Pull technology

Pull technology

Pull coding or client pull is a style of network communication, where the initial request for data originates from the client, and then is responded to by the server. The reverse is known as push technology, where the server pushes data to clients. Pull requests form the foundation of network computing, where many clients request data from centralized servers. Pull is used extensively on the Internet for HTTP page requests from websites. A push can also be simulated using multiple pulls within a short amount of time. For example, when pulling POP3 email messages from a server, a client can make regular pull requests, every few minutes. To the user, the email then appears to be pushed, as emails appear to arrive close to real-time. A trade-off of this system is that it places a heavier load on both the server and network to function correctly. Many web feeds, such as RSS are technically pulled by the client. With RSS, the user's RSS reader polls the server periodically for new content; the server does not send information to the client unrequested. This continual polling is inefficient and has contributed to the shutdown or reduction of several popular RSS feeds that could not handle the bandwidth. For solving this problem, the WebSub protocol, as another example of a push code, was devised. Podcasting is specifically a pull technology. When a new podcast episode is published to an RSS feed, it sits on the server until it is requested by a feed reader, mobile podcasting app, or directory. Directories such as Apple Podcasts (iTunes), The Blubrry Directory, and many apps' directories request the RSS feed periodically to update the Podcast's listing on those platforms. Subscribers to those RSS feeds via app or reader will get the episodes when they request the RSS feed next time, independent of when the directory listing updates.

Information extraction

Information extraction (IE) is the task of automatically extracting structured information from unstructured and/or semi-structured machine-readable documents and other electronically represented sources. Typically, this involves processing human language texts by means of natural language processing (NLP). Recent activities in multimedia document processing like automatic annotation and content extraction out of images/audio/video/documents could be seen as information extraction. Recent advances in NLP techniques have allowed for significantly improved performance compared to previous years. An example is the extraction from newswire reports of corporate mergers, such as denoted by the formal relation: MergerBetween ⁡ ( c o m p a n y 1 , c o m p a n y 2 , d a t e ) {\displaystyle \operatorname {MergerBetween} (\mathrm {company} _{1},\mathrm {company} _{2},\mathrm {date} )} , from an online news sentence such as: "Yesterday, New York based Foo Inc. announced their acquisition of Bar Corp." A broad goal of IE is to allow computation to be done on the previously unstructured data. A more specific goal is to allow automated reasoning about the logical form of the input data. Structured data is semantically well-defined data from a chosen target domain, interpreted with respect to category and context. Information extraction is the part of a greater puzzle which deals with the problem of devising automatic methods for text management, beyond its transmission, storage and display. The discipline of information retrieval (IR) has developed automatic methods, typically of a statistical flavor, for indexing large document collections and classifying documents. Another complementary approach is that of natural language processing (NLP) which has solved the problem of modelling human language processing with considerable success when taking into account the magnitude of the task. In terms of both difficulty and emphasis, IE deals with tasks in between both IR and NLP. In terms of input, IE assumes the existence of a set of documents in which each document follows a template, i.e. describes one or more entities or events in a manner that is similar to those in other documents but differing in the details. An example, consider a group of newswire articles on Latin American terrorism with each article presumed to be based upon one or more terroristic acts. We also define for any given IE task a template, which is a(or a set of) case frame(s) to hold the information contained in a single document. For the terrorism example, a template would have slots corresponding to the perpetrator, victim, and weapon of the terroristic act, and the date on which the event happened. An IE system for this problem is required to "understand" an attack article only enough to find data corresponding to the slots in this template. == History == Information extraction dates back to the late 1970s in the early days of NLP. An early commercial system from the mid-1980s was JASPER built for Reuters by the Carnegie Group Inc with the aim of providing real-time financial news to financial traders. Beginning in 1987, IE was spurred by a series of Message Understanding Conferences. MUC is a competition-based conference that focused on the following domains: MUC-1 (1987), MUC-3 (1989): Naval operations messages. MUC-3 (1991), MUC-4 (1992): Terrorism in Latin American countries. MUC-5 (1993): Joint ventures and microelectronics domain. MUC-6 (1995): News articles on management changes. MUC-7 (1998): Satellite launch reports. Considerable support came from the U.S. Defense Advanced Research Projects Agency (DARPA), who wished to automate mundane tasks performed by government analysts, such as scanning newspapers for possible links to terrorism. == Present significance == The present significance of IE pertains to the growing amount of information available in unstructured form. Tim Berners-Lee, inventor of the World Wide Web, refers to the existing Internet as the web of documents and advocates that more of the content be made available as a web of data. Until this transpires, the web largely consists of unstructured documents lacking semantic metadata. Knowledge contained within these documents can be made more accessible for machine processing by means of transformation into relational form, or by marking-up with XML tags. An intelligent agent monitoring a news data feed requires IE to transform unstructured data into something that can be reasoned with. A typical application of IE is to scan a set of documents written in a natural language and populate a database with the information extracted. == Tasks and subtasks == Applying information extraction to text is linked to the problem of text simplification in order to create a structured view of the information present in free text. The overall goal being to create a more easily machine-readable text to process the sentences. Typical IE tasks and subtasks include: Template filling: Extracting a fixed set of fields from a document, e.g. extract perpetrators, victims, time, etc. from a newspaper article about a terrorist attack. Event extraction: Given an input document, output zero or more event templates. For instance, a newspaper article might describe multiple terrorist attacks. Knowledge Base Population: Fill a database of facts given a set of documents. Typically the database is in the form of triplets, (entity 1, relation, entity 2), e.g. (Barack Obama, Spouse, Michelle Obama) Named entity recognition: recognition of known entity names (for people and organizations), place names, temporal expressions, and certain types of numerical expressions, by employing existing knowledge of the domain or information extracted from other sentences. Typically the recognition task involves assigning a unique identifier to the extracted entity. A simpler task is named entity detection, which aims at detecting entities without having any existing knowledge about the entity instances. For example, in processing the sentence "M. Smith likes fishing", named entity detection would denote detecting that the phrase "M. Smith" does refer to a person, but without necessarily having (or using) any knowledge about a certain M. Smith who is (or, "might be") the specific person whom that sentence is talking about. Coreference resolution: detection of coreference and anaphoric links between text entities. In IE tasks, this is typically restricted to finding links between previously extracted named entities. For example, "International Business Machines" and "IBM" refer to the same real-world entity. If we take the two sentences "M. Smith likes fishing. But he doesn't like biking", it would be beneficial to detect that "he" is referring to the previously detected person "M. Smith". Relationship extraction: identification of relations between entities, such as: PERSON works for ORGANIZATION (extracted from the sentence "Bill works for IBM.") PERSON located in LOCATION (extracted from the sentence "Bill is in France.") Semi-structured information extraction which may refer to any IE that tries to restore some kind of information structure that has been lost through publication, such as: Table extraction: finding and extracting tables from documents. Table information extraction : extracting information in structured manner from the tables. This task is more complex than table extraction, as table extraction is only the first step, while understanding the roles of the cells, rows, columns, linking the information inside the table and understanding the information presented in the table are additional tasks necessary for table information extraction. Comments extraction : extracting comments from the actual content of articles in order to restore the link between authors of each of the sentences Language and vocabulary analysis Terminology extraction: finding the relevant terms for a given corpus Audio extraction Template-based music extraction: finding relevant characteristic in an audio signal taken from a given repertoire; for instance time indexes of occurrences of percussive sounds can be extracted in order to represent the essential rhythmic component of a music piece. Note that this list is not exhaustive and that the exact meaning of IE activities is not commonly accepted and that many approaches combine multiple sub-tasks of IE in order to achieve a wider goal. Machine learning, statistical analysis and/or natural language processing are often used in IE. IE on non-text documents is becoming an increasingly interesting topic in research, and information extracted from multimedia documents can now be expressed in a high level structure as it is done on text. This naturally leads to the fusion of extracted information from multiple kinds of documents and sources. == World Wide Web applications == IE has been the focus of the MUC conferences. The proliferation of the Web, however, intensified the need for developing IE systems that help people

IBM Watsonx

Watsonx is a platform by IBM for building and managing artificial intelligence (AI) applications for business use. Released on May 9, 2023, the platform provides software tools and infrastructure for companies to work with both IBM's own AI models and models from third-party sources. The platform consists of three main components: watsonx.ai, a studio for training, validating, and deploying AI models; watsonx.data, a system for storing and managing data used by the models; and watsonx.governance, a toolkit to ensure AI applications are compliant with company policies and regulations. A key feature of the platform is that it can be trained on a company's private data to perform specialized tasks, a process known as fine-tuning. IBM states that this client-specific data is not used to train its own models. == History == Watsonx was introduced on May 9, 2023, at the annual IBM Think conference, as a platform that includes multiple services. Just like Watson AI computer with the similar name, Watsonx was named after Thomas J. Watson, IBM's founder and first CEO. On February 13, 2024, Anaconda partnered with IBM to embed its open-source Python packages into Watsonx. Watsonx is used at ESPN's Fantasy Football App for managing players' performance, and by Italian telecommunications company Wind Tre. It was employed to generate editorial content around nominees during the 66th Annual Grammy Awards. In 2025, Wimbledon integrated IBM watsonx generative AI into its app and website. Integrated with IBM Safer Payments, IBM watsonx has been used in banking sector fraud detection and anti-money laundering (AML) systems. == Services == === watsonx.ai === Watsonx.ai is a platform that allows AI developers to leverage a wide range of LLMs under IBM's own Granite series and others such as Facebook's LLaMA-2, free and open-source model Mistral, and many others present in the Hugging Face community. These models come pre-trained and optimized for various natural language processing (NLP) applications.The platform also allows fine-tuning with its Tuning Studio. === watsonx.data === Watsonx.data is a platform designed to assist clients in addressing issues related to data volume, complexity, cost, and governance.. The platform facilitates seamless data access, whether stored in the cloud or on-premises, through a single entry point. === watsonx.governance === Watsonx.governance is a platform that utilizes IBM's AI capabilities to implement AI lifecycle governance. This helps them manage risks and maintain compliance with evolving AI and industry regulations, while reducing AI bias through automated oversight.

KXEN Inc.

KXEN was an American software company which existed from 1998 to 2013 when it was acquired by SAP AG. == History == KXEN was founded in June 1998 by Roger Haddad and Michel Bera. It was based in San Francisco, California with offices in Paris and London. On September 10, 2013, SAP AG announced plans to acquire KXEN. On October 1, 2013, a letter to KXEN customers announced the acquisition closed. KXEN primarily marketed predictive analytics software. == Predictive analytics == InfiniteInsight is a predictive modeling suite developed by KXEN that assists analytic professionals, and business executives to extract information from data. Among other functions, InfiniteInsight is used for variable importance, classification, regression, segmentation, time series, product recommendation, as described and expressed by the Java Data Mining interface, and for social network analysis. InfiniteInsight allows prediction of a behavior or a value, the forecast of a time series or the understanding of a group of individuals with similar behavior. Advanced functions include behavioral modeling, exporting the model code into different target environments or building predictive models on top of SAS or SPSS data files. Competitors are SAS Enterprise Miner, IBM SPSS Modeler, and Statistica. Open source predictive tools like the R package or Weka are also competitors, since they provide similar features free of charge.

C4.5 algorithm

C4.5 is an algorithm used to generate a decision tree developed by Ross Quinlan. C4.5 is an extension of Quinlan's earlier ID3 algorithm. The decision trees generated by C4.5 can be used for classification, and for this reason, C4.5 is often referred to as a statistical classifier. In 2011, authors of the Weka machine learning software described the C4.5 algorithm as "a landmark decision tree program that is probably the machine learning workhorse most widely used in practice to date". It became quite popular after ranking #1 in the Top 10 Algorithms in Data Mining pre-eminent paper published by Springer LNCS in 2008. == Algorithm == C4.5 builds decision trees from a set of training data in the same way as ID3, using the concept of information entropy. The training data is a set S = s 1 , s 2 , . . . {\displaystyle S={s_{1},s_{2},...}} of already classified samples. Each sample s i {\displaystyle s_{i}} consists of a p-dimensional vector ( x 1 , i , x 2 , i , . . . , x p , i ) {\displaystyle (x_{1,i},x_{2,i},...,x_{p,i})} , where the x j {\displaystyle x_{j}} represent attribute values or features of the sample, as well as the class in which s i {\displaystyle s_{i}} falls. At each node of the tree, C4.5 chooses the attribute of the data that most effectively splits its set of samples into subsets enriched in one class or the other. The splitting criterion is the normalized information gain (difference in entropy). The attribute with the highest normalized information gain is chosen to make the decision. The C4.5 algorithm then recurses on the partitioned sublists. This algorithm has a few base cases. All the samples in the list belong to the same class. When this happens, it simply creates a leaf node for the decision tree saying to choose that class. None of the features provide any information gain. In this case, C4.5 creates a decision node higher up the tree using the expected value of the class. Instance of previously unseen class encountered. Again, C4.5 creates a decision node higher up the tree using the expected value. === Pseudocode === In pseudocode, the general algorithm for building decision trees is: Check for the above base cases. For each attribute a, find the normalized information gain ratio from splitting on a. Let a_best be the attribute with the highest normalized information gain. Create a decision node that splits on a_best. Recurse on the sublists obtained by splitting on a_best, and add those nodes as children of node. == Improvements from ID3 algorithm == C4.5 made a number of improvements to ID3. Some of these are: Handling both continuous and discrete attributes: In order to handle continuous attributes, C4.5 creates a threshold and then splits the list into those whose attribute value is above the threshold and those that are less than or equal to it. Handling training data with missing attribute values: C4.5 allows attribute values to be marked as missing. Missing attribute values are simply not used in gain and entropy calculations. Handling attributes with differing costs. Pruning trees after creation: C4.5 goes back through the tree once it's been created and attempts to remove branches that do not help by replacing them with leaf nodes. == Improvements in C5.0/See5 algorithm == Quinlan went on to create C5.0 and See5 (C5.0 for Unix/Linux, See5 for Windows) which he markets commercially. C5.0 offers a number of improvements on C4.5. Some of these are: Speed - C5.0 is significantly faster than C4.5 (several orders of magnitude) Memory usage - C5.0 is more memory efficient than C4.5 Smaller decision trees - C5.0 gets similar results to C4.5 with considerably smaller decision trees. Support for boosting - Boosting improves the trees and gives them more accuracy. Weighting - C5.0 allows you to weight different cases and misclassification types. Winnowing - a C5.0 option automatically winnows the attributes to remove those that may be unhelpful. Source for a single-threaded Linux version of C5.0 is available under the GNU General Public License (GPL).

Model compression

Model compression is a machine learning technique for reducing the size of trained models. Large models can achieve high accuracy, but often at the cost of significant resource requirements. Compression techniques aim to compress models without significant performance reduction. Smaller models require less storage space, and consume less memory and compute during inference. Compressed models enable deployment on resource-constrained devices such as smartphones, embedded systems, edge computing devices, and consumer electronics computers. Efficient inference is also valuable for large corporations that serve large model inference over an API, allowing them to reduce computational costs and improve response times for users. Model compression is not to be confused with knowledge distillation, in which a smaller "student" model is trained to imitate the input-output behavior of a larger "teacher" model (as opposed to using the "teacher"'s trained parameters or the "teacher"'s training targets). == Techniques == Several techniques are employed for model compression. === Pruning === Pruning sparsifies a large model by setting some parameters to exactly zero. This effectively reduces the number of parameters. This allows the use of sparse matrix operations, which are faster than dense matrix operations. Pruning criteria can be based on magnitudes of parameters, the statistical pattern of neural activations, Hessian values, etc. === Quantization === Quantization reduces the numerical precision of weights and activations. For example, instead of storing weights as 32-bit floating-point numbers, they can be represented using 8-bit integers. Low-precision parameters take up less space, and takes less compute to perform arithmetic with. It is also possible to quantize some parameters more aggressively than others, so for example, a less important parameter can have 8-bit precision while another, more important parameter, can have 16-bit precision. Inference with such models requires mixed-precision arithmetic. Quantized models can also be used during training (rather than after training). PyTorch implements automatic mixed-precision (AMP), which performs autocasting, gradient scaling, and loss scaling. === Low-rank factorization === Weight matrices can be approximated by low-rank matrices. Let W {\displaystyle W} be a weight matrix of shape m × n {\displaystyle m\times n} . A low-rank approximation is W ≈ U V T {\displaystyle W\approx UV^{T}} , where U {\displaystyle U} and V {\displaystyle V} are matrices of shapes m × k , n × k {\displaystyle m\times k,n\times k} . When k {\displaystyle k} is small, this both reduces the number of parameters needed to represent W {\displaystyle W} approximately, and accelerates matrix multiplication by W {\displaystyle W} . Low-rank approximations can be found by singular value decomposition (SVD). The choice of rank for each weight matrix is a hyperparameter, and jointly optimized as a mixed discrete-continuous optimization problem. The rank of weight matrices may also be pruned after training, taking into account the effect of activation functions like ReLU on the implicit rank of the weight matrices. == Training == Model compression may be decoupled from training, that is, a model is first trained without regard for how it might be compressed, then it is compressed. However, it may also be combined with training. The "train big, then compress" method trains a large model for a small number of training steps (less than it would be if it were trained to convergence), then heavily compress the model. It is found that at the same compute budget, this method results in a better model than lightly compressed, small models. In Deep Compression, the compression has three steps. First loop (pruning): prune all weights lower than a threshold, then finetune the network, then prune again, etc. Second loop (quantization): cluster weights, then enforce weight sharing among all weights in each cluster, then finetune the network, then cluster again, etc. Third step: Use Huffman coding to losslessly compress the model. The SqueezeNet paper reported that Deep Compression achieved a compression ratio of 35 on AlexNet, and a ratio of ~10 on SqueezeNets.

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 function class can shatter—that is, for which all possible binary labelings can be realized by some function in the class. 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. 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. == Definitions == === VC dimension of a set-family === Let C = { C } C ∈ C {\displaystyle {\mathcal {C}}=\{C\}_{C\in {\mathcal {C}}}} be a family of sets (also called set family, collection of sets or set of sets) and X {\displaystyle X} a set. Their intersection is defined as the following set family: C ∩ X := { C ∩ X ∣ C ∈ C } . {\displaystyle {\mathcal {C}}\cap X:=\{C\cap X\mid C\in {\mathcal {C}}\}.} Here typically X {\displaystyle X} and each C ∈ C {\displaystyle C\in {\mathcal {C}}} are subsets of a big "universe" of possibilities U {\displaystyle U} where intersection takes place. We say that a set X {\displaystyle X} is shattered by C {\displaystyle {\mathcal {C}}} if P ( X ) = C ∩ X {\displaystyle {\mathcal {P}}(X)={\mathcal {C}}\cap X} i.e. the set of intersections contains (hence is equal to) all the subsets of X {\displaystyle X} . For finite sets X {\displaystyle X} this is equivalent to | C ∩ X | = 2 | X | . {\displaystyle |{\mathcal {C}}\cap X|=2^{|X|}.} The VC dimension D {\displaystyle D} of C {\displaystyle {\mathcal {C}}} is the cardinality of the largest set that is shattered by C {\displaystyle {\mathcal {C}}} . If arbitrarily large sets can be shattered, the VC dimension of C {\displaystyle {\mathcal {C}}} is ∞ {\displaystyle \infty } . === VC dimension of a classification model === A binary classification model f {\displaystyle f} with some parameter vector θ {\displaystyle \theta } is said to shatter a set of generally positioned data points ( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\ldots ,x_{n})} if, for every assignment of labels to those points, there exists a θ {\displaystyle \theta } such that the model f {\displaystyle f} makes no errors when evaluating that set of data points. The VC dimension of a model f {\displaystyle f} is the maximum number of points that can be arranged so that f {\displaystyle f} shatters them. More formally, it is the maximum cardinal D {\displaystyle D} such that there exists a generally positioned data point set of cardinality D {\displaystyle D} that can be shattered by f {\displaystyle f} . == Examples == f {\displaystyle 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 {\displaystyle 2^{d}} different classifiers, is at most d {\displaystyle d} (this is an upper bound on the VC dimension; the Sauer–Shelah lemma gives a lower bound on the dimension). f {\displaystyle f} is a single-parametric threshold classifier on real numbers; i.e., for a certain threshold θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number is larger than θ {\displaystyle \theta } and 0 otherwise. The VC dimension of f {\displaystyle f} is 1 because: (a) It can shatter a single point. For every point x {\displaystyle x} , a classifier f θ {\displaystyle f_{\theta }} labels it as 0 if θ > x {\displaystyle \theta >x} and labels it as 1 if θ < x {\displaystyle \theta x + 2 {\displaystyle \theta >x+2} , as (1,0) if θ ∈ [ x − 4 , x − 2 ) {\displaystyle \theta \in [x-4,x-2)} , as (1,1) if θ ∈ [ x − 2 , x ] {\displaystyle \theta \in [x-2,x]} , and as (0,1) if θ ∈ ( x , x + 2 ] {\displaystyle \theta \in (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 {\displaystyle 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 {\displaystyle f} is a single-parametric sine classifier, i.e., for a certain parameter θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number x {\displaystyle x} has sin ⁡ ( θ x ) > 0 {\displaystyle \sin(\theta x)>0} and 0 otherwise. The VC dimension of f {\displaystyle f} is infinite, since it can shatter any finite subset of the set { 2 − m ∣ m ∈ N } {\displaystyle \{2^{-m}\mid m\in \mathbb {N} \}} . == Uses == === In statistical learning theory === The VC dimension can predict a probabilistic upper bound on the test error of a classification model. Vapnik proved that the probability of the test error (i.e., risk with 0–1 loss function) distancing from an upper bound (on data that is drawn i.i.d. from the same distribution as the training set) is given by: Pr ( test error ⩽ training error + 1 N [ D ( log ⁡ ( 2 N D ) + 1 ) − log ⁡ ( η 4 ) ] ) = 1 − η , {\displaystyle \Pr \left({\text{test error}}\leqslant {\text{training error}}+{\sqrt {{\frac {1}{N}}\left[D\left(\log \left({\tfrac {2N}{D}}\right)+1\right)-\log \left({\tfrac {\eta }{4}}\right)\right]}}\,\right)=1-\eta ,} where D {\displaystyle D} is the VC dimension of the classification model, 0 < η ⩽ 1 {\displaystyle 0<\eta \leqslant 1} , and N {\displaystyle N} is the size of the training set (restriction: this formula is valid when D ≪ N {\displaystyle D\ll N} . When D {\displaystyle D} is larger, the test-error may be much higher than the training-error. This is due to overfitting). The VC dimension also appears in sample-complexity bounds. A space of binary functions with VC dimension D {\displaystyle D} can be learned with: N = Θ ( D + ln ⁡ 1 δ ε 2 ) {\displaystyle N=\Theta \left({\frac {D+\ln {1 \over \delta }}{\varepsilon ^{2}}}\right)} samples, where ε {\displaystyle \varepsilon } is the learning error and δ {\displaystyle \delta } is the failure probability. Thus, the sample-complexity is a linear function of the VC dimension of the hypothesis space. === In computational geometry === The VC dimension is one of the critical parameters in the size of ε-nets, which determines the complexity of approximation algorithms based on them; range sets without finite VC dimension may not have finite ε-nets at all. == Bounds == The VC dimension of the dual set-family of C {\displaystyle {\mathcal {C}}} is strictly less than 2 vc ⁡ ( C ) + 1 {\displaystyle 2^{\operatorname {vc} ({\mathcal {C}})+1}} , and this is best possible. The VC dimension of a finite set-family C {\displaystyle {\mathcal {C}}} is at most log 2 ⁡ | C | {\displaystyle \log _{2}|{\mathcal {C}}|} . This is because | C ∩ X | ≤ | X | {\displaystyle |{\mathcal {C}}\cap X|\leq |X|} by definition. Given a set-fa