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Due to deep learning networks, such as convolutional neural networks, that are able to learn features by themselves, domain-specific-based feature engineering has become obsolete for vision and speech processing. Other examples of features in physics include the construction of dimensionless numbers such as Reynolds number in fluid dynamics; then Nusselt number in heat transfer; Archimedes number in sedimentation; construction of first approximations of the solution such as analytical strength of materials solutions in mechanics, etc.

It calculates "soft" weights for each word, more precisely for its embedding, in the context window. It can do it either in parallel (such as in transformers) or sequentially (such as recurrent neural networks). "Soft" weights can change during each runtime, in contrast to "hard" weights, which are (pre-)trained and fine-tuned and remain frozen afterwards. Attention was developed to address the weaknesses of recurrent neural networks, where words in a sentence are slowly processed one at a time. Recurrent neural networks favor more recent words at the end of a sentence while earlier words fade away in volatile neural activations. Attention gives all words equal access to any part of a sentence in a faster parallel scheme and no longer suffers the wait time of serial processing. Earlier uses attached this mechanism to a serial recurrent neural network's language translation system (below), but later uses in Transformers large language models removed the recurrent neural network and relied heavily on the faster parallel attention scheme.

It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any m n {\displaystyle \ m\times n\ } matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an m n {\displaystyle \ m\times n\ } complex matrix M is a factorization of the form M = U V , {\displaystyle \ \mathbf {M} =\mathbf {U\Sigma V^{*}} \ ,} where U is an m m {\displaystyle \ m\times m\ } complex unitary matrix, {\displaystyle \ \mathbf {\Sigma } \ } is an m n {\displaystyle \ m\times n\ } rectangular diagonal matrix with non-negative real numbers on the diagonal, V is an n n {\displaystyle n\times n} complex unitary matrix, and V {\displaystyle \ \mathbf {V^{*}} \ } is the conjugate transpose of V. Such decomposition always exists for any complex matrix. If M is real, then U and V can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted U V T {\displaystyle \ \mathbf {U\Sigma V} ^{\operatorname {T} }} . The diagonal entries i = i i {\displaystyle \ \sigma _{i}=\Sigma _{ii}\ } of {\displaystyle \ \mathbf {\Sigma } \ } are uniquely determined by M and are known as the singular values of M. The number of non-zero singular values is equal to the rank of M. The columns of U and the columns of V are called left-singular vectors and right-singular vectors of M, respectively. They form two sets of orthonormal bases u1, ..., um and v1, ..., vn , and if they are sorted so that the singular values i {\displaystyle \ \sigma _{i}\ } with value zero are all in the highest-numbered columns (or rows), the singular value decomposition can be written as M = i = 1 r i u i v i , {\displaystyle \ \mathbf {M} =\sum _{i=1}^{r}\sigma _{i}\mathbf {u} _{i}\mathbf {v} _{i}^{*}\ ,} where r min { m , n } {\displaystyle \ r\leq \min\{m,n\}\ } is the rank of M. The SVD is not unique. It is always possible to choose the decomposition so that the singular values i i {\displaystyle \Sigma _{ii}} are in descending order. In this case, {\displaystyle \mathbf {\Sigma } } (but not U and V) is uniquely determined by M. The term sometimes refers to the compact SVD, a similar decomposition M = U V {\displaystyle \ \mathbf {M} =\mathbf {U\Sigma V^{*}} \ } in which {\displaystyle \ \mathbf {\Sigma } \ } is square diagonal of size r r {\displaystyle r\times r} , where r min { m , n } {\displaystyle \ r\leq \min\{m,n\}\ } is the rank of M, and has only the non-zero singular values. In this variant, U is an m r {\displaystyle m\times r} semi-unitary matrix and V {\displaystyle \ \mathbf {V} \ } is an n r {\displaystyle n\times r} semi-unitary matrix, such that U U = V V = I r . {\displaystyle \ \mathbf {U^{*}U} =\mathbf {V^{*}V} =\mathbf {I} _{r}\ .} Mathematical applications of the SVD include computing the pseudoinverse, matrix approximation, and determining the rank, range, and null space of a matrix. The SVD is also extremely useful in all areas of science, engineering, and statistics, such as signal processing, least squares fitting of data, and process control.

It is a generalization of the logistic function to multiple dimensions, and used in multinomial logistic regression. The softmax function is often used as the last activation function of a neural network to normalize the output of a network to a probability distribution over predicted output classes, based on Luce's choice axiom.

A survey from May 2020 exposes the fact that practitioners report a dire need for better protecting machine learning systems in industrial applications.Most machine learning techniques are mostly designed to work on specific problem sets, under the assumption that the training and test data are generated from the same statistical distribution (IID). However, this assumption is often dangerously violated in practical high-stake applications, where users may intentionally supply fabricated data that violates the statistical assumption. Some of the most common attacks in adversarial machine learning include evasion attacks, data poisoning attacks, Byzantine attacks and model extraction.

It can be used in conjunction with many other types of learning algorithms to improve performance. The output of the other learning algorithms ('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 on the real line.AdaBoost is adaptive in the sense that subsequent weak learners are tweaked in favor of those instances misclassified by previous classifiers. In some problems it can be less susceptible to the overfitting problem 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 that it can also effectively combine strong base learners (such as deep 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.

Nontrivial problems can be solved only using a nonlinear activation function. Modern activation functions include the smooth version of the ReLU, the GELU, which was used in the 2018 BERT model, the logistic (sigmoid) function used in the 2012 speech recognition model developed by Hinton et al, the ReLU used in the 2012 AlexNet computer vision model and in the 2015 ResNet model.

Each connection, like the synapses in a biological brain, can transmit a signal to other neurons. An artificial neuron receives signals then processes them and can signal neurons connected to it. The "signal" at a connection is a real number, and the output of each neuron is computed by some non-linear function of the sum of its inputs. The connections are called edges. Neurons and edges typically have a weight that adjusts as learning proceeds. The weight increases or decreases the strength of the signal at a connection. Neurons may have a threshold such that a signal is sent only if the aggregate signal crosses that threshold. Typically, neurons are aggregated into layers. Different layers may perform different transformations on their inputs. Signals travel from the first layer (the input layer), to the last layer (the output layer), possibly after traversing the layers multiple times.

This can result in improved learning efficiency and prediction accuracy for the task-specific models, when compared to training the models separately. Early versions of MTL were called "hints". In a widely cited 1997 paper, Rich Caruana gave the following characterization:Multitask Learning is an approach to inductive transfer that improves generalization by using the domain information contained in the training signals of related tasks as an inductive bias. It does this by learning tasks in parallel while using a shared representation; what is learned for each task can help other tasks be learned better. In the classification context, MTL aims to improve the performance of multiple classification tasks by learning them jointly. One example is a spam-filter, which can be treated as distinct but related classification tasks across different users. To make this more concrete, consider that different people have different distributions of features which distinguish spam emails from legitimate ones, for example an English speaker may find that all emails in Russian are spam, not so for Russian speakers. Yet there is a definite commonality in this classification task across users, for example one common feature might be text related to money transfer. Solving each user's spam classification problem jointly via MTL can let the solutions inform each other and improve performance. Further examples of settings for MTL include multiclass classification and multi-label classification.Multi-task learning works because regularization induced by requiring an algorithm to perform well on a related task can be superior to regularization that prevents overfitting by penalizing all complexity uniformly. One situation where MTL may be particularly helpful is if the tasks share significant commonalities and are generally slightly under sampled. However, as discussed below, MTL has also been shown to be beneficial for learning unrelated tasks.

An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cram r in the 1920s. In optimal control, the loss is the penalty for failing to achieve a desired value. In financial risk management, the function is mapped to a monetary loss.

The goal of segmentation is to simplify and/or change the representation of an image into something that is more meaningful and easier to analyze. Image segmentation is typically used to locate objects and boundaries (lines, curves, etc.) in images. More precisely, image segmentation is the process of assigning a label to every pixel in an image such that pixels with the same label share certain characteristics. The result of image segmentation is a set of segments that collectively cover the entire image, or a set of contours extracted from the image (see edge detection). Each of the pixels in a region are similar with respect to some characteristic or computed property , such as color, intensity, or texture. Adjacent regions are significantly different color respect to the same characteristic(s). When applied to a stack of images, typical in medical imaging, the resulting contours after image segmentation can be used to create 3D reconstructions with the help of geometry reconstruction algorithms like marching cubes.

The first object in this relation is said to refer to the second object. It is called a name for the second object. The next object, the one to which the first object refers, is called the referent of the first object. A name is usually a phrase or expression, or some other symbolic representation. Its referent may be anything a material object, a person, an event, an activity, or an abstract concept. References can take on many forms, including: a thought, a sensory perception that is audible (onomatopoeia), visual (text), olfactory, or tactile, emotional state, relationship with other, spacetime coordinates, symbolic or alpha-numeric, a physical object, or an energy projection. In some cases, methods are used that intentionally hide the reference from some observers, as in cryptography.Reference feature in many spheres of human activity and knowledge, and the term adopts shades of meaning particular to the contexts in which it is used. Some of them are described in the sections below.

The training data is processed, building a function that maps new data on expected output values. An optimal scenario will allow for the algorithm to correctly determine output values for unseen instances. This requires the learning algorithm to generalize from the training data to unseen situations in a "reasonable" way (see inductive bias). This statistical quality of an algorithm is measured through the so-called generalization error.

A survey from May 2020 exposes the fact that practitioners report a dire need for better protecting machine learning systems in industrial applications.Most machine learning techniques are mostly designed to work on specific problem sets, under the assumption that the training and test data are generated from the same statistical distribution (IID). However, this assumption is often dangerously violated in practical high-stake applications, where users may intentionally supply fabricated data that violates the statistical assumption. Some of the most common attacks in adversarial machine learning include evasion attacks, data poisoning attacks, Byzantine attacks and model extraction.

Cross-validation is a resampling method that uses different portions of the data to test and train a model on different iterations. It is mainly used in settings where the goal is prediction, and one wants to estimate how accurately a predictive model will perform in practice. In a prediction problem, a model is usually given a dataset of known data on which training is run (training dataset), and a dataset of unknown data (or first seen data) against which the model is tested (called the validation dataset or testing set). The goal of cross-validation is to test the model's ability to predict new data that was not used in estimating it, in order to flag problems like overfitting or selection bias and to give an insight on how the model will generalize to an independent dataset (i.e., an unknown dataset, for instance from a real problem). One round of cross-validation involves partitioning a sample of data into complementary subsets, performing the analysis on one subset (called the training set), and validating the analysis on the other subset (called the validation set or testing set). To reduce variability, in most methods multiple rounds of cross-validation are performed using different partitions, and the validation results are combined (e.g. averaged) over the rounds to give an estimate of the model's predictive performance. In summary, cross-validation combines (averages) measures of fitness in prediction to derive a more accurate estimate of model prediction performance.

Vanishing gradients and exploding gradients, seen during backpropagation in earlier neural networks, are prevented by using regularized weights over fewer connections. For example, for each neuron in the fully-connected layer 10,000 weights would be required for processing an image sized 100 100 pixels. However, applying cascaded convolution (or cross-correlation) kernels, only 25 neurons are required to process 5x5-sized tiles. Higher-layer features are extracted from wider context windows, compared to lower-layer features. They have applications in: image and video recognition, recommender systems, image classification, image segmentation, medical image analysis, natural language processing, brain computer interfaces, and financial time series.CNNs are also known as Shift Invariant or Space Invariant Artificial Neural Networks (SIANN), based on the shared-weight architecture of the convolution kernels or filters that slide along input features and provide translation-equivariant responses known as feature maps. Counter-intuitively, most convolutional neural networks are not invariant to translation, due to the downsampling operation they apply to the input.Feed-forward neural networks are usually fully connected networks, that is, each neuron in one layer is connected to all neurons in the next layer. The "full connectivity" of these networks make them prone to overfitting data. Typical ways of regularization, or preventing overfitting, include: penalizing parameters during training (such as weight decay) or trimming connectivity (skipped connections, dropout, etc.) Robust datasets also increases the probability that CNNs will learn the generalized principles that characterize a given dataset rather than the biases of a poorly-populated set.Convolutional networks were inspired by biological processes in that the connectivity pattern between neurons resembles the organization of the animal visual cortex. Individual cortical neurons respond to stimuli only in a restricted region of the visual field known as the receptive field. The receptive fields of different neurons partially overlap such that they cover the entire visual field. CNNs use relatively little pre-processing compared to other image classification algorithms. This means that the network learns to optimize the filters (or kernels) through automated learning, whereas in traditional algorithms these filters are hand-engineered. This independence from prior knowledge and human intervention in feature extraction is a major advantage.

In general, as we increase the number of tunable parameters in a model, it becomes more flexible, and can better fit a training data set. It is said to have lower error, or bias. However, for more flexible models, there will tend to be greater variance to the model fit each time we take a set of samples to create a new training data set. It is said that there is greater variance in the model's estimated parameters. The bias variance dilemma or bias variance problem is the conflict in trying to simultaneously minimize these two sources of error that prevent supervised learning algorithms from generalizing beyond their training set: The bias error is an error from erroneous assumptions in the learning algorithm. High bias can cause an algorithm to miss the relevant relations between features and target outputs (underfitting). The variance is an error from sensitivity to small fluctuations in the training set. High variance may result from an algorithm modeling the random noise in the training data (overfitting).The bias variance decomposition is a way of analyzing a learning algorithm's expected generalization error with respect to a particular problem as a sum of three terms, the bias, variance, and a quantity called the irreducible error, resulting from noise in the problem itself.

It is characterized by using a combination of a small amount of human-labeled data (exclusively used in more expensive and time-consuming supervised learning paradigm), followed by a large amount of unlabeled data (used exclusively in unsupervised learning paradigm). In other words, the desired output values are provided only for a subset of the training data. The remaining data is unlabeled or imprecisely labeled. Intuitively, it can be seen as an exam and labeled data as sample problems that the teacher solves for the class as an aid in solving another set of problems. In the transductive setting, these unsolved problems act as exam questions. In the inductive setting, they become practice problems of the sort that will make up the exam. Technically, it could be viewed as performing clustering and then labeling the clusters with the labeled data, pushing the decision boundary away from high-density regions, or learning an underlying one-dimensional manifold where the data reside.

A hyperparameter is a parameter whose value is used to control the learning process. By contrast, the values of other parameters (typically node weights) are learned. The same kind of machine learning model can require different constraints, weights or learning rates to generalize different data patterns. These measures are called hyperparameters, and have to be tuned so that the model can optimally solve the machine learning problem. Hyperparameter optimization finds a tuple of hyperparameters that yields an optimal model which minimizes a predefined loss function on given independent data. The objective function takes a tuple of hyperparameters and returns the associated loss. Cross-validation is often used to estimate this generalization performance, and therefore choose the set of values for hyperparameters that maximize it.

For example, for image classification, knowledge gained while learning to recognize cars could be applied when trying to recognize trucks. This topic is related to the psychological literature on transfer of learning, although practical ties between the two fields are limited. Reusing/transferring information from previously learned tasks to new tasks has the potential to significantly improve learning efficiency.

Unlike a statistical ensemble in statistical mechanics, which is usually infinite, a machine learning ensemble consists of only a concrete finite set of alternative models, but typically allows for much more flexible structure to exist among those alternatives.

The adjective "deep" in deep learning refers to the use of multiple layers in the network. Methods used can be either supervised, semi-supervised or unsupervised.Deep-learning architectures such as deep neural networks, deep belief networks, deep reinforcement learning, recurrent neural networks, convolutional neural networks and transformers have been applied to fields including computer vision, speech recognition, natural language processing, machine translation, bioinformatics, drug design, medical image analysis, climate science, material inspection and board game programs, where they have produced results comparable to and in some cases surpassing human expert performance.Artificial neural networks (ANNs) were inspired by information processing and distributed communication nodes in biological systems. ANNs have various differences from biological brains. Specifically, artificial neural networks tend to be static and symbolic, while the biological brain of most living organisms is dynamic (plastic) and analog.

Because computation is postponed until a new instance is observed, these algorithms are sometimes referred to as "lazy."It is called instance-based because it constructs hypotheses directly from the training instances themselves. This means that the hypothesis complexity can grow with the data: in the worst case, a hypothesis is a list of n training items and the computational complexity of classifying a single new instance is O(n). One advantage that instance-based learning has over other methods of machine learning is its ability to adapt its model to previously unseen data. Instance-based learners may simply store a new instance or throw an old instance away. Examples of instance-based learning algorithms are the k-nearest neighbors algorithm, kernel machines and RBF networks.: ch. 8 These store (a subset of) their training set; when predicting a value/class for a new instance, they compute distances or similarities between this instance and the training instances to make a decision. To battle the memory complexity of storing all training instances, as well as the risk of overfitting to noise in the training set, instance reduction algorithms have been proposed.

In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , which is the West Semitic word for "ox". Letters that arose from alpha include the Latin letter A and the Cyrillic letter .

In such methods, during each iteration of training each of the neural networks weights receives an update proportional to the partial derivative of the error function with respect to the current weight. The problem is that in some cases, the gradient will be vanishingly small, effectively preventing the weight from changing its value. In the worst case, this may completely stop the neural network from further training. As one example of the problem cause, traditional activation functions such as the hyperbolic tangent function have gradients in the range (0,1], and backpropagation computes gradients by the chain rule. This has the effect of multiplying n of these small numbers to compute gradients of the early layers in an n-layer network, meaning that the gradient (error signal) decreases exponentially with n while the early layers train very slowly. Back-propagation allowed researchers to train supervised deep artificial neural networks from scratch, initially with little success. Hochreiter's diplom thesis of 1991 formally identified the reason for this failure in the "vanishing gradient problem", which not only affects many-layered feedforward networks, but also recurrent networks. The latter are trained by unfolding them into very deep feedforward networks, where a new layer is created for each time step of an input sequence processed by the network. (The combination of unfolding and backpropagation is termed backpropagation through time.) When activation functions are used whose derivatives can take on larger values, one risks encountering the related exploding gradient problem.

Developed at AT&T Bell Laboratories by Vladimir Vapnik with colleagues (Boser et al., 1992, Guyon et al., 1993, Cortes and Vapnik, 1995, Vapnik et al., 1997) SVMs are one of the most robust prediction methods, being based on statistical learning frameworks or VC theory proposed by Vapnik (1982, 1995) and Chervonenkis (1974). Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that assigns new examples to one category or the other, making it a non-probabilistic binary linear classifier (although methods such as Platt scaling exist to use SVM in a probabilistic classification setting). SVM maps training examples to points in space so as to maximise the width of the gap between the two categories. New examples are then mapped into that same space and predicted to belong to a category based on which side of the gap they fall. In addition to performing linear classification, SVMs can efficiently perform a non-linear classification using what is called the kernel trick, implicitly mapping their inputs into high-dimensional feature spaces. The support vector clustering algorithm, created by Hava Siegelmann and Vladimir Vapnik, applies the statistics of support vectors, developed in the support vector machines algorithm, to categorize unlabeled data. These data sets require unsupervised learning approaches, which attempt to find natural clustering of the data to groups and, then, to map new data according to these clusters.

An EA uses mechanisms inspired by biological evolution, such as reproduction, mutation, recombination, and selection. Candidate solutions to the optimization problem play the role of individuals in a population, and the fitness function determines the quality of the solutions (see also loss function). Evolution of the population then takes place after the repeated application of the above operators. Evolutionary algorithms often perform well approximating solutions to all types of problems because they ideally do not make any assumption about the underlying fitness landscape. Techniques from evolutionary algorithms applied to the modeling of biological evolution are generally limited to explorations of microevolutionary processes and planning models based upon cellular processes. In most real applications of EAs, computational complexity is a prohibiting factor. In fact, this computational complexity is due to fitness function evaluation. Fitness approximation is one of the solutions to overcome this difficulty. However, seemingly simple EA can solve often complex problems; therefore, there may be no direct link between algorithm complexity and problem complexity.

Since it influences to what extent newly acquired information overrides old information, it metaphorically represents the speed at which a machine learning model "learns". In the adaptive control literature, the learning rate is commonly referred to as gain.In setting a learning rate, there is a trade-off between the rate of convergence and overshooting. While the descent direction is usually determined from the gradient of the loss function, the learning rate determines how big a step is taken in that direction. A too high learning rate will make the learning jump over minima but a too low learning rate will either take too long to converge or get stuck in an undesirable local minimum.In order to achieve faster convergence, prevent oscillations and getting stuck in undesirable local minima the learning rate is often varied during training either in accordance to a learning rate schedule or by using an adaptive learning rate. The learning rate and its adjustments may also differ per parameter, in which case it is a diagonal matrix that can be interpreted as an approximation to the inverse of the Hessian matrix in Newton's method. The learning rate is related to the step length determined by inexact line search in quasi-Newton methods and related optimization algorithms.

It is used in natural language processing and information retrieval (IR). It disregards grammar and word order but keeps multiplicity. The bag-of-words model has also been used for computer vision.The bag-of-words model is commonly used in methods of document classification where the (frequency of) occurrence of each word is used as a feature for training a classifier.An early reference to "bag of words" in a linguistic context can be found in Zellig Harris's 1954 article on Distributional Structure.The Bag-of-words model is one example of a Vector space model.

Precision (also called positive predictive value) is the fraction of relevant instances among the retrieved instances. Written as a formula: r e l e v a n t _ r e t r i e v e d _ i n s t a n c e s a l l _ r e t r i e v e d _ i n s t a n c e s {\displaystyle {\frac {relevant\_retrieved\_instances}{all\_{\mathbf {retrieved}}\_instances}}} . Recall (also known as sensitivity) is the fraction of relevant instances that were retrieved. Written as a formula: r e l e v a n t _ r e t r i e v e d _ i n s t a n c e s a l l _ r e l e v a n t _ i n s t a n c e s {\displaystyle {\frac {relevant\_retrieved\_instances}{all\_{\mathbf {relevant}}\_instances}}} . Both precision and recall are therefore based on relevance. Consider a computer program for recognizing dogs (the relevant element) in a digital photograph. Upon processing a picture which contains ten cats and twelve dogs, the program identifies eight dogs. Of the eight elements identified as dogs, only five actually are dogs (true positives), while the other three are cats (false positives). Seven dogs were missed (false negatives), and seven cats were correctly excluded (true negatives). The program's precision is then 5/8 (true positives / selected elements) while its recall is 5/12 (true positives / relevant elements). Adopting a hypothesis-testing approach from statistics, in which, in this case, the null hypothesis is that a given item is irrelevant (i.e., not a dog), absence of type I and type II errors (i.e., perfect specificity and sensitivity of 100% each) corresponds respectively to perfect precision (no false positive) and perfect recall (no false negative). More generally, recall is simply the complement of the type II error rate (i.e., one minus the type II error rate). Precision is related to the type I error rate, but in a slightly more complicated way, as it also depends upon the prior distribution of seeing a relevant vs. an irrelevant item. The above cat and dog example contained 8 5 = 3 type I errors (false positives) out of 10 total cats (true negatives), for a type I error rate of 3/10, and 12 5 = 7 type II errors (false negatives), for a type II error rate of 7/12. Precision can be seen as a measure of quality, and recall as a measure of quantity. Higher precision means that an algorithm returns more relevant results than irrelevant ones, and high recall means that an algorithm returns most of the relevant results (whether or not irrelevant ones are also returned).

Zero-shot methods generally work by associating observed and non-observed classes through some form of auxiliary information, which encodes observable distinguishing properties of objects. For example, given a set of images of animals to be classified, along with auxiliary textual descriptions of what animals look like, an artificial intelligence model which has been trained to recognize horses, but has never been given a zebra, can still recognize a zebra when it also knows that zebras look like striped horses. This problem is widely studied in computer vision, natural language processing, and machine perception.

An autoencoder learns two functions: an encoding function that transforms the input data, and a decoding function that recreates the input data from the encoded representation. The autoencoder learns an efficient representation (encoding) for a set of data, typically for dimensionality reduction. Variants exist, aiming to force the learned representations to assume useful properties. Examples are regularized autoencoders (Sparse, Denoising and Contractive), which are effective in learning representations for subsequent classification tasks, and Variational autoencoders, with applications as generative models. Autoencoders are applied to many problems, including facial recognition, feature detection, anomaly detection and acquiring the meaning of words. Autoencoders are also generative models which can randomly generate new data that is similar to the input data (training data).

In a multi-layered network, backpropagation uses the following steps: Propagate training data through the model from input to predicted output by computing the successive hidden layers' outputs and finally the final layer's output (the feedforward step). Adjust the model weights to reduce the error relative to the weights. The error is typically the squared difference between prediction and target. For each weight, the slope or derivative of the error is found, and the weight adjusted by a negative multiple of this derivative, so as to go downslope toward the minimum-error configuration. This derivative is easy to calculate for final layer weights, and possible to calculate for one layer given the next layer's derivatives. Starting at the end, then, the derivatives are calculated layer by layer toward the beginning -- thus "backpropagation". Repeatedly update the weights until they converge or the model has undergone enough iterations.It is an efficient application of the Leibniz chain rule (1673) to such networks. It is also known as the reverse mode of automatic differentiation or reverse accumulation, due to Seppo Linnainmaa (1970). The term "back-propagating error correction" was introduced in 1962 by Frank Rosenblatt, but he did not know how to implement this, even though Henry J. Kelley had a continuous precursor of backpropagation already in 1960 in the context of control theory.Backpropagation computes the gradient of a loss function with respect to the weights of the network for a single input output example, and does so efficiently, computing the gradient one layer at a time, iterating backward from the last layer to avoid redundant calculations of intermediate terms in the chain rule; this can be derived through dynamic programming. Gradient descent, or variants such as stochastic gradient descent, are commonly used. Strictly the term backpropagation refers only to the algorithm for computing the gradient, not how the gradient is used; but the term is often used loosely to refer to the entire learning algorithm including how the gradient is used, such as by stochastic gradient descent. In 1986 David E. Rumelhart et al. published an experimental analysis of the technique. This contributed to the popularization of backpropagation and helped to initiate an active period of research in multilayer perceptrons.

Reinforcement learning is one of three basic machine learning paradigms, alongside supervised learning and unsupervised learning. Reinforcement learning differs from supervised learning in not needing labelled input/output pairs to be presented, and in not needing sub-optimal actions to be explicitly corrected. Instead the focus is on finding a balance between exploration (of uncharted territory) and exploitation (of current knowledge). The environment is typically stated in the form of a Markov decision process (MDP), because many reinforcement learning algorithms for this context use dynamic programming techniques. The main difference between the classical dynamic programming methods and reinforcement learning algorithms is that the latter do not assume knowledge of an exact mathematical model of the Markov decision process and they target large Markov decision processes where exact methods become infeasible.

Scheyvens, Griffin, Jocoy, Liu, & Bradford (2008) further noted that by utilizing learning strategies that can include small-group work, role-play and simulations, data collection and analysis, active learning is purported to increase student interest and motivation and to build students critical thinking, problem-solving and social skills . In a report from the Association for the Study of Higher Education, authors discuss a variety of methodologies for promoting active learning. They cite literature that indicates students must do more than just listen in order to learn. They must read, write, discuss, and be engaged in solving problems. This process relates to the three learning domains referred to as knowledge, skills and attitudes (KSA). This taxonomy of learning behaviors can be thought of as "the goals of the learning process." In particular, students must engage in such higher-order thinking tasks as analysis, synthesis, and evaluation.

Its flow is uni-directional, meaning that the information in the model flows in only one direction forward from the input nodes, through the hidden nodes (if any) and to the output nodes, without any cycles or loops, in contrast to recurrent neural networks, which have a bi-directional flow. Modern feedforward networks are trained using the backpropagation method and are colloquially referred to as the "vanilla" neural networks.

The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent. It is particularly useful in machine learning for minimizing the cost or loss function. Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades.A simple extension of gradient descent, stochastic gradient descent, serves as the most basic algorithm used for training most deep networks today.

It is usually employed to optimize expensive-to-evaluate functions.

Information printed directly on a container or article can also be considered labelling. Labels have many uses, including promotion and providing information on a product's origin, the manufacturer (e.g., brand name), use, safety, shelf-life and disposal, some or all of which may be governed by legislation such as that for food in the UK or United States. Methods of production and attachment to packaging are many and various and may also be subject to internationally recognised standards. In many countries, hazardous products such as poisons or flammable liquids must have a warning label.

The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models. Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis. Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications. This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine. Linear regression has many practical uses. Most applications fall into one of the following two broad categories: If the goal is error reduction in prediction or forecasting, linear regression can be used to fit a predictive model to an observed data set of values of the response and explanatory variables. After developing such a model, if additional values of the explanatory variables are collected without an accompanying response value, the fitted model can be used to make a prediction of the response. If the goal is to explain variation in the response variable that can be attributed to variation in the explanatory variables, linear regression analysis can be applied to quantify the strength of the relationship between the response and the explanatory variables, and in particular to determine whether some explanatory variables may have no linear relationship with the response at all, or to identify which subsets of explanatory variables may contain redundant information about the response.Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the "lack of fit" in some other norm (as with least absolute deviations regression), or by minimizing a penalized version of the least squares cost function as in ridge regression (L2-norm penalty) and lasso (L1-norm penalty). Conversely, the least squares approach can be used to fit models that are not linear models. Thus, although the terms "least squares" and "linear model" are closely linked, they are not synonymous.

An overfitted model is a mathematical model that contains more parameters than can be justified by the data. In a mathematical sense, these parameters represent the degree of a polynomial. The essence of overfitting is to have unknowingly extracted some of the residual variation (i.e., the noise) as if that variation represented underlying model structure. Underfitting occurs when a mathematical model cannot adequately capture the underlying structure of the data. An under-fitted model is a model where some parameters or terms that would appear in a correctly specified model are missing. Under-fitting would occur, for example, when fitting a linear model to non-linear data. Such a model will tend to have poor predictive performance. The possibility of over-fitting exists because the criterion used for selecting the model is not the same as the criterion used to judge the suitability of a model. For example, a model might be selected by maximizing its performance on some set of training data, and yet its suitability might be determined by its ability to perform well on unseen data; then over-fitting occurs when a model begins to "memorize" training data rather than "learning" to generalize from a trend. As an extreme example, if the number of parameters is the same as or greater than the number of observations, then a model can perfectly predict the training data simply by memorizing the data in its entirety. (For an illustration, see Figure 2.) Such a model, though, will typically fail severely when making predictions. The potential for overfitting depends not only on the number of parameters and data but also the conformability of the model structure with the data shape, and the magnitude of model error compared to the expected level of noise or error in the data. Even when the fitted model does not have an excessive number of parameters, it is to be expected that the fitted relationship will appear to perform less well on a new data set than on the data set used for fitting (a phenomenon sometimes known as shrinkage). In particular, the value of the coefficient of determination will shrink relative to the original data. To lessen the chance or amount of overfitting, several techniques are available (e.g., model comparison, cross-validation, regularization, early stopping, pruning, Bayesian priors, or dropout). The basis of some techniques is either (1) to explicitly penalize overly complex models or (2) to test the model's ability to generalize by evaluating its performance on a set of data not used for training, which is assumed to approximate the typical unseen data that a model will encounter.

Kingma and Max Welling. It is part of the families of probabilistic graphical models and variational Bayesian methods.Variational autoencoders are often associated with the autoencoder model because of its architectural affinity, but with significant differences in the goal and mathematical formulation. Variational autoencoders are probabilistic generative models that require neural networks as only a part of their overall structure. The neural network components are typically referred to as the encoder and decoder for the first and second component respectively. The first neural network maps the input variable to a latent space that corresponds to the parameters of a variational distribution. In this way, the encoder can produce multiple different samples that all come from the same distribution. The decoder has the opposite function, which is to map from the latent space to the input space, in order to produce or generate data points. Both networks are typically trained together with the usage of the reparameterization trick, although the variance of the noise model can be learned separately. Although this type of model was initially designed for unsupervised learning, its effectiveness has been proven for semi-supervised learning and supervised learning.

When some object X {\displaystyle X} is said to be embedded in another object Y {\displaystyle Y} , the embedding is given by some injective and structure-preserving map f : X Y {\displaystyle f:X\rightarrow Y} . The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X {\displaystyle X} and Y {\displaystyle Y} are instances. In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map f : X Y {\displaystyle f:X\rightarrow Y} is an embedding is often indicated by the use of a "hooked arrow" (U+21AA RIGHTWARDS ARROW WITH HOOK); thus: f : X Y . {\displaystyle f:X\hookrightarrow Y.} (On the other hand, this notation is sometimes reserved for inclusion maps.) Given X {\displaystyle X} and Y {\displaystyle Y} , several different embeddings of X {\displaystyle X} in Y {\displaystyle Y} may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational numbers in the real numbers, and the real numbers in the complex numbers. In such cases it is common to identify the domain X {\displaystyle X} with its image f ( X ) {\displaystyle f(X)} contained in Y {\displaystyle Y} , so that X Y {\displaystyle X\subseteq Y} .

It is also the field of study in computer science that develops and studies intelligent machines. "AI" may also refer to the machines themselves. AI technology is widely used throughout industry, government and science. Some high-profile applications are: advanced web search engines (e.g., Google Search), recommendation systems (used by YouTube, Amazon, and Netflix), understanding human speech (such as Siri and Alexa), self-driving cars (e.g., Waymo), generative or creative tools (ChatGPT and AI art), and competing at the highest level in strategic games (such as chess and Go).Artificial intelligence was founded as an academic discipline in 1956. The field went through multiple cycles of optimism followed by disappointment and loss of funding, but after 2012, when deep learning surpassed all previous AI techniques, there was a vast increase in funding and interest. The various sub-fields of AI research are centered around particular goals and the use of particular tools. The traditional goals of AI research include reasoning, knowledge representation, planning, learning, natural language processing, perception, and support for robotics. General intelligence (the ability to solve an arbitrary problem) is among the field's long-term goals. To solve these problems, AI researchers have adapted and integrated a wide range of problem-solving techniques, including search and mathematical optimization, formal logic, artificial neural networks, and methods based on statistics, operations research, and economics. AI also draws upon psychology, linguistics, philosophy, neuroscience and many other fields.

It is used for classification and regression. In both cases, the input consists of the k closest training examples in a data set. The output depends on whether k-NN is used for classification or regression: In k-NN classification, the output is a class membership. An object is classified by a plurality vote of its neighbors, with the object being assigned to the class most common among its k nearest neighbors (k is a positive integer, typically small). If k = 1, then the object is simply assigned to the class of that single nearest neighbor. In k-NN regression, the output is the property value for the object. This value is the average of the values of k nearest neighbors. If k = 1, then the output is simply assigned to the value of that single nearest neighbor.k-NN is a type of classification where the function is only approximated locally and all computation is deferred until function evaluation. Since this algorithm relies on distance for classification, if the features represent different physical units or come in vastly different scales then normalizing the training data can improve its accuracy dramatically.Both for classification and regression, a useful technique can be to assign weights to the contributions of the neighbors, so that the nearer neighbors contribute more to the average than the more distant ones. For example, a common weighting scheme consists in giving each neighbor a weight of 1/d, where d is the distance to the neighbor.The neighbors are taken from a set of objects for which the class (for k-NN classification) or the object property value (for k-NN regression) is known. This can be thought of as the training set for the algorithm, though no explicit training step is required. A peculiarity of the k-NN algorithm is that it is sensitive to the local structure of the data.

The approach is an attempt to more closely mimic biological neural organization.The idea is to add structures called "capsules" to a convolutional neural network (CNN), and to reuse output from several of those capsules to form more stable (with respect to various perturbations) representations for higher capsules. The output is a vector consisting of the probability of an observation, and a pose for that observation. This vector is similar to what is done for example when doing classification with localization in CNNs. Among other benefits, capsnets address the "Picasso problem" in image recognition: images that have all the right parts but that are not in the correct spatial relationship (e.g., in a "face", the positions of the mouth and one eye are switched). For image recognition, capsnets exploit the fact that while viewpoint changes have nonlinear effects at the pixel level, they have linear effects at the part/object level. This can be compared to inverting the rendering of an object of multiple parts.

This approach stands in contrast to traditional centralized machine learning techniques where local datasets are merged into one training session, as well as to approaches that assume that local data samples are identically distributed. Federated learning enables multiple actors to build a common, robust machine learning model without sharing data, thus addressing critical issues such as data privacy, data security, data access rights and access to heterogeneous data. Its applications engage industries including defense, telecommunications, Internet of Things, and pharmaceuticals. A major open question is when/whether federated learning is preferable to pooled data learning. Another open question concerns the trustworthiness of the devices and the impact of malicious actors on the learned model.

Formally, PCA is a statistical technique for reducing the dimensionality of a dataset. This is accomplished by linearly transforming the data into a new coordinate system where (most of) the variation in the data can be described with fewer dimensions than the initial data. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points. Principal component analysis has applications in many fields such as population genetics, microbiome studies, and atmospheric science.The principal components of a collection of points in a real coordinate space are a sequence of p {\displaystyle p} unit vectors, where the i {\displaystyle i} -th vector is the direction of a line that best fits the data while being orthogonal to the first i 1 {\displaystyle i-1} vectors. Here, a best-fitting line is defined as one that minimizes the average squared perpendicular distance from the points to the line. These directions constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Principal component analysis is the process of computing the principal components and using them to perform a change of basis on the data, sometimes using only the first few principal components and ignoring the rest. In data analysis, the first principal component of a set of p {\displaystyle p} variables, presumed to be jointly normally distributed, is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through p {\displaystyle p} iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an independent set. PCA is used in exploratory data analysis and for making predictive models. It is commonly used for dimensionality reduction by projecting each data point onto only the first few principal components to obtain lower-dimensional data while preserving as much of the data's variation as possible. The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The i {\displaystyle i} -th principal component can be taken as a direction orthogonal to the first i 1 {\displaystyle i-1} principal components that maximizes the variance of the projected data. For either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain-specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset. Robust and L1-norm-based variants of standard PCA have also been proposed.

Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. A time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements. Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as to test relationships between one or more different time series, this type of analysis is not usually called "time series analysis", which refers in particular to relationships between different points in time within a single series. Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see time reversibility). Time series analysis can be applied to real-valued, continuous data, discrete numeric data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the English language).

They are an efficient way of performing model averaging with neural networks.Dilution refers to thinning weights,while dropout refers to randomly 'dropping out', or omitting, units (both hidden and visible) during the training process of a neural network.Both trigger the same type of regularization.

It is a main task of exploratory data analysis, and a common technique for statistical data analysis, used in many fields, including pattern recognition, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning.

It is one way to display an algorithm that only contains conditional control statements. Decision trees are commonly used in operations research, specifically in decision analysis,to help identify a strategy most likely to reach a goal, but are also a popular tool in machine learning.

The most common form of regression analysis is linear regression, in which one finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion.

It uses routines, often called 'validation rules', 'validation constraints', or 'check routines', that check for correctness, meaningfulness, and security of data that are input to the system. The rules may be implemented through the automated facilities of a data dictionary, or by the inclusion of explicit application program validation logic of the computer and its application. This is distinct from formal verification, which attempts to prove or disprove the correctness of algorithms for implementing a specification or property.

Statisticians attempt to collect samples that are representative of the population. Sampling has lower costs and faster data collection compared to recording data from the entire population, and thus, it can provide insights in cases where it is infeasible to measure an entire population.

The phrase 'garbage in, garbage out' is particularly applicable to data mining and machine learning projects. Data collection methods are often loosely controlled, resulting in out-of-range values, impossible data combinations, and missing values, amongst other issues.