Nagarik App (translation: Citizen App) is a mobile application launched by the Government of Nepal to provide government-related services in a single online platform. The app was developed to facilitate an easier, systematic, and simplified delivery of government services to Nepali citizens digitally. The app was launched to play a pivotal role in revolutionizing the way citizens interact with the government. It offers government services through a single unified platform, minimizing the need for citizens to navigate multiple channels or physical offices for their diverse needs of government services. The services are added gradually according to the needs and services required. The government aims to reduce the physical queues and the need to be physically present to get services from the different government offices. One can get services online round-the-clock even during holidays. As of now, 25 services are included in the app, ranging from Police Clearance Report to Voters Card. The app contains and provides a vast range of government services. The app was launched on the occasion of the fourth National Information and Communication Technology Day, 2021 (2078 BS). The event marked a significant milestone in Nepal’s digital transformation journey. It aims to reduce all the bureaucratic hurdles that the citizens have been facing and make government services more efficient and convenient. In Oct 20, 2024, a E-Chalan was introduced for managing traffic violations in initially piloting in Kathmandu Valley. The Kathmandu Valley Traffic Police Office announced that physical licenses would no longer be confiscated for traffic rule violations. Instead, a "Digital Chit (E-Chalan)" system was implemented, allowing drivers to pay fines electronically. Integrated with the NagarikApp, the system enables police to access drivers' licenses, record violations, and update details directly in the app. == Features and Services == Inland Revenue Department (Nepal) PAN Registration Election Commission (Nepal) Voter Card Pre-Registration and Details Nepal Police Online Clearance Report Traffic Violations and Fine Payment Nepal Passport, Driving License, National Identity Card (NID), Citizenship, and Voter ID link details My Municipality (Includes contact info of the representatives, services such as ambulance, nearby police, and budget programs and plans) The Government Press ID card PF/PAN/SST/CIT statements can be viewed Nagarik Pahichan Dwar (Online bank accounts can be opened and KYC can be verified for selected banks using the QR) == Awards and honors == Each year, World Summit Award honors outstanding digital applications and solutions across various categories. The winners of the World Summit Award represent the pinnacle of innovation in their respective categories. Nagarik App was selected among 180 participants and won the World Summit Award of 2022 in Government and Citizen Engagement category. == Latest Statistics & Usage Trends (2082 BS / 2025 AD) == As of August 2025, over 1.5 million Nepali citizens have registered and actively use the Nagarik App, according to the National Information Technology Center (NITC). The majority of daily logins come from: Kathmandu Valley – 37% of total users Province 1 (Koshi) – 19% of total users Bagmati Province – 15% of total users On average, 45,000+ transactions (service requests, document verifications, and payments) are processed through the app each day. The most-used services include: PAN Card Registration – 28% of total requests Police Clearance Report – 22% Driving License Linking & E-Chalan Payment – 18% Vehicle Tax Payment – 14% Source: Internal report from NITC, July 2025 == Step-by-Step: How to Link Your Driving License with Nagarik App == Update the App – Install the latest version from Play Store or App Store. Login or Register – Ensure your SIM is registered in your own name. Go to “Transport Services” in the menu. Select “Driving License” – Enter your license number and date of birth. Verify via OTP – Sent to your registered mobile number. Confirmation – Your digital license will appear inside the app. This guide is continuously updated to reflect the latest rules from the Kathmandu Valley Traffic Police Office and changes in NITC’s backend system. For in-depth details, step-by-step tutorials, and the most recent Nagarik App updates, visit the full article on The Bipin Blog.
Security information management
Security information management (SIM) is an information security industry term for the collection of data such as log files into a central repository for trend analysis. == Overview == SIM products generally are software agents running on the computer systems that are monitored. The recorded log information is then sent to a centralized server that acts as a "security console". The console typically displays reports, charts, and graphs of that information, often in real time. Some software agents can incorporate local filters to reduce and manipulate the data that they send to the server, although typically from a forensic point of view you would collect all audit and accounting logs to ensure you can recreate a security incident. The security console is monitored by an administrator who reviews the consolidated information and takes action in response to any alerts issued. The data that is sent to the server to be correlated and analyzed are normalized by the software agents into a common form, usually XML. Those data are then aggregated in order to reduce their overall size. == Terminology == The terminology can easily be mistaken as a reference to the whole aspect of protecting one's infrastructure from any computer security breach. Due to historic reasons of terminology evolution; SIM refers to just the part of information security which consists of discovery of 'bad behavior' or policy violations by using data collection techniques. The term commonly used to represent an entire security infrastructure that protects an environment is commonly called information security management (InfoSec). Security information management is also referred to as log management and is different from SEM (security event management), but makes up a portion of a SIEM (security information and event management) solution. == Regulatory compliance == Security information management systems support compliance with regulatory frameworks that require centralized collection and analysis of security data. The Health Insurance Portability and Accountability Act (HIPAA) Security Rule requires covered entities to implement audit controls that record and examine activity in information systems containing electronic protected health information (45 CFR 164.312(b))."45 CFR § 164.312 - Technical safeguards". Legal Information Institute. Retrieved April 1, 2026. SIM platforms aggregate these audit records to support the required regular review of information system activity records (45 CFR 164.308(a)(1)(ii)(D)). The December 2024 HIPAA Security Rule NPRM proposed requiring regulated entities to deploy automated systems capable of monitoring and recording access to ePHI, including the ability to detect unauthorized access attempts in near real-time."HIPAA Security Rule To Strengthen the Cybersecurity of Electronic Protected Health Information". Federal Register. January 6, 2025. Retrieved April 1, 2026. The Payment Card Industry Data Security Standard (PCI DSS) similarly requires centralized log management and daily review of security events (Requirements 10.4 and 10.6)."PCI DSS v4.0" (PDF). PCI Security Standards Council. March 2022. Retrieved April 1, 2026. NIST Special Publication 800-53 addresses security information management through the AU (Audit and Accountability) control family, which specifies requirements for audit event generation, content, storage, and analysis."NIST SP 800-53 Rev. 5: Security and Privacy Controls". National Institute of Standards and Technology. September 2020. Retrieved April 1, 2026.
Automated Pain Recognition
Automated Pain Recognition (APR) is a method for objectively measuring pain and at the same time represents an interdisciplinary research area that comprises elements of medicine, psychology, psychobiology, and computer science. The focus is on computer-aided objective recognition of pain, implemented on the basis of machine learning. Automated pain recognition allows for the valid, reliable detection and monitoring of pain in people who are unable to communicate verbally. The underlying machine learning processes are trained and validated in advance by means of unimodal or multimodal body signals. Signals used to detect pain may include facial expressions or gestures and may also be of a (psycho-)physiological or paralinguistic nature. To date, the focus has been on identifying pain intensity, but visionary efforts are also being made to recognize the quality, site, and temporal course of pain. However, the clinical implementation of this approach is a controversial topic in the field of pain research. Critics of automated pain recognition argue that pain diagnosis can only be performed subjectively by humans. == Background == Pain diagnosis under conditions where verbal reporting is restricted - such as in verbally and/or cognitively impaired people or in patients who are sedated or mechanically ventilated - is based on behavioral observations by trained professionals. However, all known observation procedures (e.g., Zurich Observation Pain Assessment (ZOPA)); Pain Assessment in Advanced Dementia Scale (PAINAD) require a great deal of specialist expertise. These procedures can be made more difficult by perception- and interpretation-related misjudgments on the part of the observer. With regard to the differences in design, methodology, evaluation sample, and conceptualization of the phenomenon of pain, it is difficult to compare the quality criteria of the various tools. Even if trained personnel could theoretically record pain intensity several times a day using observation instruments, it would not be possible to measure it every minute or second. In this respect, the goal of automated pain recognition is to use valid, robust pain response patterns that can be recorded multimodally for a temporally dynamic, high-resolution, automated pain intensity recognition system. == Procedure == For automated pain recognition, pain-relevant parameters are usually recorded using non-invasive sensor technology, which captures data on the (physical) responses of the person in pain. This can be achieved with camera technology that captures facial expressions, gestures, or posture, while audio sensors record paralinguistic features. (Psycho-)physiological information such as muscle tone and heart rate can be collected via biopotential sensors (electrodes). Pain recognition requires the extraction of meaningful characteristics or patterns from the data collected. This is achieved using machine learning techniques that are able to provide an assessment of the pain after training (learning), e.g., "no pain," "mild pain," or "severe pain." == Parameters == Although the phenomenon of pain comprises different components (sensory discriminative, affective (emotional), cognitive, vegetative, and (psycho-)motor), automated pain recognition currently relies on the measurable parameters of pain responses. These can be divided roughly into the two main categories of "physiological responses" and "behavioral responses". === Physiological responses === In humans, pain almost always initiates autonomic nervous processes that are reflected measurably in various physiological signals. ==== Physiological signals ==== Measurements can include electrodermal activity (EDA, also skin conductance), electromyography (EMG), electrocardiogram (ECG), blood volume pulse (BVP), electroencephalogram (EEG), respiration, and body temperature, which are regulatory mechanisms of the sympathetic and parasympathetic systems. Physiological signals are mainly recorded using special non-invasive surface electrodes (for EDA, EMG, ECG, and EEG), a blood volume pulse sensor (BVP), a respiratory belt (respiration), and a thermal sensor (body temperature). Endocrinological and immunological parameters can also be recorded, but this requires measures that are somewhat invasive (e.g., blood sampling). === Behavioral responses === Behavioral responses to pain fulfil two functions: protection of the body (e.g., through protective reflexes) and external communication of the pain (e.g., as a cry for help). The responses are particularly evident in facial expressions, gestures, and paralinguistic features. ==== Facial expressions ==== Behavioral signals captured comprise facial expression patterns (expressive behavior), which are measured with the aid of video signals. Facial expression recognition is based on the everyday clinical observation that pain often manifests itself in the patient's facial expressions but that this is not necessarily always the case, since facial expressions can be inhibited through self-control. Despite the possibility that facial expressions may be influenced consciously, facial expression behavior represents an essential source of information for pain diagnosis and is thus also a source of information for automatic pain recognition. One advantage of video-based facial expression recognition is the contact-free measurement of the face, provided that it can be captured on video, which is not possible in every position (e.g., lying face down) or may be limited by bandages covering the face. Facial expression analysis relies on rapid, spontaneous, and temporary changes in neuromuscular activity that lead to visually detectable changes in the face. ==== Gestures ==== Gestures are also captured predominantly using non-contact camera technology. Motor pain responses vary and are strongly dependent on the type and cause of the pain. They range from abrupt protective reflexes (e.g., spontaneous retraction of extremities or doubling up) to agitation (pathological restlessness) and avoidance behavior (hesitant, cautious movements). ==== Paralinguistic features of language ==== Among other things, pain leads to nonverbal linguistic behavior that manifests itself in sounds such as sighing, gasping, moaning, whining, etc. Paralinguistic features are usually recorded using highly sensitive microphones. == Algorithms == After the recording, pre-processing (e.g., filtering), and extraction of relevant features, an optional information fusion can be performed. During this process, modalities from different signal sources are merged to generate new or more precise knowledge. The pain is classified using machine learning processes. The method chosen has a significant influence on the recognition rate and depends greatly on the quality and granularity of the underlying data. Similar to the field of affective computing, the following classifiers are currently being used: Support Vector Machine (SVM): The goal of an SVM is to find a clearly defined optimal hyperplane with the greatest minimal distance to two (or more) classes to be separated. The hyperplane acts as a decision function for classifying an unknown pattern. Random Forest (RF): RF is based on the composition of random, uncorrelated decision trees. An unknown pattern is judged individually by each tree and assigned to a class. The final classification of the patterns by the RF is then based on a majority decision. k-Nearest Neighbors (k-NN): The k-NN algorithm classifies an unknown object using the class label that most commonly classifies the k neighbors closest to it. Its neighbors are determined using a selected similarity measure (e.g., Euclidean distance, Jaccard coefficient, etc.). Artificial neural networks (ANNs): ANNs are inspired by biological neural networks and model their organizational principles and processes in a very simplified manner. Class patterns are learned by adjusting the weights of the individual neuronal connections. == Databases == In order to classify pain in a valid manner, it is necessary to create representative, reliable, and valid pain databases that are available to the machine learner for training. An ideal database would be sufficiently large and would consist of natural (not experimental), high-quality pain responses. However, natural responses are difficult to record and can only be obtained to a limited extent; in most cases they are characterized by suboptimal quality. The databases currently available therefore contain experimental or quasi-experimental pain responses, and each database is based on a different pain model. The following list shows a selection of the most relevant pain databases (last updated: April 2020): UNBC-McMaster Shoulder Pain BioVid Heat Pain EmoPain SenseEmotion X-ITE Pain
Taguchi loss function
The Taguchi loss function is graphical depiction of loss developed by the Japanese business statistician Genichi Taguchi to describe a phenomenon affecting the value of products produced by a company. Praised by Dr. W. Edwards Deming (the business guru of the 1980s American quality movement), it made clear the concept that quality does not suddenly plummet when, for instance, a machinist exceeds a rigid blueprint tolerance. Instead 'loss' in value progressively increases as variation increases from the intended condition. This was considered a breakthrough in describing quality, and helped fuel the continuous improvement movement. The concept of Taguchi's quality loss function was in contrast with the American concept of quality, popularly known as goal post philosophy, the concept given by American quality guru Phil Crosby. Goal post philosophy emphasizes that if a product feature doesn't meet the designed specifications it is termed as a product of poor quality (rejected), irrespective of amount of deviation from the target value (mean value of tolerance zone). This concept has similarity with the concept of scoring a 'goal' in the game of football or hockey, because a goal is counted 'one' irrespective of the location of strike of the ball in the 'goal post', whether it is in the center or towards the corner. This means that if the product dimension goes out of the tolerance limit the quality of the product drops suddenly. Through his concept of the quality loss function, Taguchi explained that from the customer's point of view this drop of quality is not sudden. The customer experiences a loss of quality the moment product specification deviates from the 'target value'. This 'loss' is depicted by a quality loss function and it follows a parabolic curve mathematically given by L = k(y–m)2, where m is the theoretical 'target value' or 'mean value' and y is the actual size of the product, k is a constant and L is the loss. This means that if the difference between 'actual size' and 'target value' i.e. (y–m) is large, loss would be more, irrespective of tolerance specifications. In Taguchi's view tolerance specifications are given by engineers and not by customers; what the customer experiences is 'loss'. This equation is true for a single product; if 'loss' is to be calculated for multiple products the loss function is given by L = k[S2 + ( y ¯ {\displaystyle {\bar {y}}} – m)2], where S2 is the 'variance of product size' and y ¯ {\displaystyle {\bar {y}}} is the average product size. == Overview == The Taguchi loss function is important for a number of reasons—primarily, to help engineers better understand the importance of designing for variation.
Error-driven learning
In reinforcement learning, error-driven learning is a method for adjusting a model's (intelligent agent's) parameters based on the difference between its output results and the ground truth. These models stand out as they depend on environmental feedback, rather than explicit labels or categories. They are based on the idea that language acquisition involves the minimization of the prediction error (MPSE). By leveraging these prediction errors, the models consistently refine expectations and decrease computational complexity. Typically, these algorithms are operated by the GeneRec algorithm. Error-driven learning has widespread applications in cognitive sciences and computer vision. These methods have also found successful application in natural language processing (NLP), including areas like part-of-speech tagging, parsing, named entity recognition (NER), machine translation (MT), speech recognition (SR), and dialogue systems. == Formal Definition == Error-driven learning models are ones that rely on the feedback of prediction errors to adjust the expectations or parameters of a model. The key components of error-driven learning include the following: A set S {\displaystyle S} of states representing the different situations that the learner can encounter. A set A {\displaystyle A} of actions that the learner can take in each state. A prediction function P ( s , a ) {\displaystyle P(s,a)} that gives the learner's current prediction of the outcome of taking action a {\displaystyle a} in state s {\displaystyle s} . An error function E ( o , p ) {\displaystyle E(o,p)} that compares the actual outcome o {\displaystyle o} with the prediction p {\displaystyle p} and produces an error value. An update rule U ( p , e ) {\displaystyle U(p,e)} that adjusts the prediction p {\displaystyle p} in light of the error e {\displaystyle e} . == Algorithms == Error-driven learning algorithms refer to a category of reinforcement learning algorithms that leverage the disparity between the real output and the expected output of a system to regulate the system's parameters. Typically applied in supervised learning, these algorithms are provided with a collection of input-output pairs to facilitate the process of generalization. The widely utilized error backpropagation learning algorithm is known as GeneRec, a generalized recirculation algorithm primarily employed for gene prediction in DNA sequences. Many other error-driven learning algorithms are derived from alternative versions of GeneRec. == Applications == === Cognitive science === Simpler error-driven learning models effectively capture complex human cognitive phenomena and anticipate elusive behaviors. They provide a flexible mechanism for modeling the brain's learning process, encompassing perception, attention, memory, and decision-making. By using errors as guiding signals, these algorithms adeptly adapt to changing environmental demands and objectives, capturing statistical regularities and structure. Furthermore, cognitive science has led to the creation of new error-driven learning algorithms that are both biologically acceptable and computationally efficient. These algorithms, including deep belief networks, spiking neural networks, and reservoir computing, follow the principles and constraints of the brain and nervous system. Their primary aim is to capture the emergent properties and dynamics of neural circuits and systems. === Computer vision === Computer vision is a complex task that involves understanding and interpreting visual data, such as images or videos. In the context of error-driven learning, the computer vision model learns from the mistakes it makes during the interpretation process. When an error is encountered, the model updates its internal parameters to avoid making the same mistake in the future. This repeated process of learning from errors helps improve the model's performance over time. For NLP to do well at computer vision, it employs deep learning techniques. This form of computer vision is sometimes called neural computer vision (NCV), since it makes use of neural networks. NCV therefore interprets visual data based on a statistical, trial and error approach and can deal with context and other subtleties of visual data. === Natural Language Processing === ==== Part-of-speech tagging ==== Part-of-speech (POS) tagging is a crucial component in Natural Language Processing (NLP). It helps resolve human language ambiguity at different analysis levels. In addition, its output (tagged data) can be used in various applications of NLP such as information extraction, information retrieval, question Answering, speech eecognition, text-to-speech conversion, partial parsing, and grammar correction. ==== Parsing ==== Parsing in NLP involves breaking down a text into smaller pieces (phrases) based on grammar rules. If a sentence cannot be parsed, it may contain grammatical errors. In the context of error-driven learning, the parser learns from the mistakes it makes during the parsing process. When an error is encountered, the parser updates its internal model to avoid making the same mistake in the future. This iterative process of learning from errors helps improve the parser's performance over time. In conclusion, error-driven learning plays a crucial role in improving the accuracy and efficiency of NLP parsers by allowing them to learn from their mistakes and adapt their internal models accordingly. ==== Named entity recognition (NER) ==== NER is the task of identifying and classifying entities (such as persons, locations, organizations, etc.) in a text. Error-driven learning can help the model learn from its false positives and false negatives and improve its recall and precision on (NER). In the context of error-driven learning, the significance of NER is quite profound. Traditional sequence labeling methods identify nested entities layer by layer. If an error occurs in the recognition of an inner entity, it can lead to incorrect identification of the outer entity, leading to a problem known as error propagation of nested entities. This is where the role of NER becomes crucial in error-driven learning. By accurately recognizing and classifying entities, it can help minimize these errors and improve the overall accuracy of the learning process. Furthermore, deep learning-based NER methods have shown to be more accurate as they are capable of assembling words, enabling them to understand the semantic and syntactic relationship between various words better. ==== Machine translation ==== Machine translation is a complex task that involves converting text from one language to another. In the context of error-driven learning, the machine translation model learns from the mistakes it makes during the translation process. When an error is encountered, the model updates its internal parameters to avoid making the same mistake in the future. This iterative process of learning from errors helps improve the model's performance over time. ==== Speech recognition ==== Speech recognition is a complex task that involves converting spoken language into written text. In the context of error-driven learning, the speech recognition model learns from the mistakes it makes during the recognition process. When an error is encountered, the model updates its internal parameters to avoid making the same mistake in the future. This iterative process of learning from errors helps improve the model's performance over time. ==== Dialogue systems ==== Dialogue systems are a popular NLP task as they have promising real-life applications. They are also complicated tasks since many NLP tasks deserving study are involved. In the context of error-driven learning, the dialogue system learns from the mistakes it makes during the dialogue process. When an error is encountered, the model updates its internal parameters to avoid making the same mistake in the future. This iterative process of learning from errors helps improve the model's performance over time. == Advantages == Error-driven learning has several advantages over other types of machine learning algorithms: They can learn from feedback and correct their mistakes, which makes them adaptive and robust to noise and changes in the data. They can handle large and high-dimensional data sets, as they do not require explicit feature engineering or prior knowledge of the data distribution. They can achieve high accuracy and performance, as they can learn complex and nonlinear relationships between the input and the output. == Limitations == Although error driven learning has its advantages, their algorithms also have the following limitations: They can suffer from overfitting, which means that they memorize the training data and fail to generalize to new and unseen data. This can be mitigated by using regularization techniques, such as adding a penalty term to the loss function, or reducing the complexity of the model. They can be sensitive to the choice of
Twproject
Twproject (say: T W Project) is a web-based project and groupware management tool created by Open Lab, an Italian software house founded in 2001. It won the 17th Jolt Productivity Award in 2007 in the project management category. In March 2019 it becomes property of Twproject company. It has widespread use in universities as a teaching tool in project management courses. It is used by Oracle Corporation, Prada, Calzedonia, General Electric and many other companies from corporations to small start-ups. == History == April 2001 - The idea of Teamwork came to Open-Lab founders from a need to overcome the PM tools used at that time. It was built in Microsoft ASP and Adobe Flash November 2002 - Open-Lab decide to move from Flash to HTML and from ASP to Java-JSP. Teamwork 2 development is started. June 2004 - Teamwork 2 released, using top open-source technologies like Hibernate, jBlooming, dynamic CSS, Ajax 7 January 2005 - Teamwork goes open source, under LGPL license; remains such until June 2006 (18 months): it is a hit application on SourceForge, with 38.000 downloads, covered by greeting but starving April 2005 - Open-Lab takes the decision to change commercial strategy to finance development of Teamwork version 3 6 June 2006 - Teamwork 3 is finally out (15 months development). New interface, many new features, agile support and much more 27 March 2007 - Teamwork wins the 2007 JOLT Productivity Awards for project management category July 2007 - Teamwork 4 development started: new interface, extended use of new HTML capabilities, JS-oriented interface, start using jQuery February 2009 - Teamwork 4.0 is out February 2010 - Teamwork 4.4: public project pages, Chinese interface. jQuery is getting more space in Teamwork December 2010 - Teamwork 4.6: released Mobile module available for iPhone, Android, BlackBerry. Intensive usage of jQuery June 2011 - Teamwork 4.7: released Issue Kanban / Organizer January 2012 - Teamwork 5.0 development started. Lighter interface, extensive usage of dynamic pages, easier installer and first time approach. Learning curve highly reduced. A jQuery Gantt editor included and released free for the community July 2012 - Teamwork 5 released and also the free online Gantt editor November 2012 - Teamwork 5.1 with new trees and improved model for staffing March 2013 - Teamwork 5.2 with stronger support for customizations and Japanese interface. April 2014 - Teamwork has changed its name in Twproject because the domain teamwork.com has been purchased by Teamwork. April 2013 - Twproject 5.4 with a redesigned more powerful Gantt chart. August 2015 - Twproject 5 finale release. September 2015 - Twproject 6 with a completely redesigned user interface. March 2019 - A new company Twproject srl has been spun off. September 2021 - Twproject 7 has been released introducing WBS based management and workload management. == Features == Project & task management (with Microsoft Project import/export), and JSON format Gantt editor. Uses jQuery Gantt components Time tracking. Several entry points: dashboard, weekly view, issues, start/stop buttons Resource planning with weekly/monthly view, work load overview, unavailability from agenda Issue tracking & planning(with Kanban), e-mail integration, task dedicated inboxes Dashboard configuration, with customizable portlets and layout Message boards Scrum module Meeting and minute management, attached documents Agenda (Integrates with iCal, Microsoft Outlook, Microsoft Entourage, and Google Calendar) Document management, remote file systems link with NTFS, FTP, SVN, S3 (Dropbox, Google drive) Mobile application for iPhone, iPad, Android, Blackberry, Windows phone == Integration == A complete JSON API is available for integrations. The applications runs in Java JDK 8+ on the Hibernate object/relational mapping. The standard distribution uses Apache Tomcat 9, but can run on any J2EE application server. Twproject is tested on these DB servers: MySQL, Oracle, SQL Server, PostgreSql, HSQLDB, but as uses Hibernate can run on many others. There is simple graphical step-by-step installer for Windows, Mac, Linux, .zip/.tar.gz/.rpm packages.
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. == Introduction == 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. == Overview of VC theory in empirical processes == === Background on empirical processes === Let ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} be a measurable space. For any measure Q {\displaystyle Q} on ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} , and any measurable functions f : X → R {\displaystyle f:{\mathcal {X}}\to \mathbf {R} } , define Q f = ∫ f d Q {\displaystyle Qf=\int fdQ} Measurability issues will be ignored here, for more technical detail see. Let F {\displaystyle {\mathcal {F}}} be a class of measurable functions f : X → R {\displaystyle f:{\mathcal {X}}\to \mathbf {R} } and define: ‖ Q ‖ F = sup { | Q f | : f ∈ F } . {\displaystyle \|Q\|_{\mathcal {F}}=\sup\{\vert Qf\vert \ :\ f\in {\mathcal {F}}\}.} Let X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} be independent, identically distributed random elements of ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} . Then define the empirical measure P n = n − 1 ∑ i = 1 n δ X i , {\displaystyle \mathbb {P} _{n}=n^{-1}\sum _{i=1}^{n}\delta _{X_{i}},} where δ here stands for the Dirac measure. The empirical measure induces a map F → R {\displaystyle {\mathcal {F}}\to \mathbf {R} } given by: f ↦ P n f = 1 n ( f ( X 1 ) + . . . + f ( X n ) ) {\displaystyle f\mapsto \mathbb {P} _{n}f={\frac {1}{n}}(f(X_{1})+...+f(X_{n}))} Now suppose P is the underlying true distribution of the data, which is unknown. Empirical Processes theory aims at identifying classes F {\displaystyle {\mathcal {F}}} for which statements such as the following hold: uniform law of large numbers: ‖ P n − P ‖ F → n 0 , {\displaystyle \|\mathbb {P} _{n}-P\|_{\mathcal {F}}{\underset {n}{\to }}0,} That is, as n → ∞ {\displaystyle n\to \infty } , | 1 n ( f ( X 1 ) + . . . + f ( X n ) ) − ∫ f d P | → 0 {\displaystyle \left|{\frac {1}{n}}(f(X_{1})+...+f(X_{n}))-\int fdP\right|\to 0} uniformly for all f ∈ F {\displaystyle f\in {\mathcal {F}}} . uniform central limit theorem: G n = n ( P n − P ) ⇝ G , in ℓ ∞ ( F ) {\displaystyle \mathbb {G} _{n}={\sqrt {n}}(\mathbb {P} _{n}-P)\rightsquigarrow \mathbb {G} ,\quad {\text{in }}\ell ^{\infty }({\mathcal {F}})} In the former case F {\displaystyle {\mathcal {F}}} is called Glivenko–Cantelli class, and in the latter case (under the assumption ∀ x , sup f ∈ F | f ( x ) − P f | < ∞ {\displaystyle \forall x,\sup \nolimits _{f\in {\mathcal {F}}}\vert f(x)-Pf\vert <\infty } ) the class F {\displaystyle {\mathcal {F}}} is called Donsker or P-Donsker. A Donsker class is Glivenko–Cantelli in probability by an application of Slutsky's theorem. These statements are true for a single f {\displaystyle f} , by standard LLN, CLT arguments under regularity conditions, and the difficulty in the Empirical Processes comes in because joint statements are being made for all f ∈ F {\displaystyle f\in {\mathcal {F}}} . Intuitively then, the set F {\displaystyle {\mathcal {F}}} cannot be too large, and as it turns out that the geometry of F {\displaystyle {\mathcal {F}}} plays a very important role. One way of measuring how big the function set F {\displaystyle {\mathcal {F}}} is to use the so-called covering numbers. The covering number N ( ε , F , ‖ ⋅ ‖ ) {\displaystyle N(\varepsilon ,{\mathcal {F}},\|\cdot \|)} is the minimal number of balls { g : ‖ g − f ‖ < ε } {\displaystyle \{g:\|g-f\|<\varepsilon \}} needed to cover the set F {\displaystyle {\mathcal {F}}} (here it is obviously assumed that there is an underlying norm on F {\displaystyle {\mathcal {F}}} ). The entropy is the logarithm of the covering number. Two sufficient conditions are provided below, under which it can be proved that the set F {\displaystyle {\mathcal {F}}} is Glivenko–Cantelli or Donsker. A class F {\displaystyle {\mathcal {F}}} is P-Glivenko–Cantelli if it is P-measurable with envelope F such that P ∗ F < ∞ {\displaystyle P^{\ast }F<\infty } and satisfies: ∀ ε > 0 sup Q N ( ε ‖ F ‖ Q , F , L 1 ( Q ) ) < ∞ . {\displaystyle \forall \varepsilon >0\quad \sup \nolimits _{Q}N(\varepsilon \|F\|_{Q},{\mathcal {F}},L_{1}(Q))<\infty .} The next condition is a version of Dudley's theorem. If F {\displaystyle {\mathcal {F}}} is a class of functions such that ∫ 0 ∞ sup Q log N ( ε ‖ F ‖ Q , 2 , F , L 2 ( Q ) ) d ε < ∞ {\displaystyle \int _{0}^{\infty }\sup \nolimits _{Q}{\sqrt {\log N\left(\varepsilon \|F\|_{Q,2},{\mathcal {F}},L_{2}(Q)\right)}}d\varepsilon <\infty } then F {\displaystyle {\mathcal {F}}} is P-Donsker for every probability measure P such that P ∗ F 2 < ∞ {\displaystyle P^{\ast }F^{2}<\infty } . In the last integral, the notation means ‖ f ‖ Q , 2 = ( ∫ | f | 2 d Q ) 1 2 {\displaystyle \|f\|_{Q,2}=\left(\int |f|^{2}dQ\right)^{\frac {1}{2}}} . === Symmetrization === The majority of the arguments about how to bound the empirical process rely on symmetrization, maximal and concentration inequalities, and chaining. Symmetrization is usually the first step of the proofs, and since it is used in many machine learning proofs on bounding empirical loss functions (including the proof of the VC inequality which is discussed in the next section). It is presented here: Consider the empirical process: f ↦ ( P n − P ) f = 1 n ∑ i = 1 n ( f ( X i ) − P f ) {\displaystyle f\mapsto (\mathbb {P} _{n}-P)f={\dfrac {1}{n}}\sum _{i=1}^{n}(f(X_{i})-Pf)} Turns out that there is a connection between the empirical and the following symmetrized process: f ↦ P n 0 f = 1 n ∑ i = 1 n ε i f ( X i ) {\displaystyle f\mapsto \mathbb {P} _{n}^{0}f={\dfrac {1}{n}}\sum _{i=1}^{n}\varepsilon _{i}f(X_{i})} The symmetrized process is a Rademacher process, conditionally on the data X i {\displaystyle X_{i}} . Therefore, it is a sub-Gaussian process by Hoeffding's inequality. Lemma (Symmetrization). For every nondecreasing, convex Φ: R → R and class of measurable functions F {\displaystyle {\mathcal {F}}} , E Φ ( ‖ P n − P ‖ F ) ≤ E Φ ( 2 ‖ P n 0 ‖ F ) {\displaystyle \mathbb {E} \Phi (\|\mathbb {P} _{n}-P\|_{\mathcal {F}})\leq \mathbb {E} \Phi \left(2\left\|\mathbb {P} _{n}^{0}\right\|_{\mathcal {F}}\right)} The proof of the Symmetrization lemma relies on introducing independent copies of the original variables X i {\displaystyle X_{i}} (sometimes referred to as a ghost sample) and replacing the inner expectation of the LHS by these copies. After an application of Jensen's inequality different signs could be introduced (hence the name symmetrization) without changing the expectation. The proof can be found below because of its instructive nature. The same proof method can be used to prove the Glivenko–Cantelli theorem. A typical way of proving empirical CLTs, first uses symmetrization to pass the empirical process to P n 0 {\displaystyle \mathbb {P} _{n}^{0}} and then argue conditionally on the data, using the fact that Rademacher processes are simple processes with nice properties. === VC Connection === It turns out that there is a fascinating connection between certain combinatorial properties of the set F {\displaystyle {\mathcal {F}}} and the entropy numbers. Uniform covering numbers can be controlled by the notion of Vapnik–Chervonenkis classes of sets – or shortly VC sets. Consider a collection C {\displaystyle {\mathcal {C}}} of subsets of the sample space X {\displaystyle