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If A {\displaystyle {\mathcal {A}}} is not consistent, then we set N ( ρ , ϵ , δ ) = ∞ {\displaystyle N(\rho ,\epsilon ,\delta )=\infty } . If there exists an algorithm for which N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} is finite, then we say that the hypothesis space H {\displaystyle {\mathcal {H}}} ... | Wikipedia - Sample complexity - Definition | 332 | 1,133 | null |
In probably approximately correct (PAC) learning, one is concerned with whether the sample complexity is polynomial, that is, whether N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} is bounded by a polynomial in 1 / ϵ {\displaystyle 1/\epsilon } and 1 / δ {\displaystyle 1/\delta } . If N ( ρ , ϵ , δ ) {\disp... | Wikipedia - Sample complexity - Definition | 151 | 538 | null |
More formally, one asks whether there exists a learning algorithm A {\displaystyle {\mathcal {A}}} , such that, for all ϵ , δ > 0 {\displaystyle \epsilon ,\delta >0} , there exists a positive integer N {\displaystyle N} such that for all n ≥ N {\displaystyle n\geq N} , we have sup ρ ( Pr ρ n [ E ( h n ) − E H ∗ ≥ ε ] )... | Wikipedia - Sample complexity - Unrestricted hypothesis space: infinite sample complexity | 306 | 675 | null |
More formally, one asks whether there exists a learning algorithm A {\displaystyle {\mathcal {A}}} , such that, for all ϵ , δ > 0 {\displaystyle \epsilon ,\delta >0} , there exists a positive integer N {\displaystyle N} such that for all n ≥ N {\displaystyle n\geq N} , we have sup ρ ( Pr ρ n [ E ( h n ) − E H ∗ ≥ ε ] )... | Wikipedia - Sample complexity - Unrestricted hypothesis space: infinite sample complexity | 364 | 912 | null |
The No Free Lunch Theorem says that without restrictions on the hypothesis space H {\displaystyle {\mathcal {H}}} , this is not the case, i.e., there always exist "bad" distributions for which the sample complexity is arbitrarily large. Thus, in order to make statements about the rate of convergence of the quantity sup... | Wikipedia - Sample complexity - Unrestricted hypothesis space: infinite sample complexity | 239 | 726 | null |
Section: Restricted hypothesis space: finite sample-complexity. The latter approach leads to concepts such as VC dimension and Rademacher complexity which control the complexity of the space H {\displaystyle {\mathcal {H}}} . A smaller hypothesis space introduces more bias into the inference process, meaning that E H ∗... | Wikipedia - Sample complexity - Restricted hypothesis space: finite sample-complexity | 263 | 1,066 | null |
Section: Restricted hypothesis space: finite sample-complexity > An example of a PAC-learnable hypothesis space. X = R d , Y = { − 1 , 1 } {\displaystyle X=\mathbb {R} ^{d},Y=\{-1,1\}} , and let H {\displaystyle {\mathcal {H}}} be the space of affine functions on X {\displaystyle X} , that is, functions of the form x ↦... | Wikipedia - Sample complexity - Restricted hypothesis space: finite sample-complexity > An example of a PAC-learnable hypothesis space | 299 | 980 | null |
Section: Restricted hypothesis space: finite sample-complexity > Sample-complexity bounds. Suppose H {\displaystyle {\mathcal {H}}} is a class of binary functions (functions to { 0 , 1 } {\displaystyle \{0,1\}} ). Then, H {\displaystyle {\mathcal {H}}} is ( ϵ , δ ) {\displaystyle (\epsilon ,\delta )} -PAC-learnable wit... | Wikipedia - Sample complexity - Restricted hypothesis space: finite sample-complexity > Sample-complexity bounds | 350 | 962 | null |
Moreover, any ( ϵ , δ ) {\displaystyle (\epsilon ,\delta )} -PAC-learning algorithm for H {\displaystyle {\mathcal {H}}} must have sample-complexity: N = Ω ( V C ( H ) + ln 1 δ ϵ ) {\displaystyle N=\Omega {\bigg (}{\frac {VC({\mathcal {H}})+\ln {1 \over \delta }}{\epsilon }}{\bigg )}} Thus, the sample-complexity is a... | Wikipedia - Sample complexity - Restricted hypothesis space: finite sample-complexity > Sample-complexity bounds | 177 | 506 | null |
Suppose H {\displaystyle {\mathcal {H}}} is a class of real-valued functions with range in [ 0 , T ] {\displaystyle [0,T]} . Then, H {\displaystyle {\mathcal {H}}} is ( ϵ , δ ) {\displaystyle (\epsilon ,\delta )} -PAC-learnable with a sample of size: N = O ( T 2 P D ( H ) ln T ϵ + ln 1 δ ϵ 2 ) {\displaystyle N=O{\b... | Wikipedia - Sample complexity - Restricted hypothesis space: finite sample-complexity > Sample-complexity bounds | 227 | 548 | null |
Article: Self-supervised learning. Self-supervised learning (SSL) is a paradigm in machine learning where a model is trained on a task using the data itself to generate supervisory signals, rather than relying on externally-provided labels. In the context of neural networks, self-supervised learning aims to leverage in... | Wikipedia - Self-supervised learning - Summary | 260 | 1,386 | null |
Section: Types > Autoassociative self-supervised learning. Autoassociative self-supervised learning is a specific category of self-supervised learning where a neural network is trained to reproduce or reconstruct its own input data. In other words, the model is tasked with learning a representation of the data that cap... | Wikipedia - Self-supervised learning - Types > Autoassociative self-supervised learning | 256 | 1,267 | null |
Section: Types > Contrastive self-supervised learning. For a binary classification task, training data can be divided into positive examples and negative examples. Positive examples are those that match the target. For example, if training a classifier to identify birds, the positive training data would include images ... | Wikipedia - Self-supervised learning - Types > Contrastive self-supervised learning | 230 | 1,155 | null |
InfoNCE (Noise-Contrastive Estimation) is a method to optimize two models jointly, based on Noise Contrastive Estimation (NCE). Given a set X = { x 1 , … x N } {\displaystyle X=\left\{x_{1},\ldots x_{N}\right\}} of N {\displaystyle N} random samples containing one positive sample from p ( x t + k ∣ c t ) {\displaystyle... | Wikipedia - Self-supervised learning - Types > Contrastive self-supervised learning | 323 | 773 | null |
Section: Comparison with other forms of machine learning. SSL belongs to supervised learning methods insofar as the goal is to generate a classified output from the input. At the same time, however, it does not require the explicit use of labeled input-output pairs. Instead, correlations, metadata embedded in the data,... | Wikipedia - Self-supervised learning - Comparison with other forms of machine learning | 306 | 1,525 | null |
Section: Examples. Self-supervised learning is particularly suitable for speech recognition. For example, Facebook developed wav2vec, a self-supervised algorithm, to perform speech recognition using two deep convolutional neural networks that build on each other. Google's Bidirectional Encoder Representations from Tran... | Wikipedia - Self-supervised learning - Examples | 286 | 1,381 | null |
Article: Semantic analysis (machine learning). In machine learning, semantic analysis of a text corpus is the task of building structures that approximate concepts from a large set of documents. It generally does not involve prior semantic understanding of the documents. Semantic analysis strategies include: Metalangua... | Wikipedia - Semantic analysis (machine learning) - Summary | 213 | 1,066 | null |
Section: Theory. Semantic folding theory draws inspiration from Douglas R. Hofstadter's Analogy as the Core of Cognition which suggests that the brain makes sense of the world by identifying and applying analogies. The theory hypothesises that semantic data must therefore be introduced to the neocortex in such a form a... | Wikipedia - Semantic folding - Theory | 163 | 923 | null |
Section: Two-dimensional semantic space. Analogous to the structure of the neocortex, Semantic Folding theory posits the implementation of a semantic space as a two-dimensional grid. This grid is populated by context-vectors in such a way as to place similar context-vectors closer to each other, for instance, by using ... | Wikipedia - Semantic folding - Two-dimensional semantic space | 312 | 1,539 | null |
Some properties of word-SDRs that are of particular interest with respect to computational semantics are: high noise resistance: As a result of similar contexts being placed closer together in the underlying map, word-SDRs are highly tolerant of false or shifted "bits". boolean logic: It is possible to manipulate word-... | Wikipedia - Semantic folding - Two-dimensional semantic space | 219 | 1,042 | null |
Section: Semantic spaces. Semantic spaces in the natural language domain aim to create representations of natural language that are capable of capturing meaning. The original motivation for semantic spaces stems from two core challenges of natural language: Vocabulary mismatch (the fact that the same meaning can be exp... | Wikipedia - Semantic folding - Semantic spaces | 320 | 1,679 | null |
"spider-web", "lighter-cigarette", as opposed to synonymous relations such as "whale-dolphin", "astronaut-driver") was achieved by explicit semantic analysis (ESA) in 2007. ESA was a novel (non-machine learning) based approach that represented words in the form of vectors with 100,000 dimensions (where each dimension r... | Wikipedia - Semantic folding - Semantic spaces | 208 | 1,075 | null |
Section: Visualization. The topological distribution over a two-dimensional grid (outlined above) lends itself to a bitmap type visualization of the semantics of any word or text, where each active semantic feature can be displayed as e.g. a pixel. As can be seen in the images shown here, this representation allows for... | Wikipedia - Semantic folding - Visualization | 262 | 1,288 | null |
Article: Weak supervision. Weak supervision (also known as semi-supervised learning) is a paradigm in machine learning, the relevance and notability of which increased with the advent of large language models due to large amount of data required to train them. It is characterized by using a combination of a small amoun... | Wikipedia - Weak supervision - Summary | 209 | 1,010 | null |
Section: Technique. More formally, semi-supervised learning assumes a set of l {\displaystyle l} independently identically distributed examples x 1 , … , x l ∈ X {\displaystyle x_{1},\dots ,x_{l}\in X} with corresponding labels y 1 , … , y l ∈ Y {\displaystyle y_{1},\dots ,y_{l}\in Y} and u {\displaystyle u} unlabeled ... | Wikipedia - Weak supervision - Technique | 298 | 1,022 | null |
Section: Assumptions > Manifold assumption. The data lie approximately on a manifold of much lower dimension than the input space. In this case learning the manifold using both the labeled and unlabeled data can avoid the curse of dimensionality. Then learning can proceed using distances and densities defined on the ma... | Wikipedia - Weak supervision - Assumptions > Manifold assumption | 157 | 833 | null |
Section: Methods > Generative models. Generative approaches to statistical learning first seek to estimate p ( x | y ) {\displaystyle p(x|y)} , the distribution of data points belonging to each class. The probability p ( y | x ) {\displaystyle p(y|x)} that a given point x {\displaystyle x} has label y {\displaystyle y}... | Wikipedia - Weak supervision - Methods > Generative models | 339 | 1,457 | null |
Gaussian mixture distributions are identifiable and commonly used for generative models. The parameterized joint distribution can be written as p ( x , y | θ ) = p ( y | θ ) p ( x | y , θ ) {\displaystyle p(x,y|\theta )=p(y|\theta )p(x|y,\theta )} by using the chain rule. Each parameter vector θ {\displaystyle \theta }... | Wikipedia - Weak supervision - Methods > Generative models | 333 | 869 | null |
Section: Methods > Low-density separation. Another major class of methods attempts to place boundaries in regions with few data points (labeled or unlabeled). One of the most commonly used algorithms is the transductive support vector machine, or TSVM (which, despite its name, may be used for inductive learning as well... | Wikipedia - Weak supervision - Methods > Low-density separation | 231 | 864 | null |
In addition to the standard hinge loss ( 1 − y f ( x ) ) + {\displaystyle (1-yf(x))_{+}} for labeled data, a loss function ( 1 − | f ( x ) | ) + {\displaystyle (1-|f(x)|)_{+}} is introduced over the unlabeled data by letting y = sign f ( x ) {\displaystyle y=\operatorname {sign} {f(x)}} . TSVM then selects f ∗ ( x ) ... | Wikipedia - Weak supervision - Methods > Low-density separation | 433 | 981 | null |
TSVM then selects f ∗ ( x ) = h ∗ ( x ) + b {\displaystyle f^{*}(x)=h^{*}(x)+b} from a reproducing kernel Hilbert space H {\displaystyle {\mathcal {H}}} by minimizing the regularized empirical risk: f ∗ = argmin f ( ∑ i = 1 l ( 1 − y i f ( x i ) ) + + λ 1 ‖ h ‖ H 2 + λ 2 ∑ i = l + 1 l + u ( 1 − | f ( x i ) | ) + ) {\di... | Wikipedia - Weak supervision - Methods > Low-density separation | 358 | 865 | null |
Section: Methods > Laplacian regularization. Laplacian regularization has been historically approached through graph-Laplacian. Graph-based methods for semi-supervised learning use a graph representation of the data, with a node for each labeled and unlabeled example. The graph may be constructed using domain knowledge... | Wikipedia - Weak supervision - Methods > Laplacian regularization | 265 | 1,031 | null |
A term is added to the standard Tikhonov regularization problem to enforce smoothness of the solution relative to the manifold (in the intrinsic space of the problem) as well as relative to the ambient input space. The minimization problem becomes argmin f ∈ H ( 1 l ∑ i = 1 l V ( f ( x i ) , y i ) + λ A ‖ f ‖ H 2 + λ I... | Wikipedia - Weak supervision - Methods > Laplacian regularization | 347 | 933 | null |
The regularization parameters λ A {\displaystyle \lambda _{A}} and λ I {\displaystyle \lambda _{I}} control smoothness in the ambient and intrinsic spaces respectively. The graph is used to approximate the intrinsic regularization term. Defining the graph Laplacian L = D − W {\displaystyle L=D-W} where D i i = ∑ j = 1 ... | Wikipedia - Weak supervision - Methods > Laplacian regularization | 332 | 766 | null |
Defining the graph Laplacian L = D − W {\displaystyle L=D-W} where D i i = ∑ j = 1 l + u W i j {\displaystyle D_{ii}=\sum _{j=1}^{l+u}W_{ij}} and f {\displaystyle \mathbf {f} } is the vector [ f ( x 1 ) … f ( x l + u ) ] {\displaystyle [f(x_{1})\dots f(x_{l+u})]} , we have f T L f = ∑ i , j = 1 l + u W i j ( f i − f j ... | Wikipedia - Weak supervision - Methods > Laplacian regularization | 345 | 855 | null |
Section: Methods > Heuristic approaches. Some methods for semi-supervised learning are not intrinsically geared to learning from both unlabeled and labeled data, but instead make use of unlabeled data within a supervised learning framework. For instance, the labeled and unlabeled examples x 1 , … , x l + u {\displaysty... | Wikipedia - Weak supervision - Methods > Heuristic approaches | 323 | 1,583 | null |
Section: In human cognition. Human responses to formal semi-supervised learning problems have yielded varying conclusions about the degree of influence of the unlabeled data. More natural learning problems may also be viewed as instances of semi-supervised learning. Much of human concept learning involves a small amoun... | Wikipedia - Weak supervision - In human cognition | 154 | 804 | null |
Article: Sequence labeling. In machine learning, sequence labeling is a type of pattern recognition task that involves the algorithmic assignment of a categorical label to each member of a sequence of observed values. A common example of a sequence labeling task is part of speech tagging, which seeks to assign a part o... | Wikipedia - Sequence labeling - Summary | 342 | 1,606 | null |
In "he sets and then knocks over the table", only the word "he" to the left is helpful (cf. "...picks up the sets and then knocks over..."). Conversely, in "... and also sets the table" only the word "the" to the right is helpful (cf. "... and also sets of books were ..."). An algorithm that proceeds from left to right... | Wikipedia - Sequence labeling - Summary | 248 | 1,094 | null |
Section: Learning setup. There are four common setups for similarity and metric distance learning. Regression similarity learning In this setup, pairs of objects are given ( x i 1 , x i 2 ) {\displaystyle (x_{i}^{1},x_{i}^{2})} together with a measure of their similarity y i ∈ R {\displaystyle y_{i}\in R} . The goal is... | Wikipedia - Similarity learning - Learning setup | 293 | 738 | null |
This is typically achieved by minimizing a regularized loss min W ∑ i l o s s ( w ; x i 1 , x i 2 , y i ) + r e g ( w ) {\displaystyle \min _{W}\sum _{i}loss(w;x_{i}^{1},x_{i}^{2},y_{i})+reg(w)} . Classification similarity learning Given are pairs of similar objects ( x i , x i + ) {\displaystyle (x_{i},x_{i}^{+})} and... | Wikipedia - Similarity learning - Learning setup | 278 | 735 | null |
The goal is again to learn a classifier that can decide if a new pair of objects is similar or not. Ranking similarity learning Given are triplets of objects ( x i , x i + , x i − ) {\displaystyle (x_{i},x_{i}^{+},x_{i}^{-})} whose relative similarity obey a predefined order: x i {\displaystyle x_{i}} is known to be mo... | Wikipedia - Similarity learning - Learning setup | 346 | 1,162 | null |
Locality sensitive hashing (LSH) Hashes input items so that similar items map to the same "buckets" in memory with high probability (the number of buckets being much smaller than the universe of possible input items). It is often applied in nearest neighbor search on large-scale high-dimensional data, e.g., image datab... | Wikipedia - Similarity learning - Learning setup | 189 | 797 | null |
Section: Metric learning. Similarity learning is closely related to distance metric learning. Metric learning is the task of learning a distance function over objects. A metric or distance function has to obey four axioms: non-negativity, identity of indiscernibles, symmetry and subadditivity (or the triangle inequalit... | Wikipedia - Similarity learning - Metric learning | 290 | 944 | null |
When W {\displaystyle W} is a symmetric positive definite matrix, D W {\displaystyle D_{W}} is a metric. Moreover, as any symmetric positive semi-definite matrix W ∈ S + d {\displaystyle W\in S_{+}^{d}} can be decomposed as W = L ⊤ L {\displaystyle W=L^{\top }L} where L ∈ R e × d {\displaystyle L\in R^{e\times d}} and ... | Wikipedia - Similarity learning - Metric learning | 292 | 644 | null |
Moreover, as any symmetric positive semi-definite matrix W ∈ S + d {\displaystyle W\in S_{+}^{d}} can be decomposed as W = L ⊤ L {\displaystyle W=L^{\top }L} where L ∈ R e × d {\displaystyle L\in R^{e\times d}} and e ≥ r a n k ( W ) {\displaystyle e\geq rank(W)} , the distance function D W {\displaystyle D_{W}} can be ... | Wikipedia - Similarity learning - Metric learning | 400 | 833 | null |
The distance D W ( x 1 , x 2 ) 2 = ‖ x 1 ′ − x 2 ′ ‖ 2 2 {\displaystyle D_{W}(x_{1},x_{2})^{2}=\|x_{1}'-x_{2}'\|_{2}^{2}} corresponds to the Euclidean distance between the transformed feature vectors x 1 ′ = L x 1 {\displaystyle x_{1}'=Lx_{1}} and x 2 ′ = L x 2 {\displaystyle x_{2}'=Lx_{2}} . Many formulations for metr... | Wikipedia - Similarity learning - Metric learning | 214 | 683 | null |
Section: Background. The field of socially assistive robotics emerged in the early 2000s, following the emergence of the field of social robots. In contrast to social robots, SARs aid users with specific goals related to behavior change rather than serving as purely social entities. The term "Socially assistive robot" ... | Wikipedia - Socially assistive robot - Background | 173 | 939 | null |
Section: Characteristics of interaction. SARs rely on artificial intelligence (AI) to generate real-time, responsive, natural, and meaningful robot behaviors during interactions with humans. The robots employ various forms of communication, such as facial expressions, gestures, body movements, and speech. In contrast t... | Wikipedia - Socially assistive robot - Characteristics of interaction | 341 | 1,844 | null |
Article: Lynda Soderholm. Lynda Soderholm is a physical chemist at the U.S. Department of Energy's (DOE) Argonne National Laboratory with a specialty in f-block elements. She is a senior scientist and the lead of the Actinide, Geochemistry & Separation Sciences Theme within Argonne's Chemical Sciences and Engineering D... | Wikipedia - Lynda Soderholm - Summary | 176 | 829 | null |
Section: Early life and education. Soderholm was awarded her PhD in 1982 by McMaster University under the direction of Prof John Greedan. Her dissertation focused on characterizing the structural and magnetic properties of a series of ternary f-ion oxides. After graduating, she was awarded a NATO postdoctoral fellow at... | Wikipedia - Lynda Soderholm - Early life and education | 157 | 803 | null |
Section: Select publications. Beno, M. A.; Soderholm, L.; Capone, D. W., II; Hinks, D. G.; Jorgensen, J. D.; Grace, J. D.; Schuller, I. K.; Segre, C. U.; Zhang, K., Structure of the single-phase high-temperature superconductor yttrium barium copper oxide (YBa2Cu3O7−δ). Appl. Phys. Lett. 1987, 51 (1), 57–9. Soderholm, L... | Wikipedia - Lynda Soderholm - Select publications | 346 | 854 | null |
Forbes, T. Z.; Burns, P. C.; Skanthakumar, S.; Soderholm, L., Synthesis, structure, and magnetism of Np2O5. J. Am. Chem. Soc. 2007, 129 (10), 2760–2761. Soderholm, L.; Almond, P. M.; Skanthakumar, S.; Wilson, R. E.; Burns, P. C., The structure of the plutonium oxide nanocluster [Pu38O56Cl54(H2O)8]14-. Angew. Chem., Int... | Wikipedia - Lynda Soderholm - Select publications | 325 | 775 | null |
2011, 50 (47), 11234–11237. Knope, K. E.; Soderholm, L., Solution and solid-state structural chemistry of actinide hydrates and their hydrolysis and condensation products. Chem. Rev. 2013, 113 (2), 944–994. Luo, G.; Bu, W.; Mihaylov, M.; Kuzmenko, I.; Schlossman, M. L.; Soderholm, L., X-ray reflectivity reveals a nonmo... | Wikipedia - Lynda Soderholm - Select publications | 243 | 681 | null |
Article: Solomonoff's theory of inductive inference. Solomonoff's theory of inductive inference proves that, under its common sense assumptions (axioms), the best possible scientific model is the shortest algorithm that generates the empirical data under consideration. In addition to the choice of data, other assumptio... | Wikipedia - Solomonoff's theory of inductive inference - Summary | 349 | 1,764 | null |
Section: Origin > Principle. Solomonoff's induction has been argued to be the computational formalization of pure Bayesianism. To understand, recall that Bayesianism derives the posterior probability P [ T | D ] {\displaystyle \mathbb {P} [T|D]} of a theory T {\displaystyle T} given data D {\displaystyle D} by applying... | Wikipedia - Solomonoff's theory of inductive inference - Origin > Principle | 349 | 1,009 | null |
In other words, any theory must define a probability distribution over observable data D {\displaystyle D} . Solomonoff's induction essentially boils down to demanding that all such probability distributions be computable. Interestingly, the set of computable probability distributions is a subset of the set of all prog... | Wikipedia - Solomonoff's theory of inductive inference - Origin > Principle | 277 | 981 | null |
Namely, we have P [ F | D ] = E T [ P [ F | T , D ] ] = ∑ T P [ F | T , D ] P [ T | D ] {\displaystyle \mathbb {P} [F|D]=\mathbb {E} _{T}[\mathbb {P} [F|T,D]]=\sum _{T}\mathbb {P} [F|T,D]\mathbb {P} [T|D]} . This quantity can be interpreted as the average predictions P [ F | T , D ] {\displaystyle \mathbb {P} [F|T,D]} ... | Wikipedia - Solomonoff's theory of inductive inference - Origin > Principle | 201 | 479 | null |
Section: Origin > Mathematical. The proof of the "razor" is based on the known mathematical properties of a probability distribution over a countable set. These properties are relevant because the infinite set of all programs is a denumerable set. The sum S of the probabilities of all programs must be exactly equal to ... | Wikipedia - Solomonoff's theory of inductive inference - Origin > Mathematical | 264 | 1,275 | null |
Section: Modern applications > Artificial intelligence. Though Solomonoff's inductive inference is not computable, several AIXI-derived algorithms approximate it in order to make it run on a modern computer. The more computing power they are given, the closer their predictions are to the predictions of inductive infere... | Wikipedia - Solomonoff's theory of inductive inference - Modern applications > Artificial intelligence | 318 | 1,434 | null |
Section: Usage. POI recommendation - generating personalized point of interest recommendations based on user preferences. Next/future location prediction - prediction of the next location a person will go to based on their historical trajectory. Zone functions classification - based on different mobility of people or P... | Wikipedia - Spatial embedding - Usage | 194 | 1,122 | null |
Section: Temporal aspect. Some of the data analyzed has a timestamp associated with it. In some cases of data analysis this information is omitted and in others it is used to divide the set into groups. The most common division is the separation of weekdays from weekends or division into hours of the day. This is parti... | Wikipedia - Spatial embedding - Temporal aspect | 162 | 837 | null |
Section: Model description. Suppose we have P possible predictors in some model. Vector γ has a length equal to P and consists of zeros and ones. This vector indicates whether a particular variable is included in the regression or not. If no specific prior information on initial inclusion probabilities of particular va... | Wikipedia - Spike-and-slab regression - Model description | 334 | 1,535 | null |
As a result, we obtain a posterior distribution of γ (variable inclusion in the model), β (regression coefficient values) and the corresponding prediction of y. The model got its name (spike-and-slab) due to the shape of the two prior distributions. The "spike" is the probability of a particular coefficient in the mode... | Wikipedia - Spike-and-slab regression - Model description | 221 | 1,017 | null |
Article: Stability (learning theory). Stability, also known as algorithmic stability, is a notion in computational learning theory of how a machine learning algorithm output is changed with small perturbations to its inputs. A stable learning algorithm is one for which the prediction does not change much when the train... | Wikipedia - Stability (learning theory) - Summary | 255 | 1,370 | null |
Section: History. A central goal in designing a machine learning system is to guarantee that the learning algorithm will generalize, or perform accurately on new examples after being trained on a finite number of them. In the 1990s, milestones were reached in obtaining generalization bounds for supervised learning algo... | Wikipedia - Stability (learning theory) - History | 340 | 1,598 | null |
Vapnik's work, using what became known as VC theory, established a relationship between generalization of a learning algorithm and properties of the hypothesis space H {\displaystyle H} of functions being learned. However, these results could not be applied to algorithms with hypothesis spaces of unbounded VC-dimension... | Wikipedia - Stability (learning theory) - History | 289 | 1,543 | null |
Section: Summary of classic results. Early 1900s - Stability in learning theory was earliest described in terms of continuity of the learning map L {\displaystyle L} , traced to Andrey Nikolayevich Tikhonov. 1979 - Devroye and Wagner observed that the leave-one-out behavior of an algorithm is related to its sensitivity... | Wikipedia - Stability (learning theory) - Summary of classic results | 329 | 1,665 | null |
Section: Preliminary definitions. We define several terms related to learning algorithms training sets, so that we can then define stability in multiple ways and present theorems from the field. A machine learning algorithm, also known as a learning map L {\displaystyle L} , maps a training data set, which is a set of ... | Wikipedia - Stability (learning theory) - Preliminary definitions | 298 | 979 | null |
from an unknown distribution D. Thus, the learning map L {\displaystyle L} is defined as a mapping from Z m {\displaystyle Z_{m}} into H {\displaystyle H} , mapping a training set S {\displaystyle S} onto a function f S {\displaystyle f_{S}} from X {\displaystyle X} to Y {\displaystyle Y} . Here, we consider only deter... | Wikipedia - Stability (learning theory) - Preliminary definitions | 305 | 926 | null |
The empirical error of f {\displaystyle f} is I S [ f ] = 1 n ∑ V ( f , z i ) {\displaystyle I_{S}[f]={\frac {1}{n}}\sum V(f,z_{i})} . The true error of f {\displaystyle f} is I [ f ] = E z V ( f , z ) {\displaystyle I[f]=\mathbb {E} _{z}V(f,z)} Given a training set S of size m, we will build, for all i = 1....,m, modi... | Wikipedia - Stability (learning theory) - Preliminary definitions | 338 | 677 | null |
Section: Definitions of stability > Point-wise Hypothesis Stability. An algorithm L {\displaystyle L} has point-wise hypothesis stability β with respect to the loss function V if the following holds: ∀ i ∈ { 1 , . . . , m } , E S [ | V ( f S , z i ) − V ( f S | i , z i ) | ] ≤ β . {\displaystyle \forall i\in \ \{1,...,... | Wikipedia - Stability (learning theory) - Definitions of stability > Point-wise Hypothesis Stability | 159 | 391 | null |
Section: Definitions of stability > Error Stability. An algorithm L {\displaystyle L} has error stability β with respect to the loss function V if the following holds: ∀ S ∈ Z m , ∀ i ∈ { 1 , . . . , m } , | E z [ V ( f S , z ) ] − E z [ V ( f S | i , z ) ] | ≤ β {\displaystyle \forall S\in Z^{m},\forall i\in \{1,...,m... | Wikipedia - Stability (learning theory) - Definitions of stability > Error Stability | 174 | 399 | null |
Section: Definitions of stability > Uniform Stability. An algorithm L {\displaystyle L} has uniform stability β with respect to the loss function V if the following holds: ∀ S ∈ Z m , ∀ i ∈ { 1 , . . . , m } , sup z ∈ Z | V ( f S , z ) − V ( f S | i , z ) | ≤ β {\displaystyle \forall S\in Z^{m},\forall i\in \{1,...,m\}... | Wikipedia - Stability (learning theory) - Definitions of stability > Uniform Stability | 184 | 456 | null |
. . , m } , P S { sup z ∈ Z | V ( f S , z ) − V ( f S | i , z ) | ≤ β } ≥ 1 − δ {\displaystyle \forall S\in Z^{m},\forall i\in \{1,...,m\},\mathbb {P} _{S}\{\sup _{z\in Z}|V(f_{S},z)-V(f_{S^{|i}},z)|\leq \beta \}\geq 1-\delta } An algorithm is said to be stable, when the value of β {\displaystyle \beta } decreases as O... | Wikipedia - Stability (learning theory) - Definitions of stability > Uniform Stability | 186 | 364 | null |
Section: Definitions of stability > Leave-one-out cross-validation (CVloo) Stability. An algorithm L {\displaystyle L} has CVloo stability β with respect to the loss function V if the following holds: ∀ i ∈ { 1 , . . . , m } , P S { | V ( f S , z i ) − V ( f S | i , z i ) | ≤ β C V } ≥ 1 − δ C V {\displaystyle \forall ... | Wikipedia - Stability (learning theory) - Definitions of stability > Leave-one-out cross-validation (CVloo) Stability | 209 | 527 | null |
, m } , P S { | I [ f S ] − 1 m ∑ i = 1 m V ( f S | i , z i ) | ≤ β E L m } ≥ 1 − δ E L m {\displaystyle \forall i\in \{1,...,m\},\mathbb {P} _{S}\{|I[f_{S}]-{\frac {1}{m}}\sum _{i=1}^{m}V(f_{S^{|i}},z_{i})|\leq \beta _{EL}^{m}\}\geq 1-\delta _{EL}^{m}} , with β E L m {\displaystyle \beta _{EL}^{m}} and δ E L m {\displ... | Wikipedia - Stability (learning theory) - Definitions of stability > Expected-leave-one-out error ( | 227 | 409 | null |
Section: Classic theorems. From Bousquet and Elisseeff (02): For symmetric learning algorithms with bounded loss, if the algorithm has Uniform Stability with the probabilistic definition above, then the algorithm generalizes. Uniform Stability is a strong condition which is not met by all algorithms but is, surprisingl... | Wikipedia - Stability (learning theory) - Classic theorems | 277 | 1,320 | null |
Section: Algorithms that are stable. This is a list of algorithms that have been shown to be stable, and the article where the associated generalization bounds are provided. Linear regression k-NN classifier with a {0-1} loss function. Support Vector Machine (SVM) classification with a bounded kernel and where the regu... | Wikipedia - Stability (learning theory) - Algorithms that are stable | 182 | 856 | null |
Section: Introduction. The goals of learning are understanding and prediction. Learning falls into many categories, including supervised learning, unsupervised learning, online learning, and reinforcement learning. From the perspective of statistical learning theory, supervised learning is best understood. Supervised l... | Wikipedia - Statistical learning theory - Introduction | 327 | 1,685 | null |
Section: Formal description. Take X {\displaystyle X} to be the vector space of all possible inputs, and Y {\displaystyle Y} to be the vector space of all possible outputs. Statistical learning theory takes the perspective that there is some unknown probability distribution over the product space Z = X × Y {\displaysty... | Wikipedia - Statistical learning theory - Formal description | 299 | 881 | null |
The training set is made up of n {\displaystyle n} samples from this probability distribution, and is notated S = { ( x 1 , y 1 ) , … , ( x n , y n ) } = { z 1 , … , z n } {\displaystyle S=\{(\mathbf {x} _{1},y_{1}),\dots ,(\mathbf {x} _{n},y_{n})\}=\{\mathbf {z} _{1},\dots ,\mathbf {z} _{n}\}} Every x i {\displaystyle... | Wikipedia - Statistical learning theory - Formal description | 293 | 825 | null |
The hypothesis space is the space of functions the algorithm will search through. Let V ( f ( x ) , y ) {\displaystyle V(f(\mathbf {x} ),y)} be the loss function, a metric for the difference between the predicted value f ( x ) {\displaystyle f(\mathbf {x} )} and the actual value y {\displaystyle y} . The expected risk ... | Wikipedia - Statistical learning theory - Formal description | 323 | 977 | null |
Section: Loss functions > Regression. The most common loss function for regression is the square loss function (also known as the L2-norm). This familiar loss function is used in Ordinary Least Squares regression. The form is: V ( f ( x ) , y ) = ( y − f ( x ) ) 2 {\displaystyle V(f(\mathbf {x} ),y)=(y-f(\mathbf {x} ))... | Wikipedia - Statistical learning theory - Loss functions > Regression | 172 | 494 | null |
Section: Loss functions > Classification. In some sense the 0-1 indicator function is the most natural loss function for classification. It takes the value 0 if the predicted output is the same as the actual output, and it takes the value 1 if the predicted output is different from the actual output. For binary classif... | Wikipedia - Statistical learning theory - Loss functions > Classification | 161 | 552 | null |
Section: Regularization. In machine learning problems, a major problem that arises is that of overfitting. Because learning is a prediction problem, the goal is not to find a function that most closely fits the (previously observed) data, but to find one that will most accurately predict output from future input. Empir... | Wikipedia - Statistical learning theory - Regularization | 321 | 1,520 | null |
One example of regularization is Tikhonov regularization. This consists of minimizing 1 n ∑ i = 1 n V ( f ( x i ) , y i ) + γ ‖ f ‖ H 2 {\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}V(f(\mathbf {x} _{i}),y_{i})+\gamma \left\|f\right\|_{\mathcal {H}}^{2}} where γ {\displaystyle \gamma } is a fixed and positive parameter, ... | Wikipedia - Statistical learning theory - Regularization | 158 | 435 | null |
Section: Bounding empirical risk. Consider a binary classifier f : X → { 0 , 1 } {\displaystyle f:{\mathcal {X}}\to \{0,1\}} . We can apply Hoeffding's inequality to bound the probability that the empirical risk deviates from the true risk to be a Sub-Gaussian distribution. P ( | R ^ ( f ) − R ( f ) | ≥ ϵ ) ≤ 2 e − 2 n... | Wikipedia - Statistical learning theory - Bounding empirical risk | 206 | 636 | null |
Therefore, a more useful result is to bound the probability of the supremum of the difference over the whole class. P ( sup f ∈ F | R ^ ( f ) − R ( f ) | ≥ ϵ ) ≤ 2 S ( F , n ) e − n ϵ 2 / 8 ≈ n d e − n ϵ 2 / 8 {\displaystyle \mathbb {P} {\bigg (}\sup _{f\in {\mathcal {F}}}|{\hat {R}}(f)-R(f)|\geq \epsilon {\bigg )}\leq... | Wikipedia - Statistical learning theory - Bounding empirical risk | 264 | 688 | null |
Article: Statistical relational learning. Statistical relational learning (SRL) is a subdiscipline of artificial intelligence and machine learning that is concerned with domain models that exhibit both uncertainty (which can be dealt with using statistical methods) and complex, relational structure. Typically, the know... | Wikipedia - Statistical relational learning - Summary | 246 | 1,348 | null |
Section: Canonical tasks. A number of canonical tasks are associated with statistical relational learning, the most common ones being. collective classification, i.e. the (simultaneous) prediction of the class of several objects given objects' attributes and their relations link prediction, i.e. predicting whether or n... | Wikipedia - Statistical relational learning - Canonical tasks | 171 | 884 | null |
Section: Representation formalisms. One of the fundamental design goals of the representation formalisms developed in SRL is to abstract away from concrete entities and to represent instead general principles that are intended to be universally applicable. Since there are countless ways in which such principles can be ... | Wikipedia - Statistical relational learning - Representation formalisms | 162 | 904 | null |
Article: Structural risk minimization. Structural risk minimization (SRM) is an inductive principle of use in machine learning. Commonly in machine learning, a generalized model must be selected from a finite data set, with the consequent problem of overfitting – the model becoming too strongly tailored to the particul... | Wikipedia - Structural risk minimization - Summary | 346 | 1,442 | null |
The SRM problem can be formulated in terms of data. Given n data points consisting of data x and labels y, the objective J ( θ ) {\displaystyle J(\theta )} is often expressed in the following manner: J ( θ ) = 1 2 n ∑ i = 1 n ( h θ ( x i ) − y i ) 2 + λ 2 ∑ j = 1 d θ j 2 {\displaystyle J(\theta )={\frac {1}{2n}}\sum _{... | Wikipedia - Structural risk minimization - Summary | 309 | 918 | null |
Article: Structured sparsity regularization. Structured sparsity regularization is a class of methods, and an area of research in statistical learning theory, that extend and generalize sparsity regularization learning methods. Both sparsity and structured sparsity regularization methods seek to exploit the assumption ... | Wikipedia - Structured sparsity regularization - Summary | 317 | 1,579 | null |
Section: Definition and related concepts > Sparsity regularization. Consider the linear kernel regularized empirical risk minimization problem with a loss function V ( y i , f ( x ) ) {\displaystyle V(y_{i},f(x))} and the ℓ 0 {\displaystyle \ell _{0}} "norm" as the regularization penalty: min w ∈ R d 1 n ∑ i = 1 n V ( ... | Wikipedia - Structured sparsity regularization - Definition and related concepts > Sparsity regularization | 348 | 843 | null |
f ( x ) = ⟨ w , x i ⟩ {\displaystyle f(x)=\langle w,x_{i}\rangle } is said to be sparse if ‖ w ‖ 0 = s < d {\displaystyle \|w\|_{0}=s<d} . Which means that the output Y {\displaystyle Y} can be described by a small subset of input variables. More generally, assume a dictionary ϕ j : X → R {\displaystyle \phi _{j}:X\rig... | Wikipedia - Structured sparsity regularization - Definition and related concepts > Sparsity regularization | 339 | 840 | null |
. . , p } } | {\displaystyle \|w\|_{0}=|\{j|w_{j}\neq 0,j\in \{1,...,p\}\}|} , where | A | {\displaystyle |A|} is the cardinality of set A {\displaystyle A} . f {\displaystyle f} is said to be sparse if ‖ f ‖ 0 = ‖ w ‖ 0 = s < d {\displaystyle \|f\|_{0}=\|w\|_{0}=s<d} . However, while using the ℓ 0 {\displaystyle \ell ... | Wikipedia - Structured sparsity regularization - Definition and related concepts > Sparsity regularization | 221 | 637 | null |
Section: Definition and related concepts > Structured sparsity regularization. Structured sparsity regularization extends and generalizes the variable selection problem that characterizes sparsity regularization. Consider the above regularized empirical risk minimization problem with a general kernel and associated fea... | Wikipedia - Structured sparsity regularization - Definition and related concepts > Structured sparsity regularization | 344 | 1,188 | null |
Section: Structures and norms > Non-overlapping groups: group Lasso. The non-overlapping group case is the most basic instance of structured sparsity. In it, an a priori partition of the coefficient vector w {\displaystyle w} in G {\displaystyle G} non-overlapping groups is assumed. Let w g {\displaystyle w_{g}} be the... | Wikipedia - Structured sparsity regularization - Structures and norms > Non-overlapping groups: group Lasso | 346 | 899 | null |
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