Pareto optimization is that, in contrast to the weighted sum, it provides all alternatives that are equivalent in terms of the objectives as an overall solution. The disadvantage is that a visualization of the alternatives becomes problematic or even impossible from four objectives on. Furthermore, the effort increases exponentially with the number of objectives. If there are more than three or four objectives, some have to be combined using the weighted sum or other aggregation methods. === Comparison of both types of assessment === With the help of the weighted sum, the total Pareto front can be obtained by a suitable choice of weights, provided that it is convex

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  Genetic representation

  In computer programming, genetic representation is a way of presenting solutions/individuals in evolutionary computation methods. The term encompasses both the concrete data structures and data types used to realize the genetic material of the candidate solutions in the form of a genome, and the relationships between search space and problem space. In the simplest case, the search space corresponds to the problem space (direct representation). The choice of problem representation is tied to the choice of genetic operators, both of which have a decisive effect on the efficiency of the optimization. Genetic representation can encode appearance, behavior, physical qualities of individuals. Difference in genetic representations is one of the major criteria drawing a line between known classes of evolutionary computation. Terminology is often analogous with natural genetics. The block of computer memory that represents one candidate solution is called an individual. The data in that block is called a chromosome. Each chromosome consists of genes. The possible values of a particular gene are called alleles. A programmer may represent all the individuals of a population using binary encoding, permutational encoding, encoding by tree, or any one of several other representations. == Representations in some popular evolutionary algorithms == Genetic algorithms (GAs) are typically linear representations; these are often, but not always, binary. Holland's original description of GA used arrays of bits. Arrays of other types and structures can be used in essentially the same way. The main property that makes these genetic representations convenient is that their parts are easily aligned due to their fixed size. This facilitates simple crossover operation. Depending on the application, variable-length representations have also been successfully used and tested in evolutionary algorithms (EA) in general and genetic algorithms in particular, although the implementation of crossover is more complex in this case. Evolution strategy uses linear real-valued representations, e.g., an array of real values. It uses mostly gaussian mutation and blending/averaging crossover. Genetic programming (GP) pioneered tree-like representations and developed genetic operators suitable for such representations. Tree-like representations are used in GP to represent and evolve functional programs with desired properties. Human-based genetic algorithm (HBGA) offers a way to avoid solving hard representation problems by outsourcing all genetic operators to outside agents, in this case, humans. The algorithm has no need for knowledge of a particular fixed genetic representation as long as there are enough external agents capable of handling those representations, allowing for free-form and evolving genetic representations. === Common genetic representations === binary array integer or real-valued array binary tree natural language parse tree directed graph == Distinction between search space and problem space == Analogous to biology, EAs distinguish between problem space (corresponds to phenotype) and search space (corresponds to genotype). The problem space contains concrete solutions to the problem being addressed, while the search space contains the encoded solutions. The mapping from search space to problem space is called genotype-phenotype mapping. The genetic operators are applied to elements of the search space, and for evaluation, elements of the search space are mapped to elements of the problem space via genotype-phenotype mapping. == Relationships between search space and problem space == The importance of an appropriate choice of search space for the success of an EA application was recognized early on. The following requirements can be placed on a suitable search space and thus on a suitable genotype-phenotype mapping: === Completeness === All possible admissible solutions must be contained in the search space. === Redundancy === When more possible genotypes exist than phenotypes, the genetic representation of the EA is called redundant. In nature, this is termed a degenerate genetic code. In the case of a redundant representation, neutral mutations are possible. These are mutations that change the genotype but do not affect the phenotype. Thus, depending on the use of the genetic operators, there may be phenotypically unchanged offspring, which can lead to unnecessary fitness determinations, among other things. Since the evaluation in real-world applications usually accounts for the lion's share of the computation time, it can slow down the optimization process. In addition, this can cause the population to have higher genotypic diversity than phenotypic diversity, which can also hinder evolutionary progress. In biology, the Neutral Theory of Molecular Evolution states that this effect plays a dominant role in natural evolution. This has motivated researchers in the EA community to examine whether neutral mutations can improve EA functioning by giving populations that have converged to a local optimum a way to escape that local optimum through genetic drift. This is discussed controversially and there are no conclusive results on neutrality in EAs. On the other hand, there are other proven measures to handle premature convergence. === Locality === The locality of a genetic representation corresponds to the degree to which distances in the search space are preserved in the problem space after genotype-phenotype mapping. That is, a representation has a high locality exactly when neighbors in the search space are also neighbors in the problem space. In order for successful schemata not to be destroyed by genotype-phenotype mapping after a minor mutation, the locality of a representation must be high. === Scaling === In genotype-phenotype mapping, the elements of the genotype can be scaled (weighted) differently. The simplest case is uniform scaling: all elements of the genotype are equally weighted in the phenotype. A common scaling is exponential. If integers are binary coded, the individual digits of the resulting binary number have exponentially different weights in representing the phenotype. Example: The number 90 is written in binary (i.e., in base two) as 1011010. If now one of the front digits is changed in the binary notation, this has a significantly greater effect on the coded number than any changes at the rear digits (the selection pressure has an exponentially greater effect on the front digits). For this reason, exponential scaling has the effect of randomly fixing the "posterior" locations in the genotype before the population gets close enough to the optimum to adjust for these subtleties. == Hybridization and repair in genotype-phenotype mapping == When mapping the genotype to the phenotype being evaluated, domain-specific knowledge can be used to improve the phenotype and/or ensure that constraints are met. This is a commonly used method to improve EA performance in terms of runtime and solution quality. It is illustrated below by two of the three examples. == Examples == === Example of a direct representation === An obvious and commonly used encoding for the traveling salesman problem and related tasks is to number the cities to be visited consecutively and store them as integers in the chromosome. The genetic operators must be suitably adapted so that they only change the order of the cities (genes) and do not cause deletions or duplications. Thus, the gene order corresponds to the city order and there is a simple one-to-one mapping. === Example of a complex genotype-phenotype mapping === In a scheduling task with heterogeneous and partially alternative resources to be assigned to a set of subtasks, the genome must contain all necessary information for the individual scheduling operations or it must be possible to derive them from it. In addition to the order of the subtasks to be executed, this includes information about the resource selection. A phenotype then consists of a list of subtasks with their start times and assigned resources. In order to be able to create this, as many allocation matrices must be created as resources can be allocated to one subtask at most. In the simplest case this is one resource, e.g., one machine, which can perform the subtask. An allocation matrix is a two-dimensional matrix, with one dimension being the available time units and the other being the resources to be allocated. Empty matrix cells indicate availability, while an entry indicates the number of the assigned subtask. The creation of allocation matrices ensures firstly that there are no inadmissible multiple allocations. Secondly, the start times of the subtasks can be read from it as well as the assigned resources. A common constraint when scheduling resources to subtasks is that a resource can only be allocated once per time unit and that the reservation must be for a contiguous period of time. To achieve this in a timely manner, which is a c

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  Mathematics of neural networks in machine learning

  An artificial neural network (ANN) or neural network combines biological principles with advanced statistics to solve problems in domains such as pattern recognition and game-play. ANNs adopt the basic model of neuron analogues connected to each other in a variety of ways. == Structure == === Neuron === A neuron with label j {\displaystyle j} receiving an input p j ( t ) {\displaystyle p_{j}(t)} from predecessor neurons consists of the following components: an activation a j ( t ) {\displaystyle a_{j}(t)} , the neuron's state, depending on a discrete time parameter, an optional threshold θ j {\displaystyle \theta _{j}} , which stays fixed unless changed by learning, an activation function f {\displaystyle f} that computes the new activation at a given time t + 1 {\displaystyle t+1} from a j ( t ) {\displaystyle a_{j}(t)} , θ j {\displaystyle \theta _{j}} and the net input p j ( t ) {\displaystyle p_{j}(t)} giving rise to the relation a j ( t + 1 ) = f ( a j ( t ) , p j ( t ) , θ j ) , {\displaystyle a_{j}(t+1)=f(a_{j}(t),p_{j}(t),\theta _{j}),} and an output function f out {\displaystyle f_{\text{out}}} computing the output from the activation o j ( t ) = f out ( a j ( t ) ) . {\displaystyle o_{j}(t)=f_{\text{out}}(a_{j}(t)).} Often the output function is simply the identity function. An input neuron has no predecessor but serves as input interface for the whole network. Similarly an output neuron has no successor and thus serves as output interface of the whole network. === Propagation function === The propagation function computes the input p j ( t ) {\displaystyle p_{j}(t)} to the neuron j {\displaystyle j} from the outputs o i ( t ) {\displaystyle o_{i}(t)} and typically has the form p j ( t ) = ∑ i o i ( t ) w i j . {\displaystyle p_{j}(t)=\sum _{i}o_{i}(t)w_{ij}.} === Bias === A bias term can be added, changing the form to the following: p j ( t ) = ∑ i o i ( t ) w i j + w 0 j , {\displaystyle p_{j}(t)=\sum _{i}o_{i}(t)w_{ij}+w_{0j},} where w 0 j {\displaystyle w_{0j}} is a bias. == Neural networks as functions == Neural network models can be viewed as defining a function that takes an input (observation) and produces an output (decision) f : X → Y {\displaystyle \textstyle f:X\rightarrow Y} or a distribution over X {\displaystyle \textstyle X} or both X {\displaystyle \textstyle X} and Y {\displaystyle \textstyle Y} . Sometimes models are intimately associated with a particular learning rule. A common use of the phrase "ANN model" is really the definition of a class of such functions (where members of the class are obtained by varying parameters, connection weights, or specifics of the architecture such as the number of neurons, number of layers or their connectivity). Mathematically, a neuron's network function f ( x ) {\displaystyle \textstyle f(x)} is defined as a composition of other functions g i ( x ) {\displaystyle \textstyle g_{i}(x)} , that can further be decomposed into other functions. This can be conveniently represented as a network structure, with arrows depicting the dependencies between functions. A widely used type of composition is the nonlinear weighted sum, where f ( x ) = K ( ∑ i w i g i ( x ) ) {\displaystyle \textstyle f(x)=K\left(\sum _{i}w_{i}g_{i}(x)\right)} , where K {\displaystyle \textstyle K} (commonly referred to as the activation function) is some predefined function, such as the hyperbolic tangent, sigmoid function, softmax function, or rectifier function. The important characteristic of the activation function is that it provides a smooth transition as input values change, i.e. a small change in input produces a small change in output. The following refers to a collection of functions g i {\displaystyle \textstyle g_{i}} as a vector g = ( g 1 , g 2 , … , g n ) {\displaystyle \textstyle g=(g_{1},g_{2},\ldots ,g_{n})} . This figure depicts such a decomposition of f {\displaystyle \textstyle f} , with dependencies between variables indicated by arrows. These can be interpreted in two ways. The first view is the functional view: the input x {\displaystyle \textstyle x} is transformed into a 3-dimensional vector h {\displaystyle \textstyle h} , which is then transformed into a 2-dimensional vector g {\displaystyle \textstyle g} , which is finally transformed into f {\displaystyle \textstyle f} . This view is most commonly encountered in the context of optimization. The second view is the probabilistic view: the random variable F = f ( G ) {\displaystyle \textstyle F=f(G)} depends upon the random variable G = g ( H ) {\displaystyle \textstyle G=g(H)} , which depends upon H = h ( X ) {\displaystyle \textstyle H=h(X)} , which depends upon the random variable X {\displaystyle \textstyle X} . This view is most commonly encountered in the context of graphical models. The two views are largely equivalent. In either case, for this particular architecture, the components of individual layers are independent of each other (e.g., the components of g {\displaystyle \textstyle g} are independent of each other given their input h {\displaystyle \textstyle h} ). This naturally enables a degree of parallelism in the implementation. Networks such as the previous one are commonly called feedforward, because their graph is a directed acyclic graph. Networks with cycles are commonly called recurrent. Such networks are commonly depicted in the manner shown at the top of the figure, where f {\displaystyle \textstyle f} is shown as dependent upon itself. However, an implied temporal dependence is not shown. == Backpropagation == Backpropagation training algorithms fall into three categories: steepest descent (with variable learning rate and momentum, resilient backpropagation); quasi-Newton (Broyden–Fletcher–Goldfarb–Shanno, one step secant); Levenberg–Marquardt and conjugate gradient (Fletcher–Reeves update, Polak–Ribiére update, Powell–Beale restart, scaled conjugate gradient). === Algorithm === Let N {\displaystyle N} be a network with e {\displaystyle e} connections, m {\displaystyle m} inputs and n {\displaystyle n} outputs. Below, x 1 , x 2 , … {\displaystyle x_{1},x_{2},\dots } denote vectors in R m {\displaystyle \mathbb {R} ^{m}} , y 1 , y 2 , … {\displaystyle y_{1},y_{2},\dots } vectors in R n {\displaystyle \mathbb {R} ^{n}} , and w 0 , w 1 , w 2 , … {\displaystyle w_{0},w_{1},w_{2},\ldots } vectors in R e {\displaystyle \mathbb {R} ^{e}} . These are called inputs, outputs and weights, respectively. The network corresponds to a function y = f N ( w , x ) {\displaystyle y=f_{N}(w,x)} which, given a weight w {\displaystyle w} , maps an input x {\displaystyle x} to an output y {\displaystyle y} . In supervised learning, a sequence of training examples ( x 1 , y 1 ) , … , ( x p , y p ) {\displaystyle (x_{1},y_{1}),\dots ,(x_{p},y_{p})} produces a sequence of weights w 0 , w 1 , … , w p {\displaystyle w_{0},w_{1},\dots ,w_{p}} starting from some initial weight w 0 {\displaystyle w_{0}} , usually chosen at random. These weights are computed in turn: first compute w i {\displaystyle w_{i}} using only ( x i , y i , w i − 1 ) {\displaystyle (x_{i},y_{i},w_{i-1})} for i = 1 , … , p {\displaystyle i=1,\dots ,p} . The output of the algorithm is then w p {\displaystyle w_{p}} , giving a new function x ↦ f N ( w p , x ) {\displaystyle x\mapsto f_{N}(w_{p},x)} . The computation is the same in each step, hence only the case i = 1 {\displaystyle i=1} is described. w 1 {\displaystyle w_{1}} is calculated from ( x 1 , y 1 , w 0 ) {\displaystyle (x_{1},y_{1},w_{0})} by considering a variable weight w {\displaystyle w} and applying gradient descent to the function w ↦ E ( f N ( w , x 1 ) , y 1 ) {\displaystyle w\mapsto E(f_{N}(w,x_{1}),y_{1})} to find a local minimum, starting at w = w 0 {\displaystyle w=w_{0}} . This makes w 1 {\displaystyle w_{1}} the minimizing weight found by gradient descent. == Learning pseudocode == To implement the algorithm above, explicit formulas are required for the gradient of the function w ↦ E ( f N ( w , x ) , y ) {\displaystyle w\mapsto E(f_{N}(w,x),y)} where the function is E ( y , y ′ ) = | y − y ′ | 2 {\displaystyle E(y,y')=|y-y'|^{2}} . The learning algorithm can be divided into two phases: propagation and weight update. === Propagation === Propagation involves the following steps: Propagation forward through the network to generate the output value(s) Calculation of the cost (error term) Propagation of the output activations back through the network using the training pattern target to generate the deltas (the difference between the targeted and actual output values) of all output and hidden neurons. === Weight update === For each weight: Multiply the weight's output delta and input activation to find the gradient of the weight. Subtract the ratio (percentage) of the weight's gradient from the weight. The learning rate is the ratio (percentage) that influences the speed and quality of learning. The greater the ratio, the faster the neuron trains, but the lower the ratio, the more accurat

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  Actor-critic algorithm

  The actor-critic algorithm (AC) is a family of reinforcement learning (RL) algorithms that combine policy-based RL algorithms such as policy gradient methods, and value-based RL algorithms such as value iteration, Q-learning, SARSA, and TD learning. An AC algorithm consists of two main components: an "actor" that determines which actions to take according to a policy function, and a "critic" that evaluates those actions according to a value function. Some AC algorithms are on-policy, some are off-policy. Some apply to either continuous or discrete action spaces. Some work in both cases. == Overview == The actor-critic methods can be understood as an improvement over pure policy gradient methods like REINFORCE via introducing a baseline. === Actor === The actor uses a policy function π ( a | s ) {\displaystyle \pi (a|s)} , while the critic estimates either the value function V ( s ) {\displaystyle V(s)} , the action-value Q-function Q ( s , a ) , {\displaystyle Q(s,a),} the advantage function A ( s , a ) {\displaystyle A(s,a)} , or any combination thereof. The actor is a parameterized function π θ {\displaystyle \pi _{\theta }} , where θ {\displaystyle \theta } are the parameters of the actor. The actor takes as argument the state of the environment s {\displaystyle s} and produces a probability distribution π θ ( ⋅ | s ) {\displaystyle \pi _{\theta }(\cdot |s)} . If the action space is discrete, then ∑ a π θ ( a | s ) = 1 {\displaystyle \sum _{a}\pi _{\theta }(a|s)=1} . If the action space is continuous, then ∫ a π θ ( a | s ) d a = 1 {\displaystyle \int _{a}\pi _{\theta }(a|s)da=1} . The goal of policy optimization is to improve the actor. That is, to find some θ {\displaystyle \theta } that maximizes the expected episodic reward J ( θ ) {\displaystyle J(\theta )} : J ( θ ) = E π θ [ ∑ t = 0 T γ t r t ] {\displaystyle J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\gamma ^{t}r_{t}\right]} where γ {\displaystyle \gamma } is the discount factor, r t {\displaystyle r_{t}} is the reward at step t {\displaystyle t} , and T {\displaystyle T} is the time-horizon (which can be infinite). The goal of policy gradient method is to optimize J ( θ ) {\displaystyle J(\theta )} by gradient ascent on the policy gradient ∇ J ( θ ) {\displaystyle \nabla J(\theta )} . As detailed on the policy gradient method page, there are many unbiased estimators of the policy gradient: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ j ≤ T ∇ θ ln ⁡ π θ ( A j | S j ) ⋅ Ψ j | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{0\leq j\leq T}\nabla _{\theta }\ln \pi _{\theta }(A_{j}|S_{j})\cdot \Psi _{j}{\Big |}S_{0}=s_{0}\right]} where Ψ j {\textstyle \Psi _{j}} is a linear sum of the following: ∑ 0 ≤ i ≤ T ( γ i R i ) {\textstyle \sum _{0\leq i\leq T}(\gamma ^{i}R_{i})} . γ j ∑ j ≤ i ≤ T ( γ i − j R i ) {\textstyle \gamma ^{j}\sum _{j\leq i\leq T}(\gamma ^{i-j}R_{i})} : the REINFORCE algorithm. γ j ∑ j ≤ i ≤ T ( γ i − j R i ) − b ( S j ) {\textstyle \gamma ^{j}\sum _{j\leq i\leq T}(\gamma ^{i-j}R_{i})-b(S_{j})} : the REINFORCE with baseline algorithm. Here b {\displaystyle b} is an arbitrary function. γ j ( R j + γ V π θ ( S j + 1 ) − V π θ ( S j ) ) {\textstyle \gamma ^{j}\left(R_{j}+\gamma V^{\pi _{\theta }}(S_{j+1})-V^{\pi _{\theta }}(S_{j})\right)} : TD(1) learning. γ j Q π θ ( S j , A j ) {\textstyle \gamma ^{j}Q^{\pi _{\theta }}(S_{j},A_{j})} . γ j A π θ ( S j , A j ) {\textstyle \gamma ^{j}A^{\pi _{\theta }}(S_{j},A_{j})} : Advantage Actor-Critic (A2C). γ j ( R j + γ R j + 1 + γ 2 V π θ ( S j + 2 ) − V π θ ( S j ) ) {\textstyle \gamma ^{j}\left(R_{j}+\gamma R_{j+1}+\gamma ^{2}V^{\pi _{\theta }}(S_{j+2})-V^{\pi _{\theta }}(S_{j})\right)} : TD(2) learning. γ j ( ∑ k = 0 n − 1 γ k R j + k + γ n V π θ ( S j + n ) − V π θ ( S j ) ) {\textstyle \gamma ^{j}\left(\sum _{k=0}^{n-1}\gamma ^{k}R_{j+k}+\gamma ^{n}V^{\pi _{\theta }}(S_{j+n})-V^{\pi _{\theta }}(S_{j})\right)} : TD(n) learning. γ j ∑ n = 1 ∞ λ n − 1 1 − λ ⋅ ( ∑ k = 0 n − 1 γ k R j + k + γ n V π θ ( S j + n ) − V π θ ( S j ) ) {\textstyle \gamma ^{j}\sum _{n=1}^{\infty }{\frac {\lambda ^{n-1}}{1-\lambda }}\cdot \left(\sum _{k=0}^{n-1}\gamma ^{k}R_{j+k}+\gamma ^{n}V^{\pi _{\theta }}(S_{j+n})-V^{\pi _{\theta }}(S_{j})\right)} : TD(λ) learning, also known as GAE (generalized advantage estimate). This is obtained by an exponentially decaying sum of the TD(n) learning terms. === Critic === In the unbiased estimators given above, certain functions such as V π θ , Q π θ , A π θ {\displaystyle V^{\pi _{\theta }},Q^{\pi _{\theta }},A^{\pi _{\theta }}} appear. These are approximated by the critic. Since these functions all depend on the actor, the critic must learn alongside the actor. The critic is learned by value-based RL algorithms. For example, if the critic is estimating the state-value function V π θ ( s ) {\displaystyle V^{\pi _{\theta }}(s)} , then it can be learned by any value function approximation method. Let the critic be a function approximator V ϕ ( s ) {\displaystyle V_{\phi }(s)} with parameters ϕ {\displaystyle \phi } . The simplest example is TD(1) learning, which trains the critic to minimize the TD(1) error: δ i = R i + γ V ϕ ( S i + 1 ) − V ϕ ( S i ) {\displaystyle \delta _{i}=R_{i}+\gamma V_{\phi }(S_{i+1})-V_{\phi }(S_{i})} The critic parameters are updated by gradient descent on the squared TD error: ϕ ← ϕ − α ∇ ϕ ( δ i ) 2 = ϕ + α δ i ∇ ϕ V ϕ ( S i ) {\displaystyle \phi \leftarrow \phi -\alpha \nabla _{\phi }(\delta _{i})^{2}=\phi +\alpha \delta _{i}\nabla _{\phi }V_{\phi }(S_{i})} where α {\displaystyle \alpha } is the learning rate. Note that the gradient is taken with respect to the ϕ {\displaystyle \phi } in V ϕ ( S i ) {\displaystyle V_{\phi }(S_{i})} only, since the ϕ {\displaystyle \phi } in γ V ϕ ( S i + 1 ) {\displaystyle \gamma V_{\phi }(S_{i+1})} constitutes a moving target, and the gradient is not taken with respect to that. This is a common source of error in implementations that use automatic differentiation, and requires "stopping the gradient" at that point. Similarly, if the critic is estimating the action-value function Q π θ {\displaystyle Q^{\pi _{\theta }}} , then it can be learned by Q-learning or SARSA. In SARSA, the critic maintains an estimate of the Q-function, parameterized by ϕ {\displaystyle \phi } , denoted as Q ϕ ( s , a ) {\displaystyle Q_{\phi }(s,a)} . The temporal difference error is then calculated as δ i = R i + γ Q θ ( S i + 1 , A i + 1 ) − Q θ ( S i , A i ) {\displaystyle \delta _{i}=R_{i}+\gamma Q_{\theta }(S_{i+1},A_{i+1})-Q_{\theta }(S_{i},A_{i})} . The critic is then updated by θ ← θ + α δ i ∇ θ Q θ ( S i , A i ) {\displaystyle \theta \leftarrow \theta +\alpha \delta _{i}\nabla _{\theta }Q_{\theta }(S_{i},A_{i})} The advantage critic can be trained by training both a Q-function Q ϕ ( s , a ) {\displaystyle Q_{\phi }(s,a)} and a state-value function V ϕ ( s ) {\displaystyle V_{\phi }(s)} , then let A ϕ ( s , a ) = Q ϕ ( s , a ) − V ϕ ( s ) {\displaystyle A_{\phi }(s,a)=Q_{\phi }(s,a)-V_{\phi }(s)} . Although, it is more common to train just a state-value function V ϕ ( s ) {\displaystyle V_{\phi }(s)} , then estimate the advantage by A ϕ ( S i , A i ) ≈ ∑ j ∈ 0 : n − 1 γ j R i + j + γ n V ϕ ( S i + n ) − V ϕ ( S i ) {\displaystyle A_{\phi }(S_{i},A_{i})\approx \sum _{j\in 0:n-1}\gamma ^{j}R_{i+j}+\gamma ^{n}V_{\phi }(S_{i+n})-V_{\phi }(S_{i})} Here, n {\displaystyle n} is a positive integer. The higher n {\displaystyle n} is, the more lower is the bias in the advantage estimation, but at the price of higher variance. The Generalized Advantage Estimation (GAE) introduces a hyperparameter λ {\displaystyle \lambda } that smoothly interpolates between Monte Carlo returns ( λ = 1 {\displaystyle \lambda =1} , high variance, no bias) and 1-step TD learning ( λ = 0 {\displaystyle \lambda =0} , low variance, high bias). This hyperparameter can be adjusted to pick the optimal bias-variance trade-off in advantage estimation. It uses an exponentially decaying average of n-step returns with λ {\displaystyle \lambda } being the decay strength. == Variants == Asynchronous Advantage Actor-Critic (A3C): Parallel and asynchronous version of A2C. Soft Actor-Critic (SAC): Incorporates entropy maximization for improved exploration. Deep Deterministic Policy Gradient (DDPG): Specialized for continuous action spaces.

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  Physical neural network

  A physical neural network is a type of artificial neural network in which an electrically adjustable material is used to emulate the function of a neural synapse or a higher-order (dendritic) neuron model. "Physical" neural network is used to emphasize the reliance on physical hardware used to emulate neurons as opposed to software-based approaches. More generally the term is applicable to other artificial neural networks in which a memristor or other electrically adjustable resistance material is used to emulate a neural synapse. == Types of physical neural networks == === ADALINE === In the 1960s Bernard Widrow and Ted Hoff developed ADALINE (Adaptive Linear Neuron) which used electrochemical cells called memistors (memory resistors) to emulate synapses of an artificial neuron. The memistors were implemented as 3-terminal devices operating based on the reversible electroplating of copper such that the resistance between two of the terminals is controlled by the integral of the current applied via the third terminal. The ADALINE circuitry was briefly commercialized by the Memistor Corporation in the 1960s enabling some applications in pattern recognition. However, since the memistors were not fabricated using integrated circuit fabrication techniques the technology was not scalable and was eventually abandoned as solid-state electronics became mature. === Analog VLSI === In 1989 Carver Mead published his book Analog VLSI and Neural Systems, which spun off perhaps the most common variant of analog neural networks. The physical realization is implemented in analog VLSI. This is often implemented as field effect transistors in low inversion. Such devices can be modelled as translinear circuits. This is a technique described by Barrie Gilbert in several papers around mid 1970th, and in particular his Translinear Circuits from 1981. With this method circuits can be analyzed as a set of well-defined functions in steady-state, and such circuits assembled into complex networks. === Physical Neural Network === Alex Nugent describes a physical neural network as one or more nonlinear neuron-like nodes used to sum signals and nanoconnections formed from nanoparticles, nanowires, or nanotubes which determine the signal strength input to the nodes. Alignment or self-assembly of the nanoconnections is determined by the history of the applied electric field performing a function analogous to neural synapses. Numerous applications for such physical neural networks are possible. For example, a temporal summation device can be composed of one or more nanoconnections having an input and an output thereof, wherein an input signal provided to the input causes one or more of the nanoconnection to experience an increase in connection strength thereof over time. Another example of a physical neural network is taught by U.S. Patent No. 7,039,619 entitled "Utilized nanotechnology apparatus using a neural network, a solution and a connection gap," which issued to Alex Nugent by the U.S. Patent & Trademark Office on May 2, 2006. A further application of physical neural network is shown in U.S. Patent No. 7,412,428 entitled "Application of hebbian and anti-hebbian learning to nanotechnology-based physical neural networks," which issued on August 12, 2008. Nugent and Molter have shown that universal computing and general-purpose machine learning are possible from operations available through simple memristive circuits operating the AHaH plasticity rule. More recently, it has been argued that also complex networks of purely memristive circuits can serve as neural networks. === Phase change neural network === In 2002, Stanford Ovshinsky described an analog neural computing medium in which phase-change material has the ability to cumulatively respond to multiple input signals. An electrical alteration of the resistance of the phase change material is used to control the weighting of the input signals. === Memristive neural network === Greg Snider of HP Labs describes a system of cortical computing with memristive nanodevices. The memristors (memory resistors) are implemented by thin film materials in which the resistance is electrically tuned via the transport of ions or oxygen vacancies within the film. DARPA's SyNAPSE project has funded IBM Research and HP Labs, in collaboration with the Boston University Department of Cognitive and Neural Systems (CNS), to develop neuromorphic architectures which may be based on memristive systems. === Protonic artificial synapses === In 2022, researchers reported the development of nanoscale brain-inspired artificial synapses, using the ion proton (H+), for 'analog deep learning'.

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  Representer theorem

  For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer f ∗ {\displaystyle f^{}} of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data. == Formal statement == The following Representer Theorem and its proof are due to Schölkopf, Herbrich, and Smola: Theorem: Consider a positive-definite real-valued kernel k : X × X → R {\displaystyle k:{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} } on a non-empty set X {\displaystyle {\mathcal {X}}} with a corresponding reproducing kernel Hilbert space H k {\displaystyle H_{k}} . Let there be given a training sample ( x 1 , y 1 ) , … , ( x n , y n ) ∈ X × R {\displaystyle (x_{1},y_{1}),\dotsc ,(x_{n},y_{n})\in {\mathcal {X}}\times \mathbb {R} } , a strictly increasing real-valued function g : [ 0 , ∞ ) → R {\displaystyle g\colon [0,\infty )\to \mathbb {R} } , and an arbitrary error function E : ( X × R 2 ) n → R ∪ { ∞ } {\displaystyle E\colon ({\mathcal {X}}\times \mathbb {R} ^{2})^{n}\to \mathbb {R} \cup \lbrace \infty \rbrace } , which together define the following regularized empirical risk functional on H k {\displaystyle H_{k}} : f ↦ E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + g ( ‖ f ‖ ) . {\displaystyle f\mapsto E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+g\left(\lVert f\rVert \right).} Then, any minimizer of the empirical risk f ∗ = argmin f ∈ H k { E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + g ( ‖ f ‖ ) } , ( ∗ ) {\displaystyle f^{}={\underset {f\in H_{k}}{\operatorname {argmin} }}\left\lbrace E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+g\left(\lVert f\rVert \right)\right\rbrace ,\quad ()} admits a representation of the form: f ∗ ( ⋅ ) = ∑ i = 1 n α i k ( ⋅ , x i ) , {\displaystyle f^{}(\cdot )=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i}),} where α i ∈ R {\displaystyle \alpha _{i}\in \mathbb {R} } for all 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} . Proof: Define a mapping φ : X → H k φ ( x ) = k ( ⋅ , x ) {\displaystyle {\begin{aligned}\varphi \colon {\mathcal {X}}&\to H_{k}\\\varphi (x)&=k(\cdot ,x)\end{aligned}}} (so that φ ( x ) = k ( ⋅ , x ) {\displaystyle \varphi (x)=k(\cdot ,x)} is itself a map X → R {\displaystyle {\mathcal {X}}\to \mathbb {R} } ). Since k {\displaystyle k} is a reproducing kernel, then φ ( x ) ( x ′ ) = k ( x ′ , x ) = ⟨ φ ( x ′ ) , φ ( x ) ⟩ , {\displaystyle \varphi (x)(x')=k(x',x)=\langle \varphi (x'),\varphi (x)\rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product on H k {\displaystyle H_{k}} . Given any x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} , one can use orthogonal projection to decompose any f ∈ H k {\displaystyle f\in H_{k}} into a sum of two functions, one lying in span ⁡ { φ ( x 1 ) , … , φ ( x n ) } {\displaystyle \operatorname {span} \left\lbrace \varphi (x_{1}),\ldots ,\varphi (x_{n})\right\rbrace } , and the other lying in the orthogonal complement: f = ∑ i = 1 n α i φ ( x i ) + v , {\displaystyle f=\sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v,} where ⟨ v , φ ( x i ) ⟩ = 0 {\displaystyle \langle v,\varphi (x_{i})\rangle =0} for all i {\displaystyle i} . The above orthogonal decomposition and the reproducing property together show that applying f {\displaystyle f} to any training point x j {\displaystyle x_{j}} produces f ( x j ) = ⟨ ∑ i = 1 n α i φ ( x i ) + v , φ ( x j ) ⟩ = ∑ i = 1 n α i ⟨ φ ( x i ) , φ ( x j ) ⟩ , {\displaystyle f(x_{j})=\left\langle \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v,\varphi (x_{j})\right\rangle =\sum _{i=1}^{n}\alpha _{i}\langle \varphi (x_{i}),\varphi (x_{j})\rangle ,} which we observe is independent of v {\displaystyle v} . Consequently, the value of the error function E {\displaystyle E} in () is likewise independent of v {\displaystyle v} . For the second term (the regularization term), since v {\displaystyle v} is orthogonal to ∑ i = 1 n α i φ ( x i ) {\displaystyle \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})} and g {\displaystyle g} is strictly monotonic, we have g ( ‖ f ‖ ) = g ( ‖ ∑ i = 1 n α i φ ( x i ) + v ‖ ) = g ( ‖ ∑ i = 1 n α i φ ( x i ) ‖ 2 + ‖ v ‖ 2 ) ≥ g ( ‖ ∑ i = 1 n α i φ ( x i ) ‖ ) . {\displaystyle {\begin{aligned}g\left(\lVert f\rVert \right)&=g\left(\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v\rVert \right)\\&=g\left({\sqrt {\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})\rVert ^{2}+\lVert v\rVert ^{2}}}\right)\\&\geq g\left(\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})\rVert \right).\end{aligned}}} Therefore, setting v = 0 {\displaystyle v=0} does not affect the first term of (), while it strictly decreases the second term. Consequently, any minimizer f ∗ {\displaystyle f^{}} in () must have v = 0 {\displaystyle v=0} , i.e., it must be of the form f ∗ ( ⋅ ) = ∑ i = 1 n α i φ ( x i ) = ∑ i = 1 n α i k ( ⋅ , x i ) , {\displaystyle f^{}(\cdot )=\sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i}),} which is the desired result. == Generalizations == The Theorem stated above is a particular example of a family of results that are collectively referred to as "representer theorems"; here we describe several such. The first statement of a representer theorem was due to Kimeldorf and Wahba for the special case in which E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) = 1 n ∑ i = 1 n ( f ( x i ) − y i ) 2 , g ( ‖ f ‖ ) = λ ‖ f ‖ 2 {\displaystyle {\begin{aligned}E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)&={\frac {1}{n}}\sum _{i=1}^{n}(f(x_{i})-y_{i})^{2},\\g(\lVert f\rVert )&=\lambda \lVert f\rVert ^{2}\end{aligned}}} for λ > 0 {\displaystyle \lambda >0} . Schölkopf, Herbrich, and Smola generalized this result by relaxing the assumption of the squared-loss cost and allowing the regularizer to be any strictly monotonically increasing function g ( ⋅ ) {\displaystyle g(\cdot )} of the Hilbert space norm. It is possible to generalize further by augmenting the regularized empirical risk functional through the addition of unpenalized offset terms. For example, Schölkopf, Herbrich, and Smola also consider the minimization f ~ ∗ = argmin ⁡ { E ( ( x 1 , y 1 , f ~ ( x 1 ) ) , … , ( x n , y n , f ~ ( x n ) ) ) + g ( ‖ f ‖ ) ∣ f ~ = f + h ∈ H k ⊕ span ⁡ { ψ p ∣ 1 ≤ p ≤ M } } , ( † ) {\displaystyle {\tilde {f}}^{}=\operatorname {argmin} \left\lbrace E\left((x_{1},y_{1},{\tilde {f}}(x_{1})),\ldots ,(x_{n},y_{n},{\tilde {f}}(x_{n}))\right)+g\left(\lVert f\rVert \right)\mid {\tilde {f}}=f+h\in H_{k}\oplus \operatorname {span} \lbrace \psi _{p}\mid 1\leq p\leq M\rbrace \right\rbrace ,\quad (\dagger )} i.e., we consider functions of the form f ~ = f + h {\displaystyle {\tilde {f}}=f+h} , where f ∈ H k {\displaystyle f\in H_{k}} and h {\displaystyle h} is an unpenalized function lying in the span of a finite set of real-valued functions { ψ p : X → R ∣ 1 ≤ p ≤ M } {\displaystyle \lbrace \psi _{p}\colon {\mathcal {X}}\to \mathbb {R} \mid 1\leq p\leq M\rbrace } . Under the assumption that the n × M {\displaystyle n\times M} matrix ( ψ p ( x i ) ) i p {\displaystyle \left(\psi _{p}(x_{i})\right)_{ip}} has rank M {\displaystyle M} , they show that the minimizer f ~ ∗ {\displaystyle {\tilde {f}}^{}} in ( † ) {\displaystyle (\dagger )} admits a representation of the form f ~ ∗ ( ⋅ ) = ∑ i = 1 n α i k ( ⋅ , x i ) + ∑ p = 1 M β p ψ p ( ⋅ ) {\displaystyle {\tilde {f}}^{}(\cdot )=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i})+\sum _{p=1}^{M}\beta _{p}\psi _{p}(\cdot )} where α i , β p ∈ R {\displaystyle \alpha _{i},\beta _{p}\in \mathbb {R} } and the β p {\displaystyle \beta _{p}} are all uniquely determined. The conditions under which a representer theorem exists were investigated by Argyriou, Micchelli, and Pontil, who proved the following: Theorem: Let X {\displaystyle {\mathcal {X}}} be a nonempty set, k {\displaystyle k} a positive-definite real-valued kernel on X × X {\displaystyle {\mathcal {X}}\times {\mathcal {X}}} with corresponding reproducing kernel Hilbert space H k {\displaystyle H_{k}} , and let R : H k → R {\displaystyle R\colon H_{k}\to \mathbb {R} } be a differentiable regularization function. Then given a training sample ( x 1 , y 1 ) , … , ( x n , y n ) ∈ X × R {\displaystyle (x_{1},y_{1}),\ldots ,(x_{n},y_{n})\in {\mathcal {X}}\times \mathbb {R} } and an arbitrary error function E : ( X × R 2 ) m → R ∪ { ∞ } {\displaystyle E\colon ({\mathcal {X}}\times \mathbb {R} ^{2})^{m}\to \mathbb {R} \cup \lbrace \infty \rbrace } , a minimizer f ∗ = argmin f ∈ H k { E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + R ( f ) } ( ‡ ) {\displaystyle f^{}={\underset {f\in H_{k}}{\operatorname {argmin} }}\left\lbrace E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+R(f)\right\rbrace \quad (\ddagger )} of the regularized empirical risk admits a repr

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  Stochastic gradient descent

  Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient (calculated from the entire data set) by an estimate thereof (calculated from a randomly selected subset of the data). Especially in high-dimensional optimization problems this reduces the very high computational burden, achieving faster iterations in exchange for a lower convergence rate. The basic idea behind stochastic approximation can be traced back to the Robbins–Monro algorithm of the 1950s. Today, stochastic gradient descent has become an important optimization method in machine learning. == Background == Both statistical estimation and machine learning consider the problem of minimizing an objective function that has the form of a sum: Q ( w ) = 1 n ∑ i = 1 n Q i ( w ) , {\displaystyle Q(w)={\frac {1}{n}}\sum _{i=1}^{n}Q_{i}(w),} where the parameter w {\displaystyle w} that minimizes Q ( w ) {\displaystyle Q(w)} is to be estimated. Each summand function Q i {\displaystyle Q_{i}} is typically associated with the i {\displaystyle i} -th observation in the data set (used for training). In classical statistics, sum-minimization problems arise in least squares and in maximum-likelihood estimation (for independent observations). The general class of estimators that arise as minimizers of sums are called M-estimators. However, in statistics, it has been long recognized that requiring even local minimization is too restrictive for some problems of maximum-likelihood estimation. Therefore, contemporary statistical theorists often consider stationary points of the likelihood function (or zeros of its derivative, the score function, and other estimating equations). The sum-minimization problem also arises for empirical risk minimization. There, Q i ( w ) {\displaystyle Q_{i}(w)} is the value of the loss function at i {\displaystyle i} -th example, and Q ( w ) {\displaystyle Q(w)} is the empirical risk. When used to minimize the above function, a standard (or "batch") gradient descent method would perform the following iterations: w := w − η ∇ Q ( w ) = w − η n ∑ i = 1 n ∇ Q i ( w ) . {\displaystyle w:=w-\eta \,\nabla Q(w)=w-{\frac {\eta }{n}}\sum _{i=1}^{n}\nabla Q_{i}(w).} The step size is denoted by η {\displaystyle \eta } (sometimes called the learning rate in machine learning) and here " := {\displaystyle :=} " denotes the update of a variable in the algorithm. In many cases, the summand functions have a simple form that enables inexpensive evaluations of the sum-function and the sum gradient. For example, in statistics, one-parameter exponential families allow economical function-evaluations and gradient-evaluations. However, in other cases, evaluating the sum-gradient may require expensive evaluations of the gradients from all summand functions. When the training set is enormous and no simple formulas exist, evaluating the sums of gradients becomes very expensive, because evaluating the gradient requires evaluating all the summand functions' gradients. To economize on the computational cost at every iteration, stochastic gradient descent samples a subset of summand functions at every step. This is very effective in the case of large-scale machine learning problems. == Iterative method == In stochastic (or "on-line") gradient descent, the true gradient of Q ( w ) {\displaystyle Q(w)} is approximated by a gradient at a single sample: w := w − η ∇ Q i ( w ) . {\displaystyle w:=w-\eta \,\nabla Q_{i}(w).} As the algorithm sweeps through the training set, it performs the above update for each training sample. Several passes can be made over the training set until the algorithm converges. If this is done, the data can be shuffled for each pass to prevent cycles. Typical implementations may use an adaptive learning rate so that the algorithm converges. In pseudocode, stochastic gradient descent can be presented as : A compromise between computing the true gradient and the gradient at a single sample is to compute the gradient against more than one training sample (called a "mini-batch") at each step. This can perform significantly better than "true" stochastic gradient descent described, because the code can make use of vectorization libraries rather than computing each step separately as was first shown in where it was called "the bunch-mode back-propagation algorithm". It may also result in smoother convergence, as the gradient computed at each step is averaged over more training samples. The convergence of stochastic gradient descent has been analyzed using the theories of convex minimization and of stochastic approximation. Briefly, when the learning rates η {\displaystyle \eta } decrease with an appropriate rate, and subject to relatively mild assumptions, stochastic gradient descent converges almost surely to a global minimum when the objective function is convex or pseudoconvex, and otherwise converges almost surely to a local minimum. This is in fact a consequence of the Robbins–Siegmund theorem. == Linear regression == Suppose we want to fit a straight line y ^ = w 1 + w 2 x {\displaystyle {\hat {y}}=w_{1}+w_{2}x} to a training set with observations ( ( x 1 , y 1 ) , ( x 2 , y 2 ) … , ( x n , y n ) ) {\displaystyle ((x_{1},y_{1}),(x_{2},y_{2})\ldots ,(x_{n},y_{n}))} and corresponding estimated responses ( y ^ 1 , y ^ 2 , … , y ^ n ) {\displaystyle ({\hat {y}}_{1},{\hat {y}}_{2},\ldots ,{\hat {y}}_{n})} using least squares. The objective function to be minimized is Q ( w ) = ∑ i = 1 n Q i ( w ) = ∑ i = 1 n ( y ^ i − y i ) 2 = ∑ i = 1 n ( w 1 + w 2 x i − y i ) 2 . {\displaystyle Q(w)=\sum _{i=1}^{n}Q_{i}(w)=\sum _{i=1}^{n}\left({\hat {y}}_{i}-y_{i}\right)^{2}=\sum _{i=1}^{n}\left(w_{1}+w_{2}x_{i}-y_{i}\right)^{2}.} The last line in the above pseudocode for this specific problem will become: [ w 1 w 2 ] ← [ w 1 w 2 ] − η [ ∂ ∂ w 1 ( w 1 + w 2 x i − y i ) 2 ∂ ∂ w 2 ( w 1 + w 2 x i − y i ) 2 ] = [ w 1 w 2 ] − η [ 2 ( w 1 + w 2 x i − y i ) 2 x i ( w 1 + w 2 x i − y i ) ] . {\displaystyle {\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}\leftarrow {\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}-\eta {\begin{bmatrix}{\frac {\partial }{\partial w_{1}}}(w_{1}+w_{2}x_{i}-y_{i})^{2}\\{\frac {\partial }{\partial w_{2}}}(w_{1}+w_{2}x_{i}-y_{i})^{2}\end{bmatrix}}={\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}-\eta {\begin{bmatrix}2(w_{1}+w_{2}x_{i}-y_{i})\\2x_{i}(w_{1}+w_{2}x_{i}-y_{i})\end{bmatrix}}.} Note that in each iteration or update step, the gradient is only evaluated at a single x i {\displaystyle x_{i}} . This is the key difference between stochastic gradient descent and batched gradient descent. In general, given a linear regression y ^ = ∑ k ∈ 1 : m w k x k {\displaystyle {\hat {y}}=\sum _{k\in 1:m}w_{k}x_{k}} problem, stochastic gradient descent behaves differently when m < n {\displaystyle m

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  Evolutionary robotics

  Evolutionary robotics is an embodied approach to Artificial Intelligence (AI) in which robots are automatically designed using Darwinian principles of natural selection. The design of a robot, or a subsystem of a robot such as a neural controller, is optimized against a behavioral goal (e.g. run as fast as possible). Usually, designs are evaluated in simulations as fabricating thousands or millions of designs and testing them in the real world is prohibitively expensive in terms of time, money, and safety. An evolutionary robotics experiment starts with a population of randomly generated robot designs. The worst performing designs are discarded and replaced with mutations and/or combinations of the better designs. This evolutionary algorithm continues until a prespecified amount of time elapses or some target performance metric is surpassed. Evolutionary robotics methods are particularly useful for engineering machines that must operate in environments in which humans have limited intuition (nanoscale, space, etc.). Evolved simulated robots can also be used as scientific tools to generate new hypotheses in biology and cognitive science, and to test old hypothesis that require experiments that have proven difficult or impossible to carry out in reality. == History == In the early 1990s, two separate European groups demonstrated different approaches to the evolution of robot control systems. Dario Floreano and Francesco Mondada at EPFL evolved controllers for the Khepera robot. Adrian Thompson, Nick Jakobi, Dave Cliff, Inman Harvey, and Phil Husbands evolved controllers for a Gantry robot at the University of Sussex. However the body of these robots was presupposed before evolution. The first simulations of evolved robots were reported by Karl Sims and Jeffrey Ventrella of the MIT Media Lab, also in the early 1990s. However these so-called virtual creatures never left their simulated worlds. The first evolved robots to be built in reality were 3D-printed by Hod Lipson and Jordan Pollack at Brandeis University at the turn of the 21st century.

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  Cellular evolutionary algorithm

  A cellular evolutionary algorithm (cEA) is a kind of evolutionary algorithm (EA) in which individuals cannot mate arbitrarily, but every one interacts with its closer neighbors on which a basic EA is applied (selection, variation, replacement). The cellular model simulates natural evolution from the point of view of the individual, which encodes a tentative optimization, learning, or search problem solution. The essential idea of this model is to provide the EA population with a special structure defined as a connected graph, in which each vertex is an individual who communicates with his nearest neighbors. Particularly, individuals are conceptually set in a toroidal mesh, and are only allowed to recombine with close individuals. This leads to a kind of locality known as "isolation by distance". The set of potential mates of an individual is called its "neighborhood". It is known that, in this kind of algorithm, similar individuals tend to cluster creating niches, and these groups operate as if they were separate sub-populations (islands). There is no clear borderline between adjacent groups, and close niches could be easily colonized by competitive niches and potentially merge solution contents during the process. Simultaneously, farther niches can be affected more slowly. == Introduction == A cellular evolutionary algorithm (cEA) usually evolves a structured bidimensional grid of individuals, although other topologies are also possible. In this grid, clusters of similar individuals are naturally created during evolution, promoting exploration in their boundaries, while exploitation is mainly performed by direct competition and merging inside them. The grid is usually 2D toroidal structure, although the number of dimensions can be easily extended (to 3D) or reduced (to 1D, e.g. a ring). The neighborhood of a particular point of the grid (where an individual is placed) is defined in terms of the Manhattan distance from it to others in the population. Each point of the grid has a neighborhood that overlaps the neighborhoods of nearby individuals. In the basic algorithm, all the neighborhoods have the same size and identical shapes. The two most commonly used neighborhoods are L5, also called the Von Neumann or NEWS (North, East, West and South) neighborhood, and C9, also known as the Moore neighborhood. Here, L stands for "linear" while C stands for "compact". In cEAs, the individuals can only interact with their neighbors in the reproductive cycle where the variation operators are applied. This reproductive cycle is executed inside the neighborhood of each individual and, generally, consists in selecting two parents among its neighbors according to a certain criterion, applying the variation operators to them (recombination and mutation for example), and replacing the considered individual by the recently created offspring following a given criterion, for instance, replace if the offspring represents a better solution than the considered individual. == Synchronous versus asynchronous == In a regular synchronous cEA, the algorithm proceeds from the very first top left individual to the right and then to the several rows by using the information in the population to create a new temporary population. After finishing with the bottom-right last individual the temporary population is full with the newly computed individuals, and the replacement step starts. In it, the old population is completely and synchronously replaced with the newly computed one according to some criterion. Usually, the replacement keeps the best individual in the same position of both populations, that is, elitism is used. According to the update policy of the population used, an asynchronous cEA may also be defined and is a well-known issue in cellular automata. In asynchronous cEAs the order in which the individuals in the grid are update changes depending on the choice of criterion: line sweep, fixed random sweep, new random sweep, and uniform choice. All four proceed using the newly computed individual (or the original if better) for the computations of its neighbors. The overlap of the neighborhoods provides an implicit mechanism of solution migration to the cEA. Since the best solutions spread smoothly through the whole population, genetic diversity in the population is preserved longer than in non structured EAs. This soft dispersion of the best solutions through the population is one of the main issues of the good tradeoff between exploration and exploitation that cEAs perform during the search. This tradeoff can be tuned (and by extension the genetic diversity level along the evolution) by modifying (for instance) the size of the neighborhood used, as the overlap degree between the neighborhoods grows according to the size of the neighborhood. A cEA can be seen as a cellular automaton (CA) with probabilistic rewritable rules, where the alphabet of the CA is equivalent to the potential number of solutions of the problem. Hence, knowledge from research in CAs can be applied to cEAs. == Parallelism == Cellular EAs are very amenable to parallelism, thus usually found in the literature of parallel metaheuristics. In particular, fine grain parallelism can be used to assign independent threads of execution to every individual, thus allowing the whole cEA to run on a concurrent or actually parallel hardware platform. In this way, large time reductions can be obtained when running cEAs on FPGAs or GPUs. However, it is important to stress that cEAs are a model of search, in many senses different from traditional EAs. Also, they can be run in sequential and parallel platforms, reinforcing the fact that the model and the implementation are two different concepts. See here for a complete description on the fundamentals for the understanding, design, and application of cEAs.

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  AdaBoost

  AdaBoost (short for Adaptive Boosting) is a statistical classification meta-algorithm formulated by Yoav Freund and Robert Schapire in 1995, who won the 2003 Gödel Prize for their work. It can be used in conjunction with many types of learning algorithm to improve performance. The output of multiple weak learners is combined into a weighted sum that represents the final output of the boosted classifier. Usually, AdaBoost is presented for binary classification, although it can be generalized to multiple classes or bounded intervals of real values. AdaBoost is adaptive in the sense that subsequent weak learners (models) are adjusted in favor of instances misclassified by previous models. In some problems, it can be less susceptible to overfitting than other learning algorithms. The individual learners can be weak, but as long as the performance of each one is slightly better than random guessing, the final model can be proven to converge to a strong learner. Although AdaBoost is typically used to combine weak base learners (such as decision stumps), it has been shown to also effectively combine strong base learners (such as deeper decision trees), producing an even more accurate model. Every learning algorithm tends to suit some problem types better than others, and typically has many different parameters and configurations to adjust before it achieves optimal performance on a dataset. AdaBoost (with decision trees as the weak learners) is often referred to as the best out-of-the-box classifier. When used with decision tree learning, information gathered at each stage of the AdaBoost algorithm about the relative 'hardness' of each training sample is fed into the tree-growing algorithm such that later trees tend to focus on harder-to-classify examples. == Training == AdaBoost refers to a particular method of training a boosted classifier. A boosted classifier is a classifier of the form F T ( x ) = ∑ t = 1 T f t ( x ) {\displaystyle F_{T}(x)=\sum _{t=1}^{T}f_{t}(x)} where each f t {\displaystyle f_{t}} is a weak learner that takes an object x {\displaystyle x} as input and returns a value indicating the class of the object. For example, in the two-class problem, the sign of the weak learner's output identifies the predicted object class and the absolute value gives the confidence in that classification. Each weak learner produces an output hypothesis h {\displaystyle h} which fixes a prediction h ( x i ) {\displaystyle h(x_{i})} for each sample in the training set. At each iteration t {\displaystyle t} , a weak learner is selected and assigned a coefficient α t {\displaystyle \alpha _{t}} such that the total training error E t {\displaystyle E_{t}} of the resulting t {\displaystyle t} -stage boosted classifier is minimized. E t = ∑ i E [ F t − 1 ( x i ) + α t h ( x i ) ] {\displaystyle E_{t}=\sum _{i}E[F_{t-1}(x_{i})+\alpha _{t}h(x_{i})]} Here F t − 1 ( x ) {\displaystyle F_{t-1}(x)} is the boosted classifier that has been built up to the previous stage of training and f t ( x ) = α t h ( x ) {\displaystyle f_{t}(x)=\alpha _{t}h(x)} is the weak learner that is being considered for addition to the final classifier. === Weighting === At each iteration of the training process, a weight w i , t {\displaystyle w_{i,t}} is assigned to each sample in the training set equal to the current error E ( F t − 1 ( x i ) ) {\displaystyle E(F_{t-1}(x_{i}))} on that sample. These weights can be used in the training of the weak learner. For instance, decision trees can be grown which favor the splitting of sets of samples with large weights. == Derivation == This derivation follows Rojas (2009): Suppose we have a data set { ( x 1 , y 1 ) , … , ( x N , y N ) } {\displaystyle \{(x_{1},y_{1}),\ldots ,(x_{N},y_{N})\}} where each item x i {\displaystyle x_{i}} has an associated class y i ∈ { − 1 , 1 } {\displaystyle y_{i}\in \{-1,1\}} , and a set of weak classifiers { k 1 , … , k L } {\displaystyle \{k_{1},\ldots ,k_{L}\}} each of which outputs a classification k j ( x i ) ∈ { − 1 , 1 } {\displaystyle k_{j}(x_{i})\in \{-1,1\}} for each item. After the ( m − 1 ) {\displaystyle (m-1)} -th iteration our boosted classifier is a linear combination of the weak classifiers of the form: C ( m − 1 ) ( x i ) = α 1 k 1 ( x i ) + ⋯ + α m − 1 k m − 1 ( x i ) , {\displaystyle C_{(m-1)}(x_{i})=\alpha _{1}k_{1}(x_{i})+\cdots +\alpha _{m-1}k_{m-1}(x_{i}),} where the class will be the sign of C ( m − 1 ) ( x i ) {\displaystyle C_{(m-1)}(x_{i})} . At the m {\displaystyle m} -th iteration we want to extend this to a better boosted classifier by adding another weak classifier k m {\displaystyle k_{m}} , with another weight α m {\displaystyle \alpha _{m}} : C m ( x i ) = C ( m − 1 ) ( x i ) + α m k m ( x i ) {\displaystyle C_{m}(x_{i})=C_{(m-1)}(x_{i})+\alpha _{m}k_{m}(x_{i})} So it remains to determine which weak classifier is the best choice for k m {\displaystyle k_{m}} , and what its weight α m {\displaystyle \alpha _{m}} should be. We define the total error E {\displaystyle E} of C m {\displaystyle C_{m}} as the sum of its exponential loss on each data point, given as follows: E = ∑ i = 1 N e − y i C m ( x i ) = ∑ i = 1 N e − y i C ( m − 1 ) ( x i ) e − y i α m k m ( x i ) {\displaystyle E=\sum _{i=1}^{N}e^{-y_{i}C_{m}(x_{i})}=\sum _{i=1}^{N}e^{-y_{i}C_{(m-1)}(x_{i})}e^{-y_{i}\alpha _{m}k_{m}(x_{i})}} Letting w i ( 1 ) = 1 {\displaystyle w_{i}^{(1)}=1} and w i ( m ) = e − y i C m − 1 ( x i ) {\displaystyle w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}} for m > 1 {\displaystyle m>1} , we have: E = ∑ i = 1 N w i ( m ) e − y i α m k m ( x i ) {\displaystyle E=\sum _{i=1}^{N}w_{i}^{(m)}e^{-y_{i}\alpha _{m}k_{m}(x_{i})}} We can split this summation between those data points that are correctly classified by k m {\displaystyle k_{m}} (so y i k m ( x i ) = 1 {\displaystyle y_{i}k_{m}(x_{i})=1} ) and those that are misclassified (so y i k m ( x i ) = − 1 {\displaystyle y_{i}k_{m}(x_{i})=-1} ): E = ∑ y i = k m ( x i ) w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) e α m = ∑ i = 1 N w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) ( e α m − e − α m ) {\displaystyle {\begin{aligned}E&=\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}e^{\alpha _{m}}\\&=\sum _{i=1}^{N}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}\left(e^{\alpha _{m}}-e^{-\alpha _{m}}\right)\end{aligned}}} Since the only part of the right-hand side of this equation that depends on k m {\displaystyle k_{m}} is ∑ y i ≠ k m ( x i ) w i ( m ) {\textstyle \sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}} , we see that the k m {\displaystyle k_{m}} that minimizes E {\displaystyle E} is the one in the set { k 1 , … , k L } {\displaystyle \{k_{1},\ldots ,k_{L}\}} that minimizes ∑ y i ≠ k m ( x i ) w i ( m ) {\textstyle \sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}} [assuming that α m > 0 {\displaystyle \alpha _{m}>0} ], i.e. the weak classifier with the lowest weighted error (with weights w i ( m ) = e − y i C m − 1 ( x i ) {\displaystyle w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}} ). To determine the desired weight α m {\displaystyle \alpha _{m}} that minimizes E {\displaystyle E} with the k m {\displaystyle k_{m}} that we just determined, we differentiate: d E d α m = d ( ∑ y i = k m ( x i ) w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) e α m ) d α m {\displaystyle {\frac {dE}{d\alpha _{m}}}={\frac {d(\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}e^{\alpha _{m}})}{d\alpha _{m}}}} The value of α m {\displaystyle \alpha _{m}} that minimizes the above expression is: α m = 1 2 ln ⁡ ( ∑ y i = k m ( x i ) w i ( m ) ∑ y i ≠ k m ( x i ) w i ( m ) ) {\displaystyle \alpha _{m}={\frac {1}{2}}\ln \left({\frac {\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}}{\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}}}\right)} We calculate the weighted error rate of the weak classifier to be ϵ m = ∑ y i ≠ k m ( x i ) w i ( m ) ∑ i = 1 N w i ( m ) {\displaystyle \epsilon _{m}={\frac {\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}}{\sum _{i=1}^{N}w_{i}^{(m)}}}} , so it follows that: α m = 1 2 ln ⁡ ( 1 − ϵ m ϵ m ) {\displaystyle \alpha _{m}={\frac {1}{2}}\ln \left({\frac {1-\epsilon _{m}}{\epsilon _{m}}}\right)} which is the negative logit function multiplied by 0.5. Due to the convexity of E {\displaystyle E} as a function of α m {\displaystyle \alpha _{m}} , this new expression for α m {\displaystyle \alpha _{m}} gives the global minimum of the loss function. Note: This derivation only applies when k m ( x i ) ∈ { − 1 , 1 } {\displaystyle k_{m}(x_{i})\in \{-1,1\}} , though it can be a good starting guess in other cases, such as when the weak learner is biased ( k m ( x ) ∈ { a , b } , a ≠ − b {\displaystyle k_{m}(x)\in \{a,b\},a\neq -b} ), has multiple leaves ( k m ( x ) ∈ { a , b , … , n } {\displaystyle k_{m}(x)\in \{a,b,\dots ,n\}} ) or is some other function k m ( x ) ∈ R {\displaystyle k_{m}(x)\in \mathbb {R} } . Thus we have derived the AdaBoost algorithm: At each

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  Witness set

  In combinatorics and computational learning theory, a witness set is a set of elements that distinguishes a given Boolean function from a given class of other Boolean functions. Let C {\displaystyle C} be a concept class over a domain X {\displaystyle X} (that is, a family of Boolean functions over X {\displaystyle X} ) and c {\displaystyle c} be a concept in X {\displaystyle X} (a single Boolean function). A subset S {\displaystyle S} of X {\displaystyle X} is a witness set for c {\displaystyle c} in X {\displaystyle X} if S {\displaystyle S} distinguishes c {\displaystyle c} from all the other functions in C {\displaystyle C} , in the sense that no other function in C {\displaystyle C} has the same values on S {\displaystyle S} . For a concept class with | C | {\displaystyle |C|} concepts, there exists a concept that has a witness of size at most log 2 ⁡ | C | {\displaystyle \log _{2}|C|} ; this bound is tight when C {\displaystyle C} consists of all Boolean functions over X {\displaystyle X} . By a result of Bondy (1972) there exists a single witness set of size at most | C | − 1 {\displaystyle |C|-1} that is valid for all concepts in C {\displaystyle C} ; this bound is tight when C {\displaystyle C} consists of the indicator functions of the empty set and some singleton sets. One way to construct this set is to interpret the concepts as bitstrings, and the domain elements as positions in these bitstrings. Then the set of positions at which a trie of the bitstrings branches forms the desired witness set. This construction is central to the operation of the fusion tree data structure. The minimum size of a witness set for c {\displaystyle c} is called the witness size or specification number and is denoted by w C ( c ) {\displaystyle w_{C}(c)} . The value max { w C ( c ) : c ∈ C } {\displaystyle \max\{w_{C}(c):c\in C\}} is called the teaching dimension of C {\displaystyle C} . It represents the number of examples of a concept that need to be presented by a teacher to a learner, in the worst case, to enable the learner to determine which concept is being presented. Witness sets have also been called teaching sets, keys, specifying sets, or discriminants. The "witness set" terminology is from Kushilevitz et al. (1996), who trace the concept of witness sets to work by Cover (1965).

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  Just This Once

  Just This Once is a 1993 romance novel written in the style of Jacqueline Susann by a Macintosh IIcx computer named "Hal" in collaboration with its programmer, Scott French. French reportedly spent $40,000 and 8 years developing an artificial intelligence program to analyze Susann's works and attempt to create a novel that Susann might have written. A legal dispute between the estate of Jacqueline Susann and the publisher resulted in a settlement to split the profits, and the book was referenced in several legal journal articles about copyright laws. The book had two small print runs totaling 35,000 copies, receiving mixed reviews. == Creation == The novel's creation spanned the fields of artificial intelligence, expert systems, and natural language processing. Scott French first scanned and analyzed portions of two books by Jacqueline Susann, Valley of the Dolls and Once Is Not Enough, to determine constituents of Susann's writing style, which French stated was the most difficult task. This analysis extracted several hundred components including frequency and type of sexual acts and sentence structure. "Once you're there, the writer's style emerges, part of her actual personality comes out, and the computer can be programmed to make a story." French also created several thousand rules to govern tone, plotting, scenes, and characters. The text generated by Hal, the computer, was intended to mimic what Susann might have written, although the output required significant editing. French credits Hal's work with "almost 100% of the plot, 100% of the theme and style." French estimates that he wrote 10% of the prose, the computer Hal wrote about 25% of the prose, and the remaining two-thirds was more of a collaboration between the two. A typical scenario to write a scene would involve Hal asking questions that French would answer (for example, Hal might ask about the "cattiness factor" involved in a meeting between two key female characters, and French would reply with a range of 1 to 10), and the computer would then generate a few sentences to which French would make minor edits. The process would repeat for the next few sentences until the scene was written. == Legal issues == Jacqueline Susann's publisher was skeptical of the legality of Just This Once, although French doubted that an author's thought processes could be copyrighted. Susann's estate reportedly threatened to sue Scott French but the parties settled out of court; the settlement involved splitting profits between the parties but the terms of the settlement were not disclosed. The publication of Just This Once raised questions in the legal profession concerning how copyright law applies to computer-generated works derived from an analysis of other copyrighted works, and whether the generation of such works infringes on copyright. The publications on this topic suggested that the copyright laws of the time were ill-equipped to deal with computer-generated creative works. == Reception == The book's publisher Steven Shragis of Carol Group said of the novel, "I'm not going to say this is a great literary work, but it's every bit as good as anything out in this field, and better than an awful lot." The novel received some positive early reviews. In USA Today, novelist Thomas Gifford compared Just This Once to another novel in the same genre, American Star by Jackie Collins. Gifford concluded: "If you do like this stuff, you'd be much, much better off with the one written by the computer." The Dead Jackie Susann Quarterly declared that Susann "would be proud. Lots of money, sleaze, disease, death, oral sex, tragedy and the good girl gone bad." Other reviews were mixed. Publishers Weekly wrote, "If the books of Jacqueline Susann and Harold Robbins seem formulaic, this debut novel of sin and success in Las Vegas outdoes them all. And that, in a way, is the point.... All novelty rests in the conceit of computer authorship, not in the story itself." Library Journal stated "French invested eight years and $50,000 in a scheme to use artificial intelligence to fulfill his authentic, if dubious, desire to generate a trashy novel a la Jacqueline Susann. Shallow, beautiful-people characters are flatly conceived and randomly accessed in a formulaic plot ... a sexy, boring morality tale. Of possible interest to computer buffs for its use of Expert Systems and the virtual promise of more worthy possibilities; others should read Susann." Kirkus Reviews wrote: "The deal here is that author French is not the author, he's just the midwife, having allegedly programmed his computer to write about our times just the way Susann would... almost perfectly capturing glamorous Jackie's turgid but E-Z reading prose style and ultrareliable mix of sex, glitz, dope 'n' despair.... One wonders, though, if French's tale spinning PC will do as well on the talkshows as Jackie did. The computer weenies have been trying to tell us for years, garbage in-garbage out."

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  Sammon mapping

  Sammon mapping or Sammon projection is an algorithm that maps a high-dimensional space to a space of lower dimensionality (see multidimensional scaling) by trying to preserve the structure of inter-point distances in high-dimensional space in the lower-dimension projection. It is particularly suited for use in exploratory data analysis. The method was proposed by John W. Sammon in 1969. It is considered a non-linear approach as the mapping cannot be represented as a linear combination of the original variables as possible in techniques such as principal component analysis, which also makes it more difficult to use for classification applications. Denote the distance between ith and jth objects in the original space by d i j ∗ {\displaystyle \scriptstyle d_{ij}^{}} , and the distance between their projections by d i j {\displaystyle \scriptstyle d_{ij}^{}} . Sammon's mapping aims to minimize the following error function, which is often referred to as Sammon's stress or Sammon's error: E = 1 ∑ i < j d i j ∗ ∑ i < j ( d i j ∗ − d i j ) 2 d i j ∗ . {\displaystyle E={\frac {1}{\sum \limits _{i Read more →

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  Influence diagram

  An influence diagram (ID) (also called a relevance diagram, decision diagram or a decision network) is a compact graphical and mathematical representation of a decision situation. It is a generalization of a Bayesian network, in which not only probabilistic inference problems but also decision making problems (following the maximum expected utility criterion) can be modeled and solved. ID was first developed in the mid-1970s by decision analysts with an intuitive semantic that is easy to understand. It is now adopted widely and becoming an alternative to the decision tree which typically suffers from exponential growth in number of branches with each variable modeled. ID is directly applicable in team decision analysis, since it allows incomplete sharing of information among team members to be modeled and solved explicitly. Extensions of ID also find their use in game theory as an alternative representation of the game tree. == Semantics == An ID is a directed acyclic graph with three types (plus one subtype) of node and three types of arc (or arrow) between nodes. Nodes: Decision node (corresponding to each decision to be made) is drawn as a rectangle. Uncertainty node (corresponding to each uncertainty to be modeled) is drawn as an oval. Deterministic node (corresponding to special kind of uncertainty that its outcome is deterministically known whenever the outcome of some other uncertainties are also known) is drawn as a double oval. Value node (corresponding to each component of additively separable Von Neumann-Morgenstern utility function) is drawn as an octagon (or diamond). Arcs: Functional arcs (ending in value node) indicate that one of the components of additively separable utility function is a function of all the nodes at their tails. Conditional arcs (ending in uncertainty node) indicate that the uncertainty at their heads is probabilistically conditioned on all the nodes at their tails. Conditional arcs (ending in deterministic node) indicate that the uncertainty at their heads is deterministically conditioned on all the nodes at their tails. Informational arcs (ending in decision node) indicate that the decision at their heads is made with the outcome of all the nodes at their tails known beforehand. Given a properly structured ID: Decision nodes and incoming information arcs collectively state the alternatives (what can be done when the outcome of certain decisions and/or uncertainties are known beforehand) Uncertainty/deterministic nodes and incoming conditional arcs collectively model the information (what are known and their probabilistic/deterministic relationships) Value nodes and incoming functional arcs collectively quantify the preference (how things are preferred over one another). Alternative, information, and preference are termed decision basis in decision analysis, they represent three required components of any valid decision situation. Formally, the semantic of influence diagram is based on sequential construction of nodes and arcs, which implies a specification of all conditional independencies in the diagram. The specification is defined by the d {\displaystyle d} -separation criterion of Bayesian network. According to this semantic, every node is probabilistically independent on its non-successor nodes given the outcome of its immediate predecessor nodes. Likewise, a missing arc between non-value node X {\displaystyle X} and non-value node Y {\displaystyle Y} implies that there exists a set of non-value nodes Z {\displaystyle Z} , e.g., the parents of Y {\displaystyle Y} , that renders Y {\displaystyle Y} independent of X {\displaystyle X} given the outcome of the nodes in Z {\displaystyle Z} . == Example == Consider the simple influence diagram representing a situation where a decision-maker is planning their vacation. There is 1 decision node (Vacation Activity), 2 uncertainty nodes (Weather Condition, Weather Forecast), and 1 value node (Satisfaction). There are 2 functional arcs (ending in Satisfaction), 1 conditional arc (ending in Weather Forecast), and 1 informational arc (ending in Vacation Activity). Functional arcs ending in Satisfaction indicate that Satisfaction is a utility function of Weather Condition and Vacation Activity. In other words, their satisfaction can be quantified if they know what the weather is like and what their choice of activity is. (Note that they do not value Weather Forecast directly) Conditional arc ending in Weather Forecast indicates their belief that Weather Forecast and Weather Condition can be dependent. Informational arc ending in Vacation Activity indicates that they will only know Weather Forecast, not Weather Condition, when making their choice. In other words, actual weather will be known after they make their choice, and only forecast is what they can count on at this stage. It also follows semantically, for example, that Vacation Activity is independent on (irrelevant to) Weather Condition given Weather Forecast is known. == Applicability to value of information == The above example highlights the power of the influence diagram in representing an extremely important concept in decision analysis known as the value of information. Consider the following three scenarios; Scenario 1: The decision-maker could make their Vacation Activity decision while knowing what Weather Condition will be like. This corresponds to adding extra informational arc from Weather Condition to Vacation Activity in the above influence diagram. Scenario 2: The original influence diagram as shown above. Scenario 3: The decision-maker makes their decision without even knowing the Weather Forecast. This corresponds to removing informational arc from Weather Forecast to Vacation Activity in the above influence diagram. Scenario 1 is the best possible scenario for this decision situation since there is no longer any uncertainty on what they care about (Weather Condition) when making their decision. Scenario 3, however, is the worst possible scenario for this decision situation since they need to make their decision without any hint (Weather Forecast) on what they care about (Weather Condition) will turn out to be. The decision-maker is usually better off (definitely no worse off, on average) to move from scenario 3 to scenario 2 through the acquisition of new information. The most they should be willing to pay for such move is called the value of information on Weather Forecast, which is essentially the value of imperfect information on Weather Condition. The applicability of this simple ID and the value of information concept is tremendous, especially in medical decision making when most decisions have to be made with imperfect information about their patients, diseases, etc. == Related concepts == Influence diagrams are hierarchical and can be defined either in terms of their structure or in greater detail in terms of the functional and numerical relation between diagram elements. An ID that is consistently defined at all levels—structure, function, and number—is a well-defined mathematical representation and is referred to as a well-formed influence diagram (WFID). WFIDs can be evaluated using reversal and removal operations to yield answers to a large class of probabilistic, inferential, and decision questions. More recent techniques have been developed by artificial intelligence researchers concerning Bayesian network inference (belief propagation). An influence diagram having only uncertainty nodes (i.e., a Bayesian network) is also called a relevance diagram. An arc connecting node A to B implies not only that "A is relevant to B", but also that "B is relevant to A" (i.e., relevance is a symmetric relationship).

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