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European Conference on Computer Vision : The European Conference on Computer Vision (ECCV) is a biennial research conference with the proceedings published by Springer Science+Business Media. Similar to ICCV in scope and quality, it is held those years which ICCV is not. It is considered to be one of the top conferences in computer vision, alongside CVPR and ICCV, with an 'A' rating from the Australian Ranking of ICT Conferences and an 'A1' rating from the Brazilian ministry of education. The acceptance rate for ECCV 2010 was 24.4% for posters and 3.3% for oral presentations. Like other top computer vision conferences, ECCV has tutorial talks, technical sessions, and poster sessions. The conference is usually spread over five to six days with the main technical program occupying three days in the middle, and tutorial and workshops, focused on specific topics, being held in the beginning and at the end. The ECCV presents the Koenderink Prize annually to recognize fundamental contributions in computer vision.
European Conference on Computer Vision : The conference is usually held in autumn in Europe.
European Conference on Computer Vision : Computer Vision and Pattern Recognition International Conference on Computer Vision == References ==
International Conference on Acoustics, Speech, and Signal Processing : ICASSP, the International Conference on Acoustics, Speech, and Signal Processing, is an annual flagship conference organized by IEEE Signal Processing Society. Ei Compendex has indexed all papers included in its proceedings. The first ICASSP was held in 1976 in Philadelphia, Pennsylvania, based on the success of a conference in Massachusetts four years earlier that had focused specifically on speech signals. As ranked by Google Scholar's h-index metric in 2016, ICASSP has the highest h-index of any conference in the Signal Processing field. The Brazilian ministry of education gave the conference an 'A1' rating based on its h-index.
International Conference on Computer Vision : The International Conference on Computer Vision (ICCV) is a research conference sponsored by the Institute of Electrical and Electronics Engineers (IEEE) held every other year. It is considered to be one of the top conferences in computer vision, alongside CVPR and ECCV, and it is held on years in which ECCV is not. The conference is usually spread over four to five days. Typically, experts in the focus areas give tutorial talks on the first day, then the technical sessions (and poster sessions in parallel) follow. Recent conferences have also had an increasing number of focused workshops and a commercial exhibition.
International Conference on Computer Vision : The conference is usually held in the Spring in various international locations.
International Conference on Computer Vision : Computer Vision and Pattern Recognition European Conference on Computer Vision == References ==
International Conference on Digital Audio Effects : The annual International Conference on Digital Audio Effects or DAFx Conference is a meeting of enthusiasts working in research areas on audio signal processing, acoustics, and music related disciplines, who come together to present and discuss their findings. The conference evolved from an EU-COST-G6 project “Digital Audio Effects” in 1998. The acronym DAFx stands for Digital Audio Effects and is also the name of a book which was written by people in the community around the conference A list of past and upcoming conferences together with an archive of all proceedings can be found at the website.
International Conference on Digital Audio Effects : DAFX, 2020 - Vienna, Austria DAFX, 2019 - Birmingham, UK DAFX, 2018 - Aveiro, Portugal DAFX, 2017 - Edinburgh, UK DAFX, 2016 - Brno, Czech Republic DAFX, 2015 - Trondheim, Norway DAFX, 2014 - Erlangen, Germany DAFX, 2013 - Maynooth, Ireland DAFX, 2012 - York, UK DAFX, 2011 - Paris, France DAFX, 2010 - Graz, Austria DAFX, 2009 - Como, Italy DAFX, 2008 - Espoo, Finland DAFX, 2007 - Bordeaux, France DAFX, 2006 - Montreal, Quebec, Canada Archived 2011-07-06 at the Wayback Machine DAFX, 2005 - Madrid, Spain DAFX, 2004 - Naples, Italy DAFX, 2003 - London, UK DAFX, 2002 - Hamburg, Germany DAFX, 2001 - Limerick, Ireland DAFX, 2000 - Verona, Italy DAFX, 1999 - Trondheim, Norway DAFX, 1998 - Barcelona, Spain
International Conference on Digital Audio Effects : International Society for Music Information Retrieval Sound and Music Computing Conference == References ==
Computational learning theory : In computer science, computational learning theory (or just learning theory) is a subfield of artificial intelligence devoted to studying the design and analysis of machine learning algorithms.
Computational learning theory : Theoretical results in machine learning mainly deal with a type of inductive learning called supervised learning. In supervised learning, an algorithm is given samples that are labeled in some useful way. For example, the samples might be descriptions of mushrooms, and the labels could be whether or not the mushrooms are edible. The algorithm takes these previously labeled samples and uses them to induce a classifier. This classifier is a function that assigns labels to samples, including samples that have not been seen previously by the algorithm. The goal of the supervised learning algorithm is to optimize some measure of performance such as minimizing the number of mistakes made on new samples. In addition to performance bounds, computational learning theory studies the time complexity and feasibility of learning. In computational learning theory, a computation is considered feasible if it can be done in polynomial time. There are two kinds of time complexity results: Positive results – Showing that a certain class of functions is learnable in polynomial time. Negative results – Showing that certain classes cannot be learned in polynomial time. Negative results often rely on commonly believed, but yet unproven assumptions, such as: Computational complexity – P ≠ NP (the P versus NP problem); Cryptographic – One-way functions exist. There are several different approaches to computational learning theory based on making different assumptions about the inference principles used to generalise from limited data. This includes different definitions of probability (see frequency probability, Bayesian probability) and different assumptions on the generation of samples. The different approaches include: Exact learning, proposed by Dana Angluin; Probably approximately correct learning (PAC learning), proposed by Leslie Valiant; VC theory, proposed by Vladimir Vapnik and Alexey Chervonenkis; Inductive inference as developed by Ray Solomonoff; Algorithmic learning theory, from the work of E. Mark Gold; Online machine learning, from the work of Nick Littlestone. While its primary goal is to understand learning abstractly, computational learning theory has led to the development of practical algorithms. For example, PAC theory inspired boosting, VC theory led to support vector machines, and Bayesian inference led to belief networks.
Computational learning theory : Error tolerance (PAC learning) Grammar induction Information theory Occam learning Stability (learning theory)
Computational learning theory : A description of some of these publications is given at important publications in machine learning.
Algorithmic learning theory : Algorithmic learning theory is a mathematical framework for analyzing machine learning problems and algorithms. Synonyms include formal learning theory and algorithmic inductive inference. Algorithmic learning theory is different from statistical learning theory in that it does not make use of statistical assumptions and analysis. Both algorithmic and statistical learning theory are concerned with machine learning and can thus be viewed as branches of computational learning theory.
Algorithmic learning theory : Unlike statistical learning theory and most statistical theory in general, algorithmic learning theory does not assume that data are random samples, that is, that data points are independent of each other. This makes the theory suitable for domains where observations are (relatively) noise-free but not random, such as language learning and automated scientific discovery. The fundamental concept of algorithmic learning theory is learning in the limit: as the number of data points increases, a learning algorithm should converge to a correct hypothesis on every possible data sequence consistent with the problem space. This is a non-probabilistic version of statistical consistency, which also requires convergence to a correct model in the limit, but allows a learner to fail on data sequences with probability measure 0 . Algorithmic learning theory investigates the learning power of Turing machines. Other frameworks consider a much more restricted class of learning algorithms than Turing machines, for example, learners that compute hypotheses more quickly, for instance in polynomial time. An example of such a framework is probably approximately correct learning .
Algorithmic learning theory : The concept was introduced in E. Mark Gold's seminal paper "Language identification in the limit". The objective of language identification is for a machine running one program to be capable of developing another program by which any given sentence can be tested to determine whether it is "grammatical" or "ungrammatical". The language being learned need not be English or any other natural language - in fact the definition of "grammatical" can be absolutely anything known to the tester. In Gold's learning model, the tester gives the learner an example sentence at each step, and the learner responds with a hypothesis, which is a suggested program to determine grammatical correctness. It is required of the tester that every possible sentence (grammatical or not) appears in the list eventually, but no particular order is required. It is required of the learner that at each step the hypothesis must be correct for all the sentences so far. A particular learner is said to be able to "learn a language in the limit" if there is a certain number of steps beyond which its hypothesis no longer changes. At this point it has indeed learned the language, because every possible sentence appears somewhere in the sequence of inputs (past or future), and the hypothesis is correct for all inputs (past or future), so the hypothesis is correct for every sentence. The learner is not required to be able to tell when it has reached a correct hypothesis, all that is required is that it be true. Gold showed that any language which is defined by a Turing machine program can be learned in the limit by another Turing-complete machine using enumeration. This is done by the learner testing all possible Turing machine programs in turn until one is found which is correct so far - this forms the hypothesis for the current step. Eventually, the correct program will be reached, after which the hypothesis will never change again (but note that the learner does not know that it won't need to change). Gold also showed that if the learner is given only positive examples (that is, only grammatical sentences appear in the input, not ungrammatical sentences), then the language can only be guaranteed to be learned in the limit if there are only a finite number of possible sentences in the language (this is possible if, for example, sentences are known to be of limited length). Language identification in the limit is a highly abstract model. It does not allow for limits of runtime or computer memory which can occur in practice, and the enumeration method may fail if there are errors in the input. However the framework is very powerful, because if these strict conditions are maintained, it allows the learning of any program known to be computable. This is because a Turing machine program can be written to mimic any program in any conventional programming language. See Church-Turing thesis.
Algorithmic learning theory : Learning theorists have investigated other learning criteria, such as the following. Efficiency: minimizing the number of data points required before convergence to a correct hypothesis. Mind Changes: minimizing the number of hypothesis changes that occur before convergence. Mind change bounds are closely related to mistake bounds that are studied in statistical learning theory. Kevin Kelly has suggested that minimizing mind changes is closely related to choosing maximally simple hypotheses in the sense of Occam’s Razor.
Algorithmic learning theory : Since 1990, there is an International Conference on Algorithmic Learning Theory (ALT), called Workshop in its first years (1990–1997). Between 1992 and 2016, proceedings were published in the LNCS series. Starting from 2017, they are published by the Proceedings of Machine Learning Research. The 34th conference will be held in Singapore in Feb 2023. The topics of the conference cover all of theoretical machine learning, including statistical and computational learning theory, online learning, active learning, reinforcement learning, and deep learning.
Algorithmic learning theory : Formal epistemology Sample exclusion dimension
Bondy's theorem : In mathematics, Bondy's theorem is a bound on the number of elements needed to distinguish the sets in a family of sets from each other. It belongs to the field of combinatorics, and is named after John Adrian Bondy, who published it in 1972.
Bondy's theorem : The theorem is as follows: Let X be a set with n elements and let A1, A2, ..., An be distinct subsets of X. Then there exists a subset S of X with n − 1 elements such that the sets Ai ∩ S are all distinct. In other words, if we have a 0-1 matrix with n rows and n columns such that each row is distinct, we can remove one column such that the rows of the resulting n × (n − 1) matrix are distinct.
Bondy's theorem : Consider the 4 × 4 matrix [ 1 1 0 1 0 1 0 1 0 0 1 1 0 1 1 0 ] 1&1&0&1\\0&1&0&1\\0&0&1&1\\0&1&1&0\end where all rows are pairwise distinct. If we delete, for example, the first column, the resulting matrix [ 1 0 1 1 0 1 0 1 1 1 1 0 ] 1&0&1\\1&0&1\\0&1&1\\1&1&0\end no longer has this property: the first row is identical to the second row. Nevertheless, by Bondy's theorem we know that we can always find a column that can be deleted without introducing any identical rows. In this case, we can delete the third column: all rows of the 3 × 4 matrix [ 1 1 1 0 1 1 0 0 1 0 1 0 ] 1&1&1\\0&1&1\\0&0&1\\0&1&0\end are distinct. Another possibility would have been deleting the fourth column.
Bondy's theorem : From the perspective of computational learning theory, Bondy's theorem can be rephrased as follows: Let C be a concept class over a finite domain X. Then there exists a subset S of X with the size at most \|C\| − 1 such that S is a witness set for every concept in C. This implies that every finite concept class C has its teaching dimension bounded by \|C\| − 1. == Notes ==
Concept class : In computational learning theory in mathematics, a concept over a domain X is a total Boolean function over X. A concept class is a class of concepts. Concept classes are a subject of computational learning theory. Concept class terminology frequently appears in model theory associated with probably approximately correct (PAC) learning. In this setting, if one takes a set Y as a set of (classifier output) labels, and X is a set of examples, the map c : X → Y , i.e. from examples to classifier labels (where Y = and where c is a subset of X), c is then said to be a concept. A concept class C is then a collection of such concepts. Given a class of concepts C, a subclass D is reachable if there exists a sample s such that D contains exactly those concepts in C that are extensions to s. Not every subclass is reachable.
Concept class : A sample s is a partial function from X to . Identifying a concept with its characteristic function mapping X to , it is a special case of a sample. Two samples are consistent if they agree on the intersection of their domains. A sample s ′ extends another sample s if the two are consistent and the domain of s is contained in the domain of s ′ .
Concept class : Suppose that C = S + ( X ) (X) . Then: the subclass \ is reachable with the sample s = ; the subclass S + ( Y ) (Y) for Y ⊆ X are reachable with a sample that maps the elements of X − Y to zero; the subclass S ( X ) , which consists of the singleton sets, is not reachable.
Concept class : Let C be some concept class. For any concept c ∈ C , we call this concept 1 / d -good for a positive integer d if, for all x ∈ X , at least 1 / d of the concepts in C agree with c on the classification of x . The fingerprint dimension F D ( C ) of the entire concept class C is the least positive integer d such that every reachable subclass C ′ ⊆ C contains a concept that is 1 / d -good for it. This quantity can be used to bound the minimum number of equivalence queries needed to learn a class of concepts according to the following inequality: F D ( C ) − 1 ≤ # E Q ( C ) ≤ ⌈ F D ( C ) ln ⁡ ( \| C \| ) ⌉ . == References ==
Distribution learning theory : The distributional learning theory or learning of probability distribution is a framework in computational learning theory. It has been proposed from Michael Kearns, Yishay Mansour, Dana Ron, Ronitt Rubinfeld, Robert Schapire and Linda Sellie in 1994 and it was inspired from the PAC-framework introduced by Leslie Valiant. In this framework the input is a number of samples drawn from a distribution that belongs to a specific class of distributions. The goal is to find an efficient algorithm that, based on these samples, determines with high probability the distribution from which the samples have been drawn. Because of its generality, this framework has been used in a large variety of different fields like machine learning, approximation algorithms, applied probability and statistics. This article explains the basic definitions, tools and results in this framework from the theory of computation point of view.
Distribution learning theory : Let X be the support of the distributions of interest. As in the original work of Kearns et al. if X is finite it can be assumed without loss of generality that X = n ^ where n is the number of bits that have to be used in order to represent any y ∈ X . We focus in probability distributions over X . There are two possible representations of a probability distribution D over X . probability distribution function (or evaluator) an evaluator E D for D takes as input any y ∈ X and outputs a real number E D [ y ] [y] which denotes the probability that of y according to D , i.e. E D [ y ] = Pr [ Y = y ] [y]=\Pr[Y=y] if Y ∼ D . generator a generator G D for D takes as input a string of truly random bits y and outputs G D [ y ] ∈ X [y]\in X according to the distribution D . Generator can be interpreted as a routine that simulates sampling from the distribution D given a sequence of fair coin tosses. A distribution D is called to have a polynomial generator (respectively evaluator) if its generator (respectively evaluator) exists and can be computed in polynomial time. Let C X a class of distribution over X, that is C X is a set such that every D ∈ C X is a probability distribution with support X . The C X can also be written as C for simplicity. Before defining learnability, it is necessary to define good approximations of a distribution D . There are several ways to measure the distance between two distribution. The three more common possibilities are Kullback-Leibler divergence Total variation distance of probability measures Kolmogorov distance The strongest of these distances is the Kullback-Leibler divergence and the weakest is the Kolmogorov distance. This means that for any pair of distributions D , D ′ : KL-distance ( D , D ′ ) ≥ TV-distance ( D , D ′ ) ≥ Kolmogorov-distance ( D , D ′ ) (D,D')\geq (D,D')\geq (D,D') Therefore, for example if D and D ′ are close with respect to Kullback-Leibler divergence then they are also close with respect to all the other distances. Next definitions hold for all the distances and therefore the symbol d ( D , D ′ ) denotes the distance between the distribution D and the distribution D ′ using one of the distances that we describe above. Although learnability of a class of distributions can be defined using any of these distances, applications refer to a specific distance. The basic input that we use in order to learn a distribution is a number of samples drawn by this distribution. For the computational point of view the assumption is that such a sample is given in a constant amount of time. So it's like having access to an oracle G E N ( D ) that returns a sample from the distribution D . Sometimes the interest is, apart from measuring the time complexity, to measure the number of samples that have to be used in order to learn a specific distribution D in class of distributions C . This quantity is called sample complexity of the learning algorithm. In order for the problem of distribution learning to be more clear consider the problem of supervised learning as defined in. In this framework of statistical learning theory a training set S = ,y_),\dots ,(x_,y_)\ and the goal is to find a target function f : X → Y that minimizes some loss function, e.g. the square loss function. More formally f = arg ⁡ min g ∫ V ( y , g ( x ) ) d ρ ( x , y ) \int V(y,g(x))d\rho (x,y) , where V ( ⋅ , ⋅ ) is the loss function, e.g. V ( y , z ) = ( y − z ) 2 and ρ ( x , y ) the probability distribution according to which the elements of the training set are sampled. If the conditional probability distribution ρ x ( y ) (y) is known then the target function has the closed form f ( x ) = ∫ y y d ρ x ( y ) yd\rho _(y) . So the set S is a set of samples from the probability distribution ρ ( x , y ) . Now the goal of distributional learning theory if to find ρ given S which can be used to find the target function f . Definition of learnability A class of distributions C is called efficiently learnable if for every ϵ > 0 and 0 < δ ≤ 1 given access to G E N ( D ) for an unknown distribution D ∈ C , there exists a polynomial time algorithm A , called learning algorithm of C , that outputs a generator or an evaluator of a distribution D ′ such that Pr [ d ( D , D ′ ) ≤ ϵ ] ≥ 1 − δ If we know that D ′ ∈ C then A is called proper learning algorithm, otherwise is called improper learning algorithm. In some settings the class of distributions C is a class with well known distributions which can be described by a set of parameters. For instance C could be the class of all the Gaussian distributions N ( μ , σ 2 ) ) . In this case the algorithm A should be able to estimate the parameters μ , σ . In this case A is called parameter learning algorithm. Obviously the parameter learning for simple distributions is a very well studied field that is called statistical estimation and there is a very long bibliography on different estimators for different kinds of simple known distributions. But distributions learning theory deals with learning class of distributions that have more complicated description.
Distribution learning theory : In their seminal work, Kearns et al. deal with the case where A is described in term of a finite polynomial sized circuit and they proved the following for some specific classes of distribution. O R gate distributions for this kind of distributions there is no polynomial-sized evaluator, unless # P ⊆ P / poly . On the other hand, this class is efficiently learnable with generator. Parity gate distributions this class is efficiently learnable with both generator and evaluator. Mixtures of Hamming Balls this class is efficiently learnable with both generator and evaluator. Probabilistic Finite Automata this class is not efficiently learnable with evaluator under the Noisy Parity Assumption which is an impossibility assumption in the PAC learning framework.
Distribution learning theory : ϵ − Covers One very common technique in order to find a learning algorithm for a class of distributions C is to first find a small ϵ − cover of C . Definition A set C ϵ is called ϵ -cover of C if for every D ∈ C there is a D ′ ∈ C ϵ such that d ( D , D ′ ) ≤ ϵ . An ϵ − cover is small if it has polynomial size with respect to the parameters that describe D . Once there is an efficient procedure that for every ϵ > 0 finds a small ϵ − cover C ϵ of C then the only left task is to select from C ϵ the distribution D ′ ∈ C ϵ that is closer to the distribution D ∈ C that has to be learned. The problem is that given D ′ , D ″ ∈ C ϵ it is not trivial how we can compare d ( D , D ′ ) and d ( D , D ″ ) in order to decide which one is the closest to D , because D is unknown. Therefore, the samples from D have to be used to do these comparisons. Obviously the result of the comparison always has a probability of error. So the task is similar with finding the minimum in a set of element using noisy comparisons. There are a lot of classical algorithms in order to achieve this goal. The most recent one which achieves the best guarantees was proposed by Daskalakis and Kamath This algorithm sets up a fast tournament between the elements of C ϵ where the winner D ∗ of this tournament is the element which is ϵ − close to D (i.e. d ( D ∗ , D ) ≤ ϵ ,D)\leq \epsilon ) with probability at least 1 − δ . In order to do so their algorithm uses O ( log ⁡ N / ϵ 2 ) ) samples from D and runs in O ( N log ⁡ N / ϵ 2 ) ) time, where N = \| C ϵ \| \| .
Distribution learning theory : Learning of simple well known distributions is a well studied field and there are a lot of estimators that can be used. One more complicated class of distributions is the distribution of a sum of variables that follow simple distributions. These learning procedure have a close relation with limit theorems like the central limit theorem because they tend to examine the same object when the sum tends to an infinite sum. Recently there are two results that described here include the learning Poisson binomial distributions and learning sums of independent integer random variables. All the results below hold using the total variation distance as a distance measure.
Distribution learning theory : Let the random variables X ∼ N ( μ 1 , Σ 1 ) ,\Sigma _) and Y ∼ N ( μ 2 , Σ 2 ) ,\Sigma _) . Define the random variable Z which takes the same value as X with probability w 1 and the same value as Y with probability w 2 = 1 − w 1 =1-w_ . Then if F 1 is the density of X and F 2 is the density of Y the density of Z is F = w 1 F 1 + w 2 F 2 F_+w_F_ . In this case Z is said to follow a mixture of Gaussians. Pearson was the first who introduced the notion of the mixtures of Gaussians in his attempt to explain the probability distribution from which he got same data that he wanted to analyze. So after doing a lot of calculations by hand, he finally fitted his data to a mixture of Gaussians. The learning task in this case is to determine the parameters of the mixture w 1 , w 2 , μ 1 , μ 2 , Σ 1 , Σ 2 ,w_,\mu _,\mu _,\Sigma _,\Sigma _ . The first attempt to solve this problem was from Dasgupta. In this work Dasgupta assumes that the two means of the Gaussians are far enough from each other. This means that there is a lower bound on the distance \| \| μ 1 − μ 2 \| \| -\mu _\|\| . Using this assumption Dasgupta and a lot of scientists after him were able to learn the parameters of the mixture. The learning procedure starts with clustering the samples into two different clusters minimizing some metric. Using the assumption that the means of the Gaussians are far away from each other with high probability the samples in the first cluster correspond to samples from the first Gaussian and the samples in the second cluster to samples from the second one. Now that the samples are partitioned the μ i , Σ i ,\Sigma _ can be computed from simple statistical estimators and w i by comparing the magnitude of the clusters. If G M is the set of all the mixtures of two Gaussians, using the above procedure theorems like the following can be proved. Theorem Let D ∈ G M with \| \| μ 1 − μ 2 \| \| ≥ c n max ( λ m a x ( Σ 1 ) , λ m a x ( Σ 2 ) ) -\mu _\|\|\geq c(\Sigma _),\lambda _(\Sigma _)) , where c > 1 / 2 and λ m a x ( A ) (A) the largest eigenvalue of A , then there is an algorithm which given ϵ > 0 , 0 < δ ≤ 1 and access to G E N ( D ) finds an approximation w i ′ , μ i ′ , Σ i ′ ,\mu '_,\Sigma '_ of the parameters such that Pr [ \| \| w i − w i ′ \| \| ≤ ϵ ] ≥ 1 − δ -w'_\|\|\leq \epsilon ]\geq 1-\delta (respectively for μ i and Σ i . The sample complexity of this algorithm is M = 2 O ( log 2 ⁡ ( 1 / ( ϵ δ ) ) ) (1/(\epsilon \delta ))) and the running time is O ( M 2 d + M d n ) d+Mdn) . The above result could also be generalized in k − mixture of Gaussians. For the case of mixture of two Gaussians there are learning results without the assumption of the distance between their means, like the following one which uses the total variation distance as a distance measure. Theorem Let F ∈ G M then there is an algorithm which given ϵ > 0 , 0 < δ ≤ 1 and access to G E N ( D ) finds w i ′ , μ i ′ , Σ i ′ ,\mu '_,\Sigma '_ such that if F ′ = w 1 ′ F 1 ′ + w 2 ′ F 2 ′ F'_+w'_F'_ , where F i ′ = N ( μ i ′ , Σ i ′ ) =N(\mu '_,\Sigma '_) then Pr [ d ( F , F ′ ) ≤ ϵ ] ≥ 1 − δ . The sample complexity and the running time of this algorithm is poly ( n , 1 / ϵ , 1 / δ , 1 / w 1 , 1 / w 2 , 1 / d ( F 1 , F 2 ) ) (n,1/\epsilon ,1/\delta ,1/w_,1/w_,1/d(F_,F_)) . The distance between F 1 and F 2 doesn't affect the quality of the result of the algorithm but just the sample complexity and the running time. == References ==
Error tolerance (PAC learning) : In PAC learning, error tolerance refers to the ability of an algorithm to learn when the examples received have been corrupted in some way. In fact, this is a very common and important issue since in many applications it is not possible to access noise-free data. Noise can interfere with the learning process at different levels: the algorithm may receive data that have been occasionally mislabeled, or the inputs may have some false information, or the classification of the examples may have been maliciously adulterated.
Error tolerance (PAC learning) : In the following, let X be our n -dimensional input space. Let H be a class of functions that we wish to use in order to learn a -valued target function f defined over X . Let D be the distribution of the inputs over X . The goal of a learning algorithm A is to choose the best function h ∈ H such that it minimizes e r r o r ( h ) = P x ∼ D ( h ( x ) ≠ f ( x ) ) (h(x)\neq f(x)) . Let us suppose we have a function s i z e ( f ) that can measure the complexity of f . Let Oracle ( x ) (x) be an oracle that, whenever called, returns an example x and its correct label f ( x ) . When no noise corrupts the data, we can define learning in the Valiant setting: Definition: We say that f is efficiently learnable using H in the Valiant setting if there exists a learning algorithm A that has access to Oracle ( x ) (x) and a polynomial p ( ⋅ , ⋅ , ⋅ , ⋅ ) such that for any 0 < ε ≤ 1 and 0 < δ ≤ 1 it outputs, in a number of calls to the oracle bounded by p ( 1 ε , 1 δ , n , size ( f ) ) ,,n,(f)\right) , a function h ∈ H that satisfies with probability at least 1 − δ the condition error ( h ) ≤ ε (h)\leq \varepsilon . In the following we will define learnability of f when data have suffered some modification.
Error tolerance (PAC learning) : In the classification noise model a noise rate 0 ≤ η < 1 2 is introduced. Then, instead of Oracle ( x ) (x) that returns always the correct label of example x , algorithm A can only call a faulty oracle Oracle ( x , η ) (x,\eta ) that will flip the label of x with probability η . As in the Valiant case, the goal of a learning algorithm A is to choose the best function h ∈ H such that it minimizes e r r o r ( h ) = P x ∼ D ( h ( x ) ≠ f ( x ) ) (h(x)\neq f(x)) . In applications it is difficult to have access to the real value of η , but we assume we have access to its upperbound η B . Note that if we allow the noise rate to be 1 / 2 , then learning becomes impossible in any amount of computation time, because every label conveys no information about the target function. Definition: We say that f is efficiently learnable using H in the classification noise model if there exists a learning algorithm A that has access to Oracle ( x , η ) (x,\eta ) and a polynomial p ( ⋅ , ⋅ , ⋅ , ⋅ ) such that for any 0 ≤ η ≤ 1 2 , 0 ≤ ε ≤ 1 and 0 ≤ δ ≤ 1 it outputs, in a number of calls to the oracle bounded by p ( 1 1 − 2 η B , 1 ε , 1 δ , n , s i z e ( f ) ) ,,,n,size(f)\right) , a function h ∈ H that satisfies with probability at least 1 − δ the condition e r r o r ( h ) ≤ ε .
Error tolerance (PAC learning) : Statistical Query Learning is a kind of active learning problem in which the learning algorithm A can decide if to request information about the likelihood P f ( x ) that a function f correctly labels example x , and receives an answer accurate within a tolerance α . Formally, whenever the learning algorithm A calls the oracle Oracle ( x , α ) (x,\alpha ) , it receives as feedback probability Q f ( x ) , such that Q f ( x ) − α ≤ P f ( x ) ≤ Q f ( x ) + α -\alpha \leq P_\leq Q_+\alpha . Definition: We say that f is efficiently learnable using H in the statistical query learning model if there exists a learning algorithm A that has access to Oracle ( x , α ) (x,\alpha ) and polynomials p ( ⋅ , ⋅ , ⋅ ) , q ( ⋅ , ⋅ , ⋅ ) , and r ( ⋅ , ⋅ , ⋅ ) such that for any 0 < ε ≤ 1 the following hold: Oracle ( x , α ) (x,\alpha ) can evaluate P f ( x ) in time q ( 1 ε , n , s i z e ( f ) ) ,n,size(f)\right) ; 1 α is bounded by r ( 1 ε , n , s i z e ( f ) ) ,n,size(f)\right) A outputs a model h such that e r r ( h ) < ε , in a number of calls to the oracle bounded by p ( 1 ε , n , s i z e ( f ) ) ,n,size(f)\right) . Note that the confidence parameter δ does not appear in the definition of learning. This is because the main purpose of δ is to allow the learning algorithm a small probability of failure due to an unrepresentative sample. Since now Oracle ( x , α ) (x,\alpha ) always guarantees to meet the approximation criterion Q f ( x ) − α ≤ P f ( x ) ≤ Q f ( x ) + α -\alpha \leq P_\leq Q_+\alpha , the failure probability is no longer needed. The statistical query model is strictly weaker than the PAC model: any efficiently SQ-learnable class is efficiently PAC learnable in the presence of classification noise, but there exist efficient PAC-learnable problems such as parity that are not efficiently SQ-learnable.
Error tolerance (PAC learning) : In the malicious classification model an adversary generates errors to foil the learning algorithm. This setting describes situations of error burst, which may occur when for a limited time transmission equipment malfunctions repeatedly. Formally, algorithm A calls an oracle Oracle ( x , β ) (x,\beta ) that returns a correctly labeled example x drawn, as usual, from distribution D over the input space with probability 1 − β , but it returns with probability β an example drawn from a distribution that is not related to D . Moreover, this maliciously chosen example may strategically selected by an adversary who has knowledge of f , β , D , or the current progress of the learning algorithm. Definition: Given a bound β B < 1 2 < for 0 ≤ β < 1 2 , we say that f is efficiently learnable using H in the malicious classification model, if there exist a learning algorithm A that has access to Oracle ( x , β ) (x,\beta ) and a polynomial p ( ⋅ , ⋅ , ⋅ , ⋅ , ⋅ ) such that for any 0 < ε ≤ 1 , 0 < δ ≤ 1 it outputs, in a number of calls to the oracle bounded by p ( 1 1 / 2 − β B , 1 ε , 1 δ , n , s i z e ( f ) ) ,,,n,size(f)\right) , a function h ∈ H that satisfies with probability at least 1 − δ the condition e r r o r ( h ) ≤ ε .
Error tolerance (PAC learning) : In the nonuniform random attribute noise model the algorithm is learning a Boolean function, a malicious oracle Oracle ( x , ν ) (x,\nu ) may flip each i -th bit of example x = ( x 1 , x 2 , … , x n ) ,x_,\ldots ,x_) independently with probability ν i ≤ ν \leq \nu . This type of error can irreparably foil the algorithm, in fact the following theorem holds: In the nonuniform random attribute noise setting, an algorithm A can output a function h ∈ H such that e r r o r ( h ) < ε only if ν < 2 ε .
Growth function : The growth function, also called the shatter coefficient or the shattering number, measures the richness of a set family or class of functions. It is especially used in the context of statistical learning theory, where it is used to study properties of statistical learning methods. The term 'growth function' was coined by Vapnik and Chervonenkis in their 1968 paper, where they also proved many of its properties. It is a basic concept in machine learning.
Growth function : 1. The domain is the real line R . The set-family H contains all the half-lines (rays) from a given number to positive infinity, i.e., all sets of the form \mid x\in \mathbb \ for some x 0 ∈ R \in \mathbb . For any set C of m real numbers, the intersection H ∩ C contains m + 1 sets: the empty set, the set containing the largest element of C , the set containing the two largest elements of C , and so on. Therefore: Growth ⁡ ( H , m ) = m + 1 (H,m)=m+1 .: Ex.1 The same is true whether H contains open half-lines, closed half-lines, or both. 2. The domain is the segment [ 0 , 1 ] . The set-family H contains all the open sets. For any finite set C of m real numbers, the intersection H ∩ C contains all possible subsets of C . There are 2 m such subsets, so Growth ⁡ ( H , m ) = 2 m (H,m)=2^ . : Ex.2 3. The domain is the Euclidean space R n ^ . The set-family H contains all the half-spaces of the form: x ⋅ ϕ ≥ 1 , where ϕ is a fixed vector. Then Growth ⁡ ( H , m ) = Comp ⁡ ( n , m ) (H,m)=\operatorname (n,m) , where Comp is the number of components in a partitioning of an n-dimensional space by m hyperplanes.: Ex.3 4. The domain is the real line R . The set-family H contains all the real intervals, i.e., all sets of the form ,x_]\|x\in \mathbb \ for some x 0 , x 1 ∈ R ,x_\in \mathbb . For any set C of m real numbers, the intersection H ∩ C contains all runs of between 0 and m consecutive elements of C . The number of such runs is ( m + 1 2 ) + 1 +1 , so Growth ⁡ ( H , m ) = ( m + 1 2 ) + 1 (H,m)=+1 .
Growth function : The main property that makes the growth function interesting is that it can be either polynomial or exponential - nothing in-between. The following is a property of the intersection-size:: Lem.1 If, for some set C m of size m , and for some number n ≤ m , \| H ∩ C m \| ≥ Comp ⁡ ( n , m ) \|\geq \operatorname (n,m) - then, there exists a subset C n ⊆ C m \subseteq C_ of size n such that \| H ∩ C n \| = 2 n \|=2^ . This implies the following property of the Growth function.: Th.1 For every family H there are two cases: The exponential case: Growth ⁡ ( H , m ) = 2 m (H,m)=2^ identically. The polynomial case: Growth ⁡ ( H , m ) (H,m) is majorized by Comp ⁡ ( n , m ) ≤ m n + 1 (n,m)\leq m^+1 , where n is the smallest integer for which Growth ⁡ ( H , n ) < 2 n (H,n)<2^ .
Growth function : Let Ω be a set on which a probability measure Pr is defined. Let H be family of subsets of Ω (= a family of events). Suppose we choose a set C m that contains m elements of Ω , where each element is chosen at random according to the probability measure P , independently of the others (i.e., with replacements). For each event h ∈ H , we compare the following two quantities: Its relative frequency in C m , i.e., \| h ∩ C m \| / m \|/m ; Its probability Pr [ h ] . We are interested in the difference, D ( h , C m ) := \| \| h ∩ C m \| / m − Pr [ h ] \| ):=\|h\cap C_\|/m-\Pr[h] . This difference satisfies the following upper bound: Pr [ ∀ h ∈ H : D ( h , C m ) ≤ 8 ( ln ⁡ Growth ⁡ ( H , 2 m ) + ln ⁡ ( 4 / δ ) ) m ] > 1 − δ )\leq (H,2m)+\ln(4/\delta )) \over m\right]~~~~>~~~~1-\delta which is equivalent to:: Th.2 Pr [ ∀ h ∈ H : D ( h , C m ) ≤ ε ] > 1 − 4 ⋅ Growth ⁡ ( H , 2 m ) ⋅ exp ⁡ ( − ε 2 ⋅ m / 8 ) \forall h\in H:D(h,C_)\leq \varepsilon ~~~~>~~~~1-4\cdot \operatorname (H,2m)\cdot \exp(-\varepsilon ^\cdot m/8) In words: the probability that for all events in H , the relative-frequency is near the probability, is lower-bounded by an expression that depends on the growth-function of H . A corollary of this is that, if the growth function is polynomial in m (i.e., there exists some n such that Growth ⁡ ( H , m ) ≤ m n + 1 (H,m)\leq m^+1 ), then the above probability approaches 1 as m → ∞ . I.e, the family H enjoys uniform convergence in probability. == References ==
Induction of regular languages : In computational learning theory, induction of regular languages refers to the task of learning a formal description (e.g. grammar) of a regular language from a given set of example strings. Although E. Mark Gold has shown that not every regular language can be learned this way (see language identification in the limit), approaches have been investigated for a variety of subclasses. They are sketched in this article. For learning of more general grammars, see Grammar induction.
Induction of regular languages : A regular language is defined as a (finite or infinite) set of strings that can be described by one of the mathematical formalisms called "finite automaton", "regular grammar", or "regular expression", all of which have the same expressive power. Since the latter formalism leads to shortest notations, it shall be introduced and used here. Given a set Σ of symbols (a.k.a. alphabet), a regular expression can be any of ∅ (denoting the empty set of strings), ε (denoting the singleton set containing just the empty string), a (where a is any character in Σ; denoting the singleton set just containing the single-character string a), r + s (where r and s are, in turn, simpler regular expressions; denoting their set's union) r ⋅ s (denoting the set of all possible concatenations of strings from r's and s's set), r + (denoting the set of n-fold repetitions of strings from r's set, for any n ≥ 1), or r * (similarly denoting the set of n-fold repetitions, but also including the empty string, seen as 0-fold repetition). For example, using Σ = , the regular expression (0+1+ε)⋅(0+1) denotes the set of all binary numbers with one or two digits (leading zero allowed), while 1⋅(0+1)⋅0 denotes the (infinite) set of all even binary numbers (no leading zeroes). Given a set of strings (also called "positive examples"), the task of regular language induction is to come up with a regular expression that denotes a set containing all of them. As an example, given , a "natural" description could be the regular expression 1⋅0, corresponding to the informal characterization "a 1 followed by arbitrarily many (maybe even none) 0's". However, (0+1)* and 1+(1⋅0)+(1⋅0⋅0) is another regular expression, denoting the largest (assuming Σ = ) and the smallest set containing the given strings, and called the trivial overgeneralization and undergeneralization, respectively. Some approaches work in an extended setting where also a set of "negative example" strings is given; then, a regular expression is to be found that generates all of the positive, but none of the negative examples.
Induction of regular languages : Dupont et al. have shown that the set of all structurally complete finite automata generating a given input set of example strings forms a lattice, with the trivial undergeneralized and the trivial overgeneralized automaton as bottom and top element, respectively. Each member of this lattice can be obtained by factoring the undergeneralized automaton by an appropriate equivalence relation. For the above example string set , the picture shows at its bottom the undergeneralized automaton Aa,b,c,d in grey, consisting of states a, b, c, and d. On the state set , a total of 15 equivalence relations exist, forming a lattice. Mapping each equivalence E to the corresponding quotient automaton language L(Aa,b,c,d / E) obtains the partially ordered set shown in the picture. Each node's language is denoted by a regular expression. The language may be recognized by quotient automata w.r.t. different equivalence relations, all of which are shown below the node. An arrow between two nodes indicates that the lower node's language is a proper subset of the higher node's. If both positive and negative example strings are given, Dupont et al. build the lattice from the positive examples, and then investigate the separation border between automata that generate some negative example and such that do not. Most interesting are those automata immediately below the border. In the picture, separation borders are shown for the negative example strings 11 (green), 1001 (blue), 101 (cyan), and 0 (red). Coste and Nicolas present an own search method within the lattice, which they relate to Mitchell's version space paradigm. To find the separation border, they use a graph coloring algorithm on the state inequality relation induced by the negative examples. Later, they investigate several ordering relations on the set of all possible state fusions. Kudo and Shimbo use the representation by automaton factorizations to give a unique framework for the following approaches (sketched below): k-reversible languages and the "tail clustering" follow-up approach, Successor automata and the predecessor-successor method, and pumping-based approaches (framework-integration challenged by Luzeaux, however). Each of these approaches is shown to correspond to a particular kind of equivalence relations used for factorization.
Induction of regular languages : Finding common patterns in DNA and RNA structure descriptions (Bioinformatics) Modelling natural language acquisition by humans Learning of structural descriptions from structured example documents, in particular Document Type Definitions (DTD) from SGML documents Learning the structure of music pieces Obtaining compact representations of finite languages Classifying and retrieving documents Generating of context-dependent correction rules for English grammatical errors
Language identification in the limit : Language identification in the limit is a formal model for inductive inference of formal languages, mainly by computers (see machine learning and induction of regular languages). It was introduced by E. Mark Gold in a technical report and a journal article with the same title. In this model, a teacher provides to a learner some presentation (i.e. a sequence of strings) of some formal language. The learning is seen as an infinite process. Each time the learner reads an element of the presentation, it should provide a representation (e.g. a formal grammar) for the language. Gold defines that a learner can identify in the limit a class of languages if, given any presentation of any language in the class, the learner will produce only a finite number of wrong representations, and then stick with the correct representation. However, the learner need not be able to announce its correctness; and the teacher might present a counterexample to any representation arbitrarily long after. Gold defined two types of presentations: Text (positive information): an enumeration of all strings the language consists of. Complete presentation (positive and negative information): an enumeration of all possible strings, each with a label indicating if the string belongs to the language or not.
Language identification in the limit : This model is an early attempt to formally capture the notion of learnability. Gold's journal article introduces for contrast the stronger models Finite identification (where the learner has to announce correctness after a finite number of steps), and Fixed-time identification (where correctness has to be reached after an apriori-specified number of steps). A weaker formal model of learnability is the Probably approximately correct learning (PAC) model, introduced by Leslie Valiant in 1984.
Language identification in the limit : It is instructive to look at concrete examples (in the tables) of learning sessions the definition of identification in the limit speaks about. A fictitious session to learn a regular language L over the alphabet from text presentation:In each step, the teacher gives a string belonging to L, and the learner answers a guess for L, encoded as a regular expression. In step 3, the learner's guess is not consistent with the strings seen so far; in step 4, the teacher gives a string repeatedly. After step 6, the learner sticks to the regular expression (ab+ba)*. If this happens to be a description of the language L the teacher has in mind, it is said that the learner has learned that language.If a computer program for the learner's role would exist that was able to successfully learn each regular language, that class of languages would be identifiable in the limit. Gold has shown that this is not the case. A particular learning algorithm always guessing L to be just the union of all strings seen so far:If L is a finite language, the learner will eventually guess it correctly, however, without being able to tell when. Although the guess didn't change during step 3 to 6, the learner couldn't be sure to be correct.Gold has shown that the class of finite languages is identifiable in the limit, however, this class is neither finitely nor fixed-time identifiable. Learning from complete presentation by telling:In each step, the teacher gives a string and tells whether it belongs to L (green) or not (red, struck-out). Each possible string is eventually classified in this way by the teacher. Learning from complete presentation by request:The learner gives a query string, the teacher tells whether it belongs to L (yes) or not (no); the learner then gives a guess for L, followed by the next query string. In this example, the learner happens to query in each step just the same string as given by the teacher in example 3.In general, Gold has shown that each language class identifiable in the request-presentation setting is also identifiable in the telling-presentation setting, since the learner, instead of querying a string, just needs to wait until it is eventually given by the teacher.
Language identification in the limit : More formally, a language L is a nonempty set, and its elements are called sentences. a language family is a set of languages. a language-learning environment E for a language L is a stream of sentences from L , such that each sentence in L appears at least once. a language learner is a function f that sends a list of sentences to a language. This is interpreted as saying that, after seeing sentences a 1 , a 2 . . . , a n ,a_...,a_ in that order, the language learner guesses that the language that produces the sentences should be f ( a 1 , . . . , a n ) ,...,a_) . Note that the learner is not obliged to be correct — it could very well guess a language that does not even contain a 1 , . . . , a n ,...,a_ . a language learner f learns a language L in environment E = ( a 1 , a 2 , . . . ) ,a_,...) if the learner always guesses L after seeing enough examples from the environment. a language learner f learns a language L if it learns L in any environment E for L . a language family is learnable if there exists a language learner that can learn all languages in the family. Notes: In the context of Gold's theorem, sentences need only be distinguishable. They need not be anything in particular, such as finite strings (as usual in formal linguistics). Learnability is not a concept for individual languages. Any individual language L could be learned by a trivial learner that always guesses L . Learnability is not a concept for individual learners. A language family is learnable iff there exists some learner that can learn the family. It does not matter how well the learner performs for learning languages outside the family. Gold's theorem is easily bypassed if negative examples are allowed. In particular, the language family ,L_,...,L_\ can be learned by a learner that always guesses L ∞ until it receives the first negative example ¬ a n , where a n ∈ L n + 1 ∖ L n \in L_\setminus L_ , at which point it always guesses L n .
Language identification in the limit : Dana Angluin gave the characterizations of learnability from text (positive information) in a 1980 paper. If a learner is required to be effective, then an indexed class of recursive languages is learnable in the limit if there is an effective procedure that uniformly enumerates tell-tales for each language in the class (Condition 1). It is not hard to see that if an ideal learner (i.e., an arbitrary function) is allowed, then an indexed class of languages is learnable in the limit if each language in the class has a tell-tale (Condition 2).
Language identification in the limit : The table shows which language classes are identifiable in the limit in which learning model. On the right-hand side, each language class is a superclass of all lower classes. Each learning model (i.e. type of presentation) can identify in the limit all classes below it. In particular, the class of finite languages is identifiable in the limit by text presentation (cf. Example 2 above), while the class of regular languages is not. Pattern Languages, introduced by Dana Angluin in another 1980 paper, are also identifiable by normal text presentation; they are omitted in the table, since they are above the singleton and below the primitive recursive language class, but incomparable to the classes in between.
Language identification in the limit : Condition 1 in Angluin's paper is not always easy to verify. Therefore, people come up with various sufficient conditions for the learnability of a language class. See also Induction of regular languages for learnable subclasses of regular languages.
Language identification in the limit : A bound over the number of hypothesis changes that occur before convergence.
Language identification in the limit : Finite thickness implies finite elasticity; the converse is not true. Finite elasticity and conservatively learnable implies the existence of a mind change bound. [1] Finite elasticity and M-finite thickness implies the existence of a mind change bound. However, M-finite thickness alone does not imply the existence of a mind change bound; neither does the existence of a mind change bound imply M-finite thickness. [2] Existence of a mind change bound implies learnability; the converse is not true. If we allow for noncomputable learners, then finite elasticity implies the existence of a mind change bound; the converse is not true. If there is no accumulation order for a class of languages, then there is a language (not necessarily in the class) that has infinite cross property within the class, which in turn implies infinite elasticity of the class.
Language identification in the limit : If a countable class of recursive languages has a mind change bound for noncomputable learners, does the class also have a mind change bound for computable learners, or is the class unlearnable by a computable learner?
Natarajan dimension : In the theory of Probably Approximately Correct Machine Learning, the Natarajan dimension characterizes the complexity of learning a set of functions, generalizing from the Vapnik-Chervonenkis dimension for boolean functions to multi-class functions. Originally introduced as the Generalized Dimension by Natarajan, it was subsequently renamed the Natarajan Dimension by Haussler and Long.
Natarajan dimension : Let H be a set of functions from a set X to a set Y . H shatters a set C ⊂ X if there exist two functions f 0 , f 1 ∈ H ,f_\in H such that For every x ∈ C , f 0 ( x ) ≠ f 1 ( x ) (x)\neq f_(x) . For every B ⊂ C , there exists a function h ∈ H such that for all x ∈ B , h ( x ) = f 0 ( x ) (x) and for all x ∈ C − B , h ( x ) = f 1 ( x ) (x) . The Natarajan dimension of H is the maximal cardinality of a set shattered by H . It is easy to see that if \| Y \| = 2 , the Natarajan dimension collapses to the Vapnik Chervonenkis dimension. Shalev-Shwartz and Ben-David present comprehensive material on multi-class learning and the Natarajan dimension, including uniform convergence and learnability. == References ==
Occam learning : In computational learning theory, Occam learning is a model of algorithmic learning where the objective of the learner is to output a succinct representation of received training data. This is closely related to probably approximately correct (PAC) learning, where the learner is evaluated on its predictive power of a test set. Occam learnability implies PAC learning, and for a wide variety of concept classes, the converse is also true: PAC learnability implies Occam learnability.
Occam learning : Occam Learning is named after Occam's razor, which is a principle stating that, given all other things being equal, a shorter explanation for observed data should be favored over a lengthier explanation. The theory of Occam learning is a formal and mathematical justification for this principle. It was first shown by Blumer, et al. that Occam learning implies PAC learning, which is the standard model of learning in computational learning theory. In other words, parsimony (of the output hypothesis) implies predictive power.
Occam learning : The succinctness of a concept c in concept class C can be expressed by the length s i z e ( c ) of the shortest bit string that can represent c in C . Occam learning connects the succinctness of a learning algorithm's output to its predictive power on unseen data. Let C and H be concept classes containing target concepts and hypotheses respectively. Then, for constants α ≥ 0 and 0 ≤ β < 1 , a learning algorithm L is an ( α , β ) -Occam algorithm for C using H iff, given a set S = ,\dots ,x_\ of m samples labeled according to a concept c ∈ C , L outputs a hypothesis h ∈ H such that h is consistent with c on S (that is, h ( x ) = c ( x ) , ∀ x ∈ S ), and s i z e ( h ) ≤ ( n ⋅ s i z e ( c ) ) α m β m^ where n is the maximum length of any sample x ∈ S . An Occam algorithm is called efficient if it runs in time polynomial in n , m , and s i z e ( c ) . We say a concept class C is Occam learnable with respect to a hypothesis class H if there exists an efficient Occam algorithm for C using H . .
Occam learning : Occam learnability implies PAC learnability, as the following theorem of Blumer, et al. shows:
Occam learning : We first prove the Cardinality version. Call a hypothesis h ∈ H bad if e r r o r ( h ) ≥ ϵ , where again e r r o r ( h ) is with respect to the true concept c and the underlying distribution D . The probability that a set of samples S is consistent with h is at most ( 1 − ϵ ) m , by the independence of the samples. By the union bound, the probability that there exists a bad hypothesis in H n , m _ is at most \| H n , m \| ( 1 − ϵ ) m _\|(1-\epsilon )^ , which is less than δ if log ⁡ \| H n , m \| ≤ O ( ϵ m ) − log ⁡ 1 δ _\|\leq O(\epsilon m)-\log . This concludes the proof of the second theorem above. Using the second theorem, we can prove the first theorem. Since we have a ( α , β ) -Occam algorithm, this means that any hypothesis output by L can be represented by at most ( n ⋅ s i z e ( c ) ) α m β m^ bits, and thus log ⁡ \| H n , m \| ≤ ( n ⋅ s i z e ( c ) ) α m β _\|\leq (n\cdot size(c))^m^ . This is less than O ( ϵ m ) − log ⁡ 1 δ if we set m ≥ a ( 1 ϵ log ⁡ 1 δ + ( ( n ⋅ s i z e ( c ) ) α ) ϵ ) 1 1 − β ) \log +\left()\right)^\right) for some constant a > 0 . Thus, by the Cardinality version Theorem, L will output a consistent hypothesis h with probability at least 1 − δ . This concludes the proof of the first theorem above.
Occam learning : Though Occam and PAC learnability are equivalent, the Occam framework can be used to produce tighter bounds on the sample complexity of classical problems including conjunctions, conjunctions with few relevant variables, and decision lists.
Occam learning : Occam algorithms have also been shown to be successful for PAC learning in the presence of errors, probabilistic concepts, function learning and Markovian non-independent examples.
Occam learning : Structural risk minimization Computational learning theory
Occam learning : Blumer, A.; Ehrenfeucht, A.; Haussler, D.; Warmuth, M. K. Learnability and the Vapnik-Chervonenkis dimension. Journal of the ACM, 36(4):929–865, 1989.
Probably approximately correct learning : In computational learning theory, probably approximately correct (PAC) learning is a framework for mathematical analysis of machine learning. It was proposed in 1984 by Leslie Valiant. In this framework, the learner receives samples and must select a generalization function (called the hypothesis) from a certain class of possible functions. The goal is that, with high probability (the "probably" part), the selected function will have low generalization error (the "approximately correct" part). The learner must be able to learn the concept given any arbitrary approximation ratio, probability of success, or distribution of the samples. The model was later extended to treat noise (misclassified samples). An important innovation of the PAC framework is the introduction of computational complexity theory concepts to machine learning. In particular, the learner is expected to find efficient functions (time and space requirements bounded to a polynomial of the example size), and the learner itself must implement an efficient procedure (requiring an example count bounded to a polynomial of the concept size, modified by the approximation and likelihood bounds).
Probably approximately correct learning : In order to give the definition for something that is PAC-learnable, we first have to introduce some terminology. For the following definitions, two examples will be used. The first is the problem of character recognition given an array of n bits encoding a binary-valued image. The other example is the problem of finding an interval that will correctly classify points within the interval as positive and the points outside of the range as negative. Let X be a set called the instance space or the encoding of all the samples. In the character recognition problem, the instance space is X = n ^ . In the interval problem the instance space, X , is the set of all bounded intervals in R , where R denotes the set of all real numbers. A concept is a subset c ⊂ X . One concept is the set of all patterns of bits in X = n ^ that encode a picture of the letter "P". An example concept from the second example is the set of open intervals, \ , each of which contains only the positive points. A concept class C is a collection of concepts over X . This could be the set of all subsets of the array of bits that are skeletonized 4-connected (width of the font is 1). Let EX ⁡ ( c , D ) (c,D) be a procedure that draws an example, x , using a probability distribution D and gives the correct label c ( x ) , that is 1 if x ∈ c and 0 otherwise. Now, given 0 < ϵ , δ < 1 , assume there is an algorithm A and a polynomial p in 1 / ϵ , 1 / δ (and other relevant parameters of the class C ) such that, given a sample of size p drawn according to EX ⁡ ( c , D ) (c,D) , then, with probability of at least 1 − δ , A outputs a hypothesis h ∈ C that has an average error less than or equal to ϵ on X with the same distribution D . Further if the above statement for algorithm A is true for every concept c ∈ C and for every distribution D over X , and for all 0 < ϵ , δ < 1 then C is (efficiently) PAC learnable (or distribution-free PAC learnable). We can also say that A is a PAC learning algorithm for C .
Probably approximately correct learning : Under some regularity conditions these conditions are equivalent: The concept class C is PAC learnable. The VC dimension of C is finite. C is a uniformly Glivenko-Cantelli class. C is compressible in the sense of Littlestone and Warmuth
Probably approximately correct learning : Occam learning Data mining Error tolerance (PAC learning) Sample complexity
Probably approximately correct learning : M. Kearns, U. Vazirani. An Introduction to Computational Learning Theory. MIT Press, 1994. A textbook. M. Mohri, A. Rostamizadeh, and A. Talwalkar. Foundations of Machine Learning. MIT Press, 2018. Chapter 2 contains a detailed treatment of PAC-learnability. Readable through open access from the publisher. D. Haussler. Overview of the Probably Approximately Correct (PAC) Learning Framework. An introduction to the topic. L. Valiant. Probably Approximately Correct. Basic Books, 2013. In which Valiant argues that PAC learning describes how organisms evolve and learn. Littlestone, N.; Warmuth, M. K. (June 10, 1986). "Relating Data Compression and Learnability" (PDF). Archived from the original (PDF) on 2017-08-09. Moran, Shay; Yehudayoff, Amir (2015). "Sample compression schemes for VC classes". arXiv:1503.06960 [cs.LG].
Representer theorem : For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer 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.
Representer theorem : 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 \times \to \mathbb on a non-empty set X with a corresponding reproducing kernel Hilbert space H k . Let there be given a training sample ( x 1 , y 1 ) , … , ( x n , y n ) ∈ X × R ,y_),\dotsc ,(x_,y_)\in \times \mathbb , a strictly increasing real-valued function g : [ 0 , ∞ ) → R , and an arbitrary error function E : ( X × R 2 ) n → R ∪ \times \mathbb ^)^\to \mathbb \cup \lbrace \infty \rbrace , which together define the following regularized empirical risk functional on H k : f ↦ E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + g ( ‖ f ‖ ) . ,y_,f(x_)),\ldots ,(x_,y_,f(x_))\right)+g\left(\lVert f\rVert \right). Then, any minimizer of the empirical risk f ∗ = argmin f ∈ H k , ( ∗ ) = \left\lbrace E\left((x_,y_,f(x_)),\ldots ,(x_,y_,f(x_))\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 ) , (\cdot )=\sum _^\alpha _k(\cdot ,x_), where α i ∈ R \in \mathbb for all 1 ≤ i ≤ n . Proof: Define a mapping φ : X → H k φ ( x ) = k ( ⋅ , x ) \varphi \colon &\to H_\\\varphi (x)&=k(\cdot ,x)\end (so that φ ( x ) = k ( ⋅ , x ) is itself a map X → R \to \mathbb ). Since k is a reproducing kernel, then φ ( x ) ( x ′ ) = k ( x ′ , x ) = ⟨ φ ( x ′ ) , φ ( x ) ⟩ , where ⟨ ⋅ , ⋅ ⟩ is the inner product on H k . Given any x 1 , … , x n ,\ldots ,x_ , one can use orthogonal projection to decompose any f ∈ H k into a sum of two functions, one lying in span ⁡ \left\lbrace \varphi (x_),\ldots ,\varphi (x_)\right\rbrace , and the other lying in the orthogonal complement: f = ∑ i = 1 n α i φ ( x i ) + v , ^\alpha _\varphi (x_)+v, where ⟨ v , φ ( x i ) ⟩ = 0 )\rangle =0 for all i . The above orthogonal decomposition and the reproducing property together show that applying f to any training point 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 ) ⟩ , )=\left\langle \sum _^\alpha _\varphi (x_)+v,\varphi (x_)\right\rangle =\sum _^\alpha _\langle \varphi (x_),\varphi (x_)\rangle , which we observe is independent of v . Consequently, the value of the error function E in () is likewise independent of v . For the second term (the regularization term), since v is orthogonal to ∑ i = 1 n α i φ ( x i ) ^\alpha _\varphi (x_) and 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 ) ‖ ) . g\left(\lVert f\rVert \right)&=g\left(\lVert \sum _^\alpha _\varphi (x_)+v\rVert \right)\\&=g\left(^\alpha _\varphi (x_)\rVert ^+\lVert v\rVert ^\right)\\&\geq g\left(\lVert \sum _^\alpha _\varphi (x_)\rVert \right).\end Therefore, setting v = 0 does not affect the first term of (), while it strictly decreases the second term. Consequently, any minimizer f ∗ in () must have 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 ) , (\cdot )=\sum _^\alpha _\varphi (x_)=\sum _^\alpha _k(\cdot ,x_), which is the desired result.
Representer theorem : 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 E\left((x_,y_,f(x_)),\ldots ,(x_,y_,f(x_))\right)&=\sum _^(f(x_)-y_)^,\\g(\lVert f\rVert )&=\lambda \lVert f\rVert ^\end for λ > 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 ( ⋅ ) 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 ⁡ , ( † ) ^=\operatorname \left\lbrace E\left((x_,y_,(x_)),\ldots ,(x_,y_,(x_))\right)+g\left(\lVert f\rVert \right)\mid =f+h\in H_\oplus \operatorname \lbrace \psi _\mid 1\leq p\leq M\rbrace \right\rbrace ,\quad (\dagger ) i.e., we consider functions of the form f ~ = f + h =f+h , where f ∈ H k and h is an unpenalized function lying in the span of a finite set of real-valued functions \colon \to \mathbb \mid 1\leq p\leq M\rbrace . Under the assumption that the n × M matrix ( ψ p ( x i ) ) i p (x_)\right)_ has rank M , they show that the minimizer f ~ ∗ ^ in ( † ) admits a representation of the form f ~ ∗ ( ⋅ ) = ∑ i = 1 n α i k ( ⋅ , x i ) + ∑ p = 1 M β p ψ p ( ⋅ ) ^(\cdot )=\sum _^\alpha _k(\cdot ,x_)+\sum _^\beta _\psi _(\cdot ) where α i , β p ∈ R ,\beta _\in \mathbb and the β 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 be a nonempty set, k a positive-definite real-valued kernel on X × X \times with corresponding reproducing kernel Hilbert space H k , and let R : H k → R \to \mathbb be a differentiable regularization function. Then given a training sample ( x 1 , y 1 ) , … , ( x n , y n ) ∈ X × R ,y_),\ldots ,(x_,y_)\in \times \mathbb and an arbitrary error function E : ( X × R 2 ) m → R ∪ \times \mathbb ^)^\to \mathbb \cup \lbrace \infty \rbrace , a minimizer f ∗ = argmin f ∈ H k ( ‡ ) = \left\lbrace E\left((x_,y_,f(x_)),\ldots ,(x_,y_,f(x_))\right)+R(f)\right\rbrace \quad (\ddagger ) of the regularized empirical risk admits a representation of the form f ∗ ( ⋅ ) = ∑ i = 1 n α i k ( ⋅ , x i ) , (\cdot )=\sum _^\alpha _k(\cdot ,x_), where α i ∈ R \in \mathbb for all 1 ≤ i ≤ n , if and only if there exists a nondecreasing function h : [ 0 , ∞ ) → R for which R ( f ) = h ( ‖ f ‖ ) . Effectively, this result provides a necessary and sufficient condition on a differentiable regularizer R ( ⋅ ) under which the corresponding regularized empirical risk minimization ( ‡ ) will have a representer theorem. In particular, this shows that a broad class of regularized risk minimizations (much broader than those originally considered by Kimeldorf and Wahba) have representer theorems.
Representer theorem : Representer theorems are useful from a practical standpoint because they dramatically simplify the regularized empirical risk minimization problem ( ‡ ) . In most interesting applications, the search domain H k for the minimization will be an infinite-dimensional subspace of L 2 ( X ) () , and therefore the search (as written) does not admit implementation on finite-memory and finite-precision computers. In contrast, the representation of f ∗ ( ⋅ ) (\cdot ) afforded by a representer theorem reduces the original (infinite-dimensional) minimization problem to a search for the optimal n -dimensional vector of coefficients α = ( α 1 , … , α n ) ∈ R n ,\ldots ,\alpha _)\in \mathbb ^ ; α can then be obtained by applying any standard function minimization algorithm. Consequently, representer theorems provide the theoretical basis for the reduction of the general machine learning problem to algorithms that can actually be implemented on computers in practice. The following provides an example of how to solve for the minimizer whose existence is guaranteed by the representer theorem. This method works for any positive definite kernel K , and allows us to transform a complicated (possibly infinite dimensional) optimization problem into a simple linear system that can be solved numerically. Assume that we are using a least squares error function and a regularization function g ( x ) = λ x 2 for some λ > 0 . By the representer theorem, the minimizer has the form for some α ∗ = ( α 1 ∗ , … , α n ∗ ) ∈ R n =(\alpha _^,\dots ,\alpha _^)\in \mathbb ^ . Noting that we see that α ∗ has the form where A i j = k ( x i , x j ) =k(x_,x_) and y = ( y 1 , … , y n ) ,\dots ,y_) . This can be factored out and simplified to Since A ⊺ A + λ A A+\lambda A is positive definite, there is indeed a single global minimum for this expression. Let F ( α ) = α ⊺ ( A ⊺ A + λ A ) α − 2 α ⊺ A ⊺ y (A^A+\lambda A)\alpha -2\alpha ^A^y and note that F is convex. Then α ∗ , the global minimum, can be solved by setting ∇ α F = 0 F=0 . Recalling that all positive definite matrices are invertible, we see that so the minimizer may be found via a linear solve.
Representer theorem : Mercer's theorem Kernel methods
Representer theorem : Argyriou, Andreas; Micchelli, Charles A.; Pontil, Massimiliano (2009). "When Is There a Representer Theorem? Vector Versus Matrix Regularizers". Journal of Machine Learning Research. 10 (Dec): 2507–2529. Cucker, Felipe; Smale, Steve (2002). "On the Mathematical Foundations of Learning". Bulletin of the American Mathematical Society. 39 (1): 1–49. doi:10.1090/S0273-0979-01-00923-5. MR 1864085. Kimeldorf, George S.; Wahba, Grace (1970). "A correspondence between Bayesian estimation on stochastic processes and smoothing by splines". The Annals of Mathematical Statistics. 41 (2): 495–502. doi:10.1214/aoms/1177697089. Schölkopf, Bernhard; Herbrich, Ralf; Smola, Alex J. (2001). "A Generalized Representer Theorem". Computational Learning Theory. Lecture Notes in Computer Science. Vol. 2111. pp. 416–426. CiteSeerX 10.1.1.42.8617. doi:10.1007/3-540-44581-1_27. ISBN 978-3-540-42343-0.
Sample exclusion dimension : In computational learning theory, sample exclusion dimensions arise in the study of exact concept learning with queries. In algorithmic learning theory, a concept over a domain X is a Boolean function over X. Here we only consider finite domains. A partial approximation S of a concept c is a Boolean function over Y ⊆ X such that c is an extension to S. Let C be a class of concepts and c be a concept (not necessarily in C). Then a specifying set for c w.r.t. C, denoted by S is a partial approximation S of c such that C contains at most one extension to S. If we have observed a specifying set for some concept w.r.t. C, then we have enough information to verify a concept in C with at most one more mind change. The exclusion dimension, denoted by XD(C), of a concept class is the maximum of the size of the minimum specifying set of c' with respect to C, where c' is a concept not in C. == References ==
Shattered set : A class of sets is said to shatter another set if it is possible to "pick out" any element of that set using intersection. The concept of shattered sets plays an important role in Vapnik–Chervonenkis theory, also known as VC-theory. Shattering and VC-theory are used in the study of empirical processes as well as in statistical computational learning theory.
Shattered set : Suppose A is a set and C is a class of sets. The class C shatters the set A if for each subset a of A, there is some element c of C such that a = c ∩ A . Equivalently, C shatters A when their intersection is equal to A's power set: P(A) = . We employ the letter C to refer to a "class" or "collection" of sets, as in a Vapnik–Chervonenkis class (VC-class). The set A is often assumed to be finite because, in empirical processes, we are interested in the shattering of finite sets of data points.
Shattered set : We will show that the class of all discs in the plane (two-dimensional space) does not shatter every set of four points on the unit circle, yet the class of all convex sets in the plane does shatter every finite set of points on the unit circle. Let A be a set of four points on the unit circle and let C be the class of all discs. To test where C shatters A, we attempt to draw a disc around every subset of points in A. First, we draw a disc around the subsets of each isolated point. Next, we try to draw a disc around every subset of point pairs. This turns out to be doable for adjacent points, but impossible for points on opposite sides of the circle. Any attempt to include those points on the opposite side will necessarily include other points not in that pair. Hence, any pair of opposite points cannot be isolated out of A using intersections with class C and so C does not shatter A. As visualized below: Because there is some subset which can not be isolated by any disc in C, we conclude then that A is not shattered by C. And, with a bit of thought, we can prove that no set of four points is shattered by this C. However, if we redefine C to be the class of all elliptical discs, we find that we can still isolate all the subsets from above, as well as the points that were formerly problematic. Thus, this specific set of 4 points is shattered by the class of elliptical discs. Visualized below: With a bit of thought, we could generalize that any set of finite points on a unit circle could be shattered by the class of all convex sets (visualize connecting the dots).
Shattered set : To quantify the richness of a collection C of sets, we use the concept of shattering coefficients (also known as the growth function). For a collection C of sets s ⊂ Ω , Ω being any space, often a sample space, we define the nth shattering coefficient of C as S C ( n ) = max ∀ x 1 , x 2 , … , x n ∈ Ω card ⁡ ∩ s , s ∈ C (n)=\max _,x_,\dots ,x_\in \Omega \operatorname \,x_,\dots ,x_\\cap s,s\in C\ where card denotes the cardinality of the set and x 1 , x 2 , … , x n ∈ Ω ,x_,\dots ,x_\in \Omega is any set of n points,. S C ( n ) (n) is the largest number of subsets of any set A of n points that can be formed by intersecting A with the sets in collection C. For example, if set A contains 3 points, its power set, P ( A ) , contains 2 3 = 8 =8 elements. If C shatters A, its shattering coefficient(3) would be 8 and S_C(2) would be 2 2 = 4 =4 . However, if one of those sets in P ( A ) cannot be obtained through intersections in c, then S_C(3) would only be 7. If none of those sets can be obtained, S_C(3) would be 0. Additionally, if S_C(2)=3, for example, then there is an element in the set of all 2-point sets from A that cannot be obtained from intersections with C. It follows from this that S_C(3) would also be less than 8 (i.e. C would not shatter A) because we have already located a "missing" set in the smaller power set of 2-point sets. This example illustrates some properties of S C ( n ) (n) : S C ( n ) ≤ 2 n (n)\leq 2^ for all n because ⊆ P ( A ) \subseteq P(A) for any A ⊆ Ω . If S C ( n ) = 2 n (n)=2^ , that means there is a set of cardinality n, which can be shattered by C. If S C ( N ) < 2 N (N)<2^ for some N > 1 then S C ( n ) < 2 n (n)<2^ for all n ≥ N . The third property means that if C cannot shatter any set of cardinality N then it can not shatter sets of larger cardinalities.
Shattered set : If A cannot be shattered by C, there will be a smallest value of n that makes the shatter coefficient(n) less than 2 n because as n gets larger, there are more sets that could be missed. Alternatively, there is also a largest value of n for which the S_C(n) is still 2 n , because as n gets smaller, there are fewer sets that could be omitted. The extreme of this is S_C(0) (the shattering coefficient of the empty set), which must always be 2 0 = 1 =1 . These statements lends themselves to defining the VC dimension of a class C as: V C ( C ) = min n \(n)<2^\\, or, alternatively, as V C 0 ( C ) = max n . (C)=\(n)=2^\.\, Note that V C ( C ) = V C 0 ( C ) + 1. (C)+1. . The VC dimension is usually defined as VC_0, the largest cardinality of points chosen that will still shatter A (i.e. n such that S C ( n ) = 2 n (n)=2^ ). Altneratively, if for any n there is a set of cardinality n which can be shattered by C, then S C ( n ) = 2 n (n)=2^ for all n and the VC dimension of this class C is infinite. A class with finite VC dimension is called a Vapnik–Chervonenkis class or VC class. A class C is uniformly Glivenko–Cantelli if and only if it is a VC class.
Shattered set : Sauer–Shelah lemma, relating the cardinality of a family of sets to the size of its largest shattered set
Shattered set : Wencour, R. S.; Dudley, R. M. (1981), "Some special Vapnik–Chervonenkis classes", Discrete Mathematics, 33 (3): 313–318, doi:10.1016/0012-365X(81)90274-0. Steele, J. M. (1975), Combinatorial Entropy and Uniform Limit Laws, Ph.D. thesis, Stanford University, Mathematics Department Steele, J. M. (1978), "Empirical discrepancies and subadditive processes", Annals of Probability, 6 (1): 118–227, doi:10.1214/aop/1176995615, JSTOR 2242865.
Shattered set : Origin of "Shattered sets" terminology, by J. Steele
Vapnik–Chervonenkis dimension : In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the size (capacity, complexity, expressive power, richness, or flexibility) of a class of sets. The notion can be extended to classes of binary functions. It is defined as the cardinality of the largest set of points that the algorithm can shatter, which means the algorithm can always learn a perfect classifier for any labeling of at least one configuration of those data points. It was originally defined by Vladimir Vapnik and Alexey Chervonenkis. Informally, the capacity of a classification model is related to how complicated it can be. For example, consider the thresholding of a high-degree polynomial: if the polynomial evaluates above zero, that point is classified as positive, otherwise as negative. A high-degree polynomial can be wiggly, so that it can fit a given set of training points well. But one can expect that the classifier will make errors on other points, because it is too wiggly. Such a polynomial has a high capacity. A much simpler alternative is to threshold a linear function. This function may not fit the training set well, because it has a low capacity. This notion of capacity is made rigorous below.
Vapnik–Chervonenkis dimension : f is a constant classifier (with no parameters); Its VC dimension is 0 since it cannot shatter even a single point. In general, the VC dimension of a finite classification model, which can return at most 2 d different classifiers, is at most d (this is an upper bound on the VC dimension; the Sauer–Shelah lemma gives a lower bound on the dimension). f is a single-parametric threshold classifier on real numbers; i.e., for a certain threshold θ , the classifier f θ returns 1 if the input number is larger than θ and 0 otherwise. The VC dimension of f is 1 because: (a) It can shatter a single point. For every point x , a classifier f θ labels it as 0 if θ > x and labels it as 1 if θ < x . (b) It cannot shatter all the sets with two points. For every set of two numbers, if the smaller is labeled 1, then the larger must also be labeled 1, so not all labelings are possible. f is a single-parametric interval classifier on real numbers; i.e., for a certain parameter θ , the classifier f θ returns 1 if the input number is in the interval [ θ , θ + 4 ] and 0 otherwise. The VC dimension of f is 2 because: (a) It can shatter some sets of two points. E.g., for every set , a classifier f θ labels it as (0,0) if θ < x − 4 or if θ > x + 2 , as (1,0) if θ ∈ [ x − 4 , x − 2 ) , as (1,1) if θ ∈ [ x − 2 , x ] , and as (0,1) if θ ∈ ( x , x + 2 ] . (b) It cannot shatter any set of three points. For every set of three numbers, if the smallest and the largest are labeled 1, then the middle one must also be labeled 1, so not all labelings are possible. f is a straight line as a classification model on points in a two-dimensional plane (this is the model used by a perceptron). The line should separate positive data points from negative data points. There exist sets of 3 points that can indeed be shattered using this model (any 3 points that are not collinear can be shattered). However, no set of 4 points can be shattered: by Radon's theorem, any four points can be partitioned into two subsets with intersecting convex hulls, so it is not possible to separate one of these two subsets from the other. Thus, the VC dimension of this particular classifier is 3. It is important to remember that while one can choose any arrangement of points, the arrangement of those points cannot change when attempting to shatter for some label assignment. Note, only 3 of the 23 = 8 possible label assignments are shown for the three points. f is a single-parametric sine classifier, i.e., for a certain parameter θ , the classifier f θ returns 1 if the input number x has sin ⁡ ( θ x ) > 0 and 0 otherwise. The VC dimension of f is infinite, since it can shatter any finite subset of the set \mid m\in \mathbb \ .: 57
Vapnik–Chervonenkis dimension : The VC dimension of the dual set-family of F is strictly less than 2 vc ⁡ ( F ) + 1 ()+1 , and this is best possible. The VC dimension of a finite set-family H is at most log 2 ⁡ \| H \| \|H\| .: 56 This is because \| H ∩ C \| ≤ \| H \| by definition. Given a set-family H , define H s as a set-family that contains all intersections of s elements of H . Then:: 57 VCDim ⁡ ( H s ) ≤ VCDim ⁡ ( H ) ⋅ ( 2 s log 2 ⁡ ( 3 s ) ) (H_)\leq \operatorname (H)\cdot (2s\log _(3s)) Given a set-family H and an element h 0 ∈ H \in H , define H Δ h 0 := :=\\mid h\in H\ where Δ denotes symmetric set difference. Then:: 58 VCDim ⁡ ( H Δ h 0 ) = VCDim ⁡ ( H ) (H\,\Delta h_)=\operatorname (H)
Vapnik–Chervonenkis dimension : The VC dimension is defined for spaces of binary functions (functions to ). Several generalizations have been suggested for spaces of non-binary functions. For multi-class functions (e.g., functions to ), the Natarajan dimension, and its generalization the DS dimension can be used. For real-valued functions (e.g., functions to a real interval, [0,1]), the Graph dimension or Pollard's pseudo-dimension can be used. The Rademacher complexity provides similar bounds to the VC, and can sometimes provide more insight than VC dimension calculations into such statistical methods such as those using kernels. The Memory Capacity (sometimes Memory Equivalent Capacity) gives a lower bound capacity, rather than an upper bound (see for example: Artificial neural network#Capacity) and therefore indicates the point of potential overfitting.
Vapnik–Chervonenkis dimension : Growth function Sauer–Shelah lemma, a bound on the number of sets in a set system in terms of the VC dimension. Karpinski–Macintyre theorem, a bound on the VC dimension of general Pfaffian formulas.
Vapnik–Chervonenkis dimension : Moore, Andrew. "VC dimension tutorial" (PDF). Vapnik, Vladimir (2000). The nature of statistical learning theory. Springer. Blumer, A.; Ehrenfeucht, A.; Haussler, D.; Warmuth, M. K. (1989). "Learnability and the Vapnik–Chervonenkis dimension" (PDF). Journal of the ACM. 36 (4): 929–865. doi:10.1145/76359.76371. S2CID 1138467. Burges, Christopher. "Tutorial on SVMs for Pattern Recognition" (PDF). Microsoft. (containing information also for VC dimension) Chazelle, Bernard. "The Discrepancy Method". Natarajan, B.K. (1989). "On Learning sets and functions". Machine Learning. 4: 67–97. doi:10.1007/BF00114804. Ben-David, Shai; Cesa-Bianchi, Nicolò; Long, Philip M. (1992). "Characterizations of learnability for classes of -valued functions". Proceedings of the fifth annual workshop on Computational learning theory – COLT '92. p. 333. doi:10.1145/130385.130423. ISBN 089791497X. Brukhim, Nataly; Carmon, Daniel; Dinur, Irit; Moran, Shay; Yehudayoff, Amir (2022). "A Characterization of Multiclass Learnability". 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS). arXiv:2203.01550. Pollard, D. (1984). Convergence of Stochastic Processes. Springer. ISBN 9781461252542. Anthony, Martin; Bartlett, Peter L. (2009). Neural Network Learning: Theoretical Foundations. ISBN 9780521118620. Morgenstern, Jamie H.; Roughgarden, Tim (2015). On the Pseudo-Dimension of Nearly Optimal Auctions. NIPS. arXiv:1506.03684. Bibcode:2015arXiv150603684M. Karpinski, Marek; Macintyre, Angus (February 1997). "Polynomial Bounds for VC Dimension of Sigmoidal and General Pfaffian Neural Networks". Journal of Computer and System Sciences. 54 (1): 169–176. doi:10.1006/jcss.1997.1477.
Vapnik–Chervonenkis theory : Vapnik–Chervonenkis theory (also known as VC theory) was developed during 1960–1990 by Vladimir Vapnik and Alexey Chervonenkis. The theory is a form of computational learning theory, which attempts to explain the learning process from a statistical point of view.
Vapnik–Chervonenkis theory : VC theory covers at least four parts (as explained in The Nature of Statistical Learning Theory): Theory of consistency of learning processes What are (necessary and sufficient) conditions for consistency of a learning process based on the empirical risk minimization principle? Nonasymptotic theory of the rate of convergence of learning processes How fast is the rate of convergence of the learning process? Theory of controlling the generalization ability of learning processes How can one control the rate of convergence (the generalization ability) of the learning process? Theory of constructing learning machines How can one construct algorithms that can control the generalization ability? VC Theory is a major subbranch of statistical learning theory. One of its main applications in statistical learning theory is to provide generalization conditions for learning algorithms. From this point of view, VC theory is related to stability, which is an alternative approach for characterizing generalization. In addition, VC theory and VC dimension are instrumental in the theory of empirical processes, in the case of processes indexed by VC classes. Arguably these are the most important applications of the VC theory, and are employed in proving generalization. Several techniques will be introduced that are widely used in the empirical process and VC theory. The discussion is mainly based on the book Weak Convergence and Empirical Processes: With Applications to Statistics.
Vapnik–Chervonenkis theory : A similar setting is considered, which is more common to machine learning. Let X is a feature space and Y = =\ . A function f : X → Y \to is called a classifier. Let F be a set of classifiers. Similarly to the previous section, define the shattering coefficient (also known as growth function): S ( F , n ) = max x 1 , … , x n \| \| ,n)=\max _,\ldots ,x_\|\),\ldots ,f(x_)),f\in \\| Note here that there is a 1:1 go between each of the functions in F and the set on which the function is 1. We can thus define C to be the collection of subsets obtained from the above mapping for every f ∈ F . Therefore, in terms of the previous section the shattering coefficient is precisely max x 1 , … , x n Δ n ( C , x 1 , … , x n ) ,\ldots ,x_\Delta _(,x_,\ldots ,x_) . This equivalence together with Sauer's Lemma implies that S ( F , n ) ,n) is going to be polynomial in n, for sufficiently large n provided that the collection C has a finite VC-index. Let D n = =\,Y_),\ldots ,(X_,Y_)\ is an observed dataset. Assume that the data is generated by an unknown probability distribution P X Y . Define R ( f ) = P ( f ( X ) ≠ Y ) to be the expected 0/1 loss. Of course since P X Y is unknown in general, one has no access to R ( f ) . However the empirical risk, given by: R ^ n ( f ) = 1 n ∑ i = 1 n I ( f ( X i ) ≠ Y i ) _(f)=\sum _^\mathbb (f(X_)\neq Y_) can certainly be evaluated. Then one has the following Theorem:
Vapnik–Chervonenkis theory : See references in articles: Richard M. Dudley, empirical processes, Shattered set. Bousquet, Olivier; Elisseeff, Andr´e (1 March 2002). "Stability and Generalization". The Journal of Machine Learning Research. 2: 499–526. doi:10.1162/153244302760200704. S2CID 1157797. Retrieved 10 December 2022. Vapnik, V.; Chervonenkis, A. (2004). "On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities". Theory Probab. Appl. 16 (2): 264–280. doi:10.1137/1116025.
Win–stay, lose–switch : In psychology, game theory, statistics, and machine learning, win–stay, lose–switch (also win–stay, lose–shift) is a heuristic learning strategy used to model learning in decision situations. It was first invented as an improvement over randomization in bandit problems. It was later applied to the prisoner's dilemma in order to model the evolution of altruism. The learning rule bases its decision only on the outcome of the previous play. Outcomes are divided into successes (wins) and failures (losses). If the play on the previous round resulted in a success, then the agent plays the same strategy on the next round. Alternatively, if the play resulted in a failure the agent switches to another action. A large-scale empirical study of players of the game rock, paper, scissors shows that a variation of this strategy is adopted by real-world players of the game, instead of the Nash equilibrium strategy of choosing entirely at random between the three options.

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