Stephen Wolfram ( WUUL-frəm; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer algebra and theoretical physics. In 2012, he was named a fellow of the American Mathematical Society. As a businessman, Wolfram is the founder and CEO of the software company Wolfram Research, where he works as chief designer of Mathematica and the Wolfram Alpha answer engine. == Early life == === Family === Stephen Wolfram was born in London in 1959 to Hugo and Sybil Wolfram, both German Jewish refugees to the United Kingdom. His maternal grandmother was British psychoanalyst Kate Friedlander. Wolfram's father, Hugo Wolfram, was a textile manufacturer and served as managing director of the Lurex Company—makers of the fabric Lurex. Wolfram's mother, Sybil Wolfram, was a Fellow and Tutor in Philosophy at Lady Margaret Hall at University of Oxford from 1964 to 1993. Wolfram is married to a mathematician. They have four children together. === Education === Wolfram was educated at Eton College, but left prematurely in 1976. As a young child, Wolfram had difficulties learning arithmetic. He entered St. John's College, Oxford, at age 17 and left in 1978 without graduating to attend the California Institute of Technology the following year, where he received a PhD in particle physics in 1980. Wolfram's thesis committee was composed of Richard Feynman, Peter Goldreich, Frank J. Sciulli, and Steven Frautschi, and chaired by Richard D. Field. == Early career == Wolfram, at the age of 15, began research in applied quantum field theory and particle physics and published scientific papers in peer-reviewed scientific journals; by the time he left Oxford, he had published ten such papers. Following his PhD, Wolfram joined the faculty at Caltech and became the youngest recipient of a MacArthur Fellowship in 1981, at age 21. == Later career == === Complex systems and cellular automata === In 1983, Wolfram left for the School of Natural Sciences of the Institute for Advanced Study in Princeton. By that time, he was no longer interested in particle physics. Instead, he began pursuing investigations into cellular automata, mainly with computer simulations. He produced a series of papers investigating the class of elementary cellular automata, conceiving the Wolfram code, a naming system for one-dimensional cellular automata, and a classification scheme for the complexity of their behaviour. He conjectured that the Rule 110 cellular automaton might be Turing complete, which a research assistant to Wolfram, Matthew Cook, later proved correct. Wolfram sued Cook and temporarily blocked publication of the work on Rule 110 for allegedly violating a non-disclosure agreement until Wolfram could publish the work in his controversial book A New Kind of Science. Wolfram's cellular-automata work came to be cited in more than 10,000 papers. In the mid-1980s, Wolfram worked on simulations of physical processes (such as turbulent fluid flow) with cellular automata on the Connection Machine alongside Richard Feynman and helped initiate the field of complex systems. In 1984, he was a participant in the Founding Workshops of the Santa Fe Institute, along with Nobel laureates Murray Gell-Mann, Manfred Eigen, and Philip Warren Anderson, and future laureate Frank Wilczek. In 1986, he founded the Center for Complex Systems Research (CCSR) at the University of Illinois Urbana–Champaign. In 1987, he founded the journal Complex Systems. === Symbolic Manipulation Program === Wolfram led the development of the computer algebra system SMP (Symbolic Manipulation Program) in the Caltech physics department during 1979–1981. A dispute with the administration over the intellectual property rights regarding SMP—patents, copyright, and faculty involvement in commercial ventures—eventually led him to resign from Caltech. SMP was further developed and marketed commercially by Inference Corp. of Los Angeles during 1983–1988. === Mathematica === In 1986, Wolfram left the Institute for Advanced Study for the University of Illinois Urbana–Champaign, where he had founded their Center for Complex Systems Research, and started to develop the computer algebra system Mathematica, which was released on 23 June 1988, when he left academia. In 1987, he founded Wolfram Research, which continues to develop and market the program. === A New Kind of Science === From 1992 to 2002, Wolfram worked on his controversial book A New Kind of Science, which presents an empirical study of simple computational systems. Additionally, it argues that for fundamental reasons these types of systems, rather than traditional mathematics, are needed to model and understand complexity in nature. Wolfram's conclusion is that the universe is discrete in its nature, and runs on fundamental laws that can be described as simple programs. He predicts that a realization of this within scientific communities will have a revolutionary influence on physics, chemistry, biology, and most other scientific areas, hence the book's title. The book was met with skepticism and criticism that Wolfram took credit for the work of others and made conclusions without evidence to support them. === Wolfram Alpha computational knowledge engine === In March 2009, Wolfram announced Wolfram Alpha, an answer engine. Wolfram Alpha launched in May 2009, and a paid-for version with extra features launched in February 2012 that was met with criticism for its high price, which later dropped from $50 to $2. The engine is based on natural language processing and a large library of rules-based algorithms. The application programming interface allows other applications to extend and enhance Wolfram Alpha. === Touchpress === In 2010, Wolfram co-founded Touchpress with Theodore Gray, Max Whitby, and John Cromie. The company specialised in creating in-depth premium apps and games covering a wide range of educational subjects designed for children, parents, students, and educators. Touchpress published more than 100 apps. The company is no longer active. === Wolfram Language === In March 2014, at the annual South by Southwest (SXSW) event, Wolfram officially announced the Wolfram Language as a new general multi-paradigm programming language, though it was previously available through Mathematica and not an entirely new programming language. The documentation for the language was pre-released in October 2013 to coincide with the bundling of Mathematica and the Wolfram Language on every Raspberry Pi computer with some controversy because of the proprietary nature of the Wolfram Language. While the Wolfram Language has existed for over 30 years as the primary programming language used in Mathematica, it was not officially named until 2014, and is not widely used. === Wolfram Physics Project === In April 2020, Wolfram announced the "Wolfram Physics Project" as an effort to reduce and explain all the laws of physics within a paradigm of a hypergraph that is transformed by minimal rewriting rules that obey the Church–Rosser property. The effort is a continuation of the ideas he originally described in A New Kind of Science. Wolfram claims that "From an extremely simple model, we're able to reproduce special relativity, general relativity and the core results of quantum mechanics." Physicists are generally unimpressed with Wolfram's claim, and say his results are non-quantitative and arbitrary. == Personal interests and activities == Wolfram has a log of personal analytics, including emails received and sent, keystrokes made, meetings and events attended, recordings of phone calls, and even physical movement dating back to the 1980s. In the preface of A New Kind of Science, he noted that he recorded over 100 million keystrokes and 100 mouse miles. He has said that personal analytics "can give us a whole new dimension to experiencing our lives." Wolfram was a scientific consultant for the 2016 film Arrival. He and his son Christopher Wolfram wrote some of the code featured on screen, such as the code in graphics depicting an analysis of the alien logograms, for which they used the Wolfram Language.
MeeMix
MeeMix Ltd is a company specializing in personalizing media-related content recommendations, discovery and advertising for the telecommunication industry, founded in 2006. On January 1, 2008, MeeMix launched meemix.com, a public personalized internet radio serving as an online testbed for the development of music taste-prediction technologies. Subsequently, MeeMix released in 2009 a line of Business-to-business commercial services intended to personalize media recommendations, discovery and advertising. MeeMix hybrid taste-prediction technology relies on integrating machine learning algorithms, digital signal processing, behavior analysis, metadata analysis and collaborative filtering, and is provided via API web service. In August 2009, MeeMix was announced as Innovator Nominee in the GSM Association’s Mobile Innovation Grand Prix worldwide contest. As of 2013, MeeMix no longer features internet radios on meemix.com. On Sep 28, 2014, meemix.com went offline.
Very large database
A very large database, (originally written very large data base) or VLDB, is a database that contains a very large amount of data, so much that it can require specialized architectural, management, processing and maintenance methodologies. == Definition == The vague adjectives of very and large allow for a broad and subjective interpretation, but attempts at defining a metric and threshold have been made. Early metrics were the size of the database in a canonical form via database normalization or the time for a full database operation like a backup. Technology improvements have continually changed what is considered very large. One definition has suggested that a database has become a VLDB when it is "too large to be maintained within the window of opportunity… the time when the database is quiet". == Sizes of a VLDB database == There is no absolute amount of data that can be cited. For example, one cannot say that any database with more than 1 TB of data is considered a VLDB. This absolute amount of data has varied over time as computer processing, storage and backup methods have become better able to handle larger amounts of data. That said, VLDB issues may start to appear when 1 TB is approached, and are more than likely to have appeared as 30 TB or so is exceeded. == VLDB challenges == Key areas where a VLDB may present challenges include configuration, storage, performance, maintenance, administration, availability and server resources. === Configuration === Careful configuration of databases that lie in the VLDB realm is necessary to alleviate or reduce issues raised by VLDB databases. === Administration === The complexities of managing a VLDB can increase exponentially for the database administrator as database size increases. === Availability and maintenance === When dealing with VLDB operations relating to maintenance and recovery such as database reorganizations and file copies which were quite practical on a non-VLDB take very significant amounts of time and resources for a VLDB database. In particular it typically infeasible to meet a typical recovery time objective (RTO), the maximum expected time a database is expected to be unavailable due to interruption, by methods which involve copying files from disk or other storage archives. To overcome these issues techniques such as clustering, cloned/replicated/standby databases, file-snapshots, storage snapshots or a backup manager may help achieve the RTO and availability, although individual methods may have limitations, caveats, license, and infrastructure requirements while some may risk data loss and not meet the recovery point objective (RPO). For many systems only geographically remote solutions may be acceptable. ==== Backup and recovery ==== Best practice is for backup and recovery to be architectured in terms of the overall availability and business continuity solution. === Performance === Given the same infrastructure there may typically be a decrease in performance, that is increase in response time as database size increases. Some accesses will simply have more data to process (scan) which will take proportionally longer (linear time); while the indexes used to access data may grow slightly in height requiring perhaps an extra storage access to reach the data (sub-linear time). Other effects can be caching becoming less efficient because proportionally less data can be cached and while some indexes such as the B+ automatically sustain well with growth others such as a hash table may need to be rebuilt. Should an increase in database size cause the number of accessors of the database to increase then more server and network resources may be consumed, and the risk of contention will increase. Some solutions to regaining performance include partitioning, clustering, possibly with sharding, or use of a database machine. ==== Partitioning ==== Partitioning may be able assist the performance of bulk operations on a VLDB including backup and recovery., bulk movements due to information lifecycle management (ILM), reducing contention as well as allowing optimization of some query processing. === Storage === In order to satisfy needs of a VLDB the database storage needs to have low access latency and contention, high throughput, and high availability. === Server resources === The increasing size of a VLDB may put pressure on server and network resources and a bottleneck may appear that may require infrastructure investment to resolve. == Relationship to big data == VLDB is not the same as big data, but the storage aspect of big data may involve a VLDB database. That said some of the storage solutions supporting big data were designed from the start to support large volumes of data, so database administrators may not encounter VLDB issues that older versions of traditional RDBMS's might encounter.
External memory algorithm
In computing, external memory algorithms or out-of-core algorithms are algorithms that are designed to process data that are too large to fit into a computer's main memory at once. Such algorithms must be optimized to efficiently fetch and access data stored in slow bulk memory (auxiliary memory) such as hard drives or tape drives, or when memory is on a computer network. External memory algorithms are analyzed in the external memory model. == Model == External memory algorithms are analyzed in an idealized model of computation called the external memory model (or I/O model, or disk access model). The external memory model is an abstract machine similar to the RAM machine model, but with a cache in addition to main memory. The model captures the fact that read and write operations are much faster in a cache than in main memory, and that reading long contiguous blocks is faster than reading randomly using a disk read-and-write head. The running time of an algorithm in the external memory model is defined by the number of reads and writes to memory required. The model was introduced by Alok Aggarwal and Jeffrey Vitter in 1988. The external memory model is related to the cache-oblivious model, but algorithms in the external memory model may know both the block size and the cache size. For this reason, the model is sometimes referred to as the cache-aware model. The model consists of a processor with an internal memory or cache of size M, connected to an unbounded external memory. Both the internal and external memory are divided into blocks of size B. One input/output or memory transfer operation consists of moving a block of B contiguous elements from external to internal memory, and the running time of an algorithm is determined by the number of these input/output operations. == Algorithms == Algorithms in the external memory model take advantage of the fact that retrieving one object from external memory retrieves an entire block of size B. This property is sometimes referred to as locality. Searching for an element among N objects is possible in the external memory model using a B-tree with branching factor B. Using a B-tree, searching, insertion, and deletion can be achieved in O ( log B N ) {\displaystyle O(\log _{B}N)} time (in Big O notation). Information theoretically, this is the minimum running time possible for these operations, so using a B-tree is asymptotically optimal. External sorting is sorting in an external memory setting. External sorting can be done via distribution sort, which is similar to quicksort, or via a M B {\displaystyle {\tfrac {M}{B}}} -way merge sort. Both variants achieve the asymptotically optimal runtime of O ( N B log M B N B ) {\displaystyle O\left({\frac {N}{B}}\log _{\frac {M}{B}}{\frac {N}{B}}\right)} to sort N objects. This bound also applies to the fast Fourier transform in the external memory model. The permutation problem is to rearrange N elements into a specific permutation. This can either be done either by sorting, which requires the above sorting runtime, or inserting each element in order and ignoring the benefit of locality. Thus, permutation can be done in O ( min ( N , N B log M B N B ) ) {\displaystyle O\left(\min \left(N,{\frac {N}{B}}\log _{\frac {M}{B}}{\frac {N}{B}}\right)\right)} time. == Applications == The external memory model captures the memory hierarchy, which is not modeled in other common models used in analyzing data structures, such as the random-access machine, and is useful for proving lower bounds for data structures. The model is also useful for analyzing algorithms that work on datasets too big to fit in internal memory. A typical example is geographic information systems, especially digital elevation models, where the full data set easily exceeds several gigabytes or even terabytes of data. This methodology extends beyond general purpose CPUs and also includes GPU computing as well as classical digital signal processing. In general-purpose computing on graphics processing units (GPGPU), powerful graphics cards (GPUs) with little memory (compared with the more familiar system memory, which is most often referred to simply as RAM) are utilized with relatively slow CPU-to-GPU memory transfer (when compared with computation bandwidth). == History == An early use of the term "out-of-core" as an adjective is in 1962 in reference to devices that are other than the core memory of an IBM 360. An early use of the term "out-of-core" with respect to algorithms appears in 1971.
Pseudonymization
Pseudonymization is a data management and de-identification procedure by which personally identifiable information fields within a data record are replaced by one or more artificial identifiers, or pseudonyms. A single pseudonym for each replaced field or collection of replaced fields makes the data record less identifiable while remaining suitable for data analysis and data processing. Pseudonymization (or pseudonymisation, the spelling under European guidelines) is one way to comply with the European Union's General Data Protection Regulation (GDPR) demands for secure data storage of personal information. Pseudonymized data can be restored to its original state with the addition of information which allows individuals to be re-identified. In contrast, anonymization is intended to prevent re-identification of individuals within the dataset. Clause 18, Module Four, footnote 2 of the Adoption by the European Commission of the Implementing Decisions (EU) 2021/914 "requires rendering the data anonymous in such a way that the individual is no longer identifiable by anyone ... and that this process is irreversible." == Impact of Schrems II ruling == The European Data Protection Supervisor (EDPS) on 9 December 2021 highlighted pseudonymization as the top technical supplementary measure for Schrems II compliance. Less than two weeks later, the EU Commission highlighted pseudonymization as an essential element of the equivalency decision for South Korea, which is the status that was lost by the United States under the Schrems II ruling by the Court of Justice of the European Union (CJEU). The importance of GDPR-compliant pseudonymization increased dramatically in June 2021 when the European Data Protection Board (EDPB) and the European Commission highlighted GDPR-compliant pseudonymization as the state-of-the-art technical supplementary measure for the ongoing lawful use of EU personal data when using third country (i.e., non-EU) cloud processors or remote service providers under the "Schrems II" ruling by the CJEU. Under the GDPR and final EDPB Schrems II Guidance, the term pseudonymization requires a new protected "state" of data, producing a protected outcome that: Protects direct, indirect, and quasi-identifiers, together with characteristics and behaviors; Protects at the record and data set level versus only the field level so that the protection travels wherever the data goes, including when it is in use; and Protects against unauthorized re-identification via the mosaic effect by generating high entropy (uncertainty) levels by dynamically assigning different tokens at different times for various purposes. The combination of these protections is necessary to prevent the re-identification of data subjects without the use of additional information kept separately, as required under GDPR Article 4(5) and as further underscored by paragraph 85(4) of the final EDPB Schrems II guidance: Article 4(5) "Definitions" of the GDPR defines pseudonymization as "the processing of personal data in such a manner that the personal data can no longer be attributed to a specific data subject without the use of additional information, provided that such additional information is kept separately and is subject to technical and organisational measures to ensure that the personal data are not attributed to an identified or identifiable natural person." "Use Case 2: Transfer of pseudonymised Data Paragraph 85(4)" of the final EDPB Schrems II Guidance requires that “the controller has established by means of a thorough analysis of the data in question – taking into account any information that the public authorities of the recipient country may be expected to possess and use – that the pseudonymised personal data cannot be attributed to an identified or identifiable natural person even if cross-referenced with such information." GDPR-compliant pseudonymization requires that data is "anonymous" in the strictest EU sense of the word – globally anonymous – but for the additional information held separately and made available under controlled conditions as authorized by the data controller for permitted re-identification of individual data subjects. Clause 18, Module Four, footnote 2 of the Adoption by the European Commission of the Implementing Decision (EU) 2021/914 "requires rendering the data anonymous in such a way that the individual is no longer identifiable by anyone, in line with recital 26 of Regulation (EU) 2016/679, and that this process is irreversible." Before the Schrems II ruling, pseudonymization was a technique used by security experts or government officials to hide personally identifiable information to maintain data structure and privacy of information. Some common examples of sensitive information include postal code, location of individuals, names of individuals, race and gender, etc. After the Schrems II ruling, GDPR-compliant pseudonymization must satisfy the above-noted elements as an "outcome" versus merely a technique. == Data fields == The choice of which data fields are to be pseudonymized is partly subjective. Less selective fields, such as birth date or postal code are often also included because they are usually available from other sources and therefore make a record easier to identify. Pseudonymizing these less identifying fields removes most of their analytic value and is therefore normally accompanied by the introduction of new derived and less identifying forms, such as year of birth or a larger postal code region. Data fields that are less identifying, such as date of attendance, are usually not pseudonymized. This is because too much statistical utility is lost in doing so, not because the data cannot be identified. For example, given prior knowledge of a few attendance dates it is easy to identify someone's data in a pseudonymized dataset by selecting only those people with that pattern of dates. This is an example of an inference attack. The weakness of pre-GDPR pseudonymized data to inference attacks is commonly overlooked. A famous example is the AOL search data scandal. The AOL example of unauthorized re-identification did not require access to separately kept "additional information" that was under the control of the data controller as is now required for GDPR-compliant pseudonymization, outlined below under the section "New Definition for Pseudonymization Under GDPR". Protecting statistically useful pseudonymized data from re-identification requires: a sound information security base controlling the risk that the analysts, researchers or other data workers cause a privacy breach The pseudonym allows tracking back of data to its origins, which distinguishes pseudonymization from anonymization, where all person-related data that could allow backtracking has been purged. Pseudonymization is an issue in, for example, patient-related data that has to be passed on securely between clinical centers. The application of pseudonymization to e-health intends to preserve the patient's privacy and data confidentiality. It allows primary use of medical records by authorized health care providers and privacy preserving secondary use by researchers. In the US, HIPAA provides guidelines on how health care data must be handled and data de-identification or pseudonymization is one way to simplify HIPAA compliance. However, plain pseudonymization for privacy preservation often reaches its limits when genetic data are involved (see also genetic privacy). Due to the identifying nature of genetic data, depersonalization is often not sufficient to hide the corresponding person. Potential solutions are the combination of pseudonymization with fragmentation and encryption. An example of application of pseudonymization procedure is creation of datasets for de-identification research by replacing identifying words with words from the same category (e.g. replacing a name with a random name from the names dictionary), however, in this case it is in general not possible to track data back to its origins. == New definition under GDPR == Effective as of May 25, 2018, the EU General Data Protection Regulation (GDPR) defines pseudonymization for the very first time at the EU level in Article 4(5). Under Article 4(5) definitional requirements, data is pseudonymized if it cannot be attributed to a specific data subject without the use of separately kept "additional information". Pseudonymized data embodies the state of the art in Data Protection by Design and by Default because it requires protection of both direct and indirect identifiers (not just direct). GDPR Data Protection by Design and by Default principles as embodied in pseudonymization require protection of both direct and indirect identifiers so that personal data is not cross-referenceable (or re-identifiable) via the "mosaic effect" without access to "additional information" that is kept separately by the controller. Because access to separately kept "additional information" is required
Labeled data
Labeled data is a group of samples that have been tagged with one or more labels. Labeling typically takes a set of unlabeled data and augments each piece of it with informative tags called judgments. For example, a data label might indicate whether a photo contains a horse or a cow, which words were uttered in an audio recording, what type of action is being performed in a video, what the topic of a news article is, what the overall sentiment of a tweet is, or whether a dot in an X-ray is a tumor. Labels can be obtained by having humans make judgments about a given piece of unlabeled data. Labeled data is significantly more expensive to obtain than the raw unlabeled data. The quality of labeled data directly influences the performance of supervised machine learning models in operation, as these models learn from the provided labels. == Crowdsourced labeled data == In 2006, Fei-Fei Li, the co-director of the Stanford Human-Centered AI Institute, initiated research to improve the artificial intelligence models and algorithms for image recognition by significantly enlarging the training data. The researchers downloaded millions of images from the World Wide Web and a team of undergraduates started to apply labels for objects to each image. In 2007, Li outsourced the data labeling work on Amazon Mechanical Turk, an online marketplace for digital piece work. The 3.2 million images that were labeled by more than 49,000 workers formed the basis for ImageNet, one of the largest hand-labeled database for outline of object recognition. == Automated data labelling == After obtaining a labeled dataset, machine learning models can be applied to the data so that new unlabeled data can be presented to the model and a likely label can be guessed or predicted for that piece of unlabeled data. == Challenges == === Data-driven bias === Algorithmic decision-making is subject to programmer-driven bias as well as data-driven bias. Training data that relies on bias labeled data will result in prejudices and omissions in a predictive model, despite the machine learning algorithm being legitimate. The labeled data used to train a specific machine learning algorithm needs to be a statistically representative sample to not bias the results. For example, in facial recognition systems underrepresented groups are subsequently often misclassified if the labeled data available to train has not been representative of the population,. In 2018, a study by Joy Buolamwini and Timnit Gebru demonstrated that two facial analysis datasets that have been used to train facial recognition algorithms, IJB-A and Adience, are composed of 79.6% and 86.2% lighter skinned humans respectively. === Human error and inconsistency === Human annotators are prone to errors and biases when labeling data. This can lead to inconsistent labels and affect the quality of the data set. The inconsistency can affect the machine learning model's ability to generalize well. === Domain expertise === Certain fields, such as legal document analysis or medical imaging, require annotators with specialized domain knowledge. Without the expertise, the annotations or labeled data may be inaccurate, negatively impacting the machine learning model's performance in a real-world scenario.
Lai–Robbins lower bound
The Lai–Robbins lower bound gives an asymptotic lower bound on the regret that any uniformly good algorithm must incur in the stochastic multi-armed bandit problem. The original result was proved by Tze Leung Lai and Herbert Robbins in 1985 for parametric exponential families. Later work extended the statement to more general classes of distributions. == Multi-armed bandit problem == The multi-armed bandit problem (MAB) is a sequential game in which the player must trade off exploration (to learn) and exploitation (to earn). The player chooses among K {\displaystyle K} actions (arms) with unknown distributions ν = ( ν 1 , … , ν K ) {\displaystyle \nu =(\nu _{1},\dots ,\nu _{K})} . The player is assumed to know a class of distributions D {\displaystyle {\mathcal {D}}} such that for every k {\displaystyle k} one has ν k ∈ D {\displaystyle \nu _{k}\in {\mathcal {D}}} (for example, D {\displaystyle {\mathcal {D}}} may be the family of Gaussian or Bernoulli distributions). At each round t = 1 , … , T {\displaystyle t=1,\dots ,T} the player selects (pulls) an arm a t {\displaystyle a_{t}} and observes a reward X t ∼ ν a t {\displaystyle X_{t}\sim \nu _{a_{t}}} . We denote N a ( t ) := ∑ s = 1 t 1 { a s = a } {\displaystyle N_{a}(t):=\sum _{s=1}^{t}\mathbf {1} _{\{a_{s}=a\}}} the number of times arm a {\displaystyle a} has been pulled in the first t {\displaystyle t} rounds, μ ( ν ) := ( μ 1 , … , μ K ) {\displaystyle \mu (\nu ):=(\mu _{1},\dots ,\mu _{K})} the vector of arm means, where μ k = E X ∼ ν k [ X ] {\displaystyle \mu _{k}=\mathbb {E} _{X\sim \nu _{k}}[X]} , μ ∗ := max a μ a {\displaystyle \mu ^{}:=\max _{a}\mu _{a}} the highest mean Δ a := μ ∗ − μ a ≥ 0 {\displaystyle \Delta _{a}:=\mu ^{}-\mu _{a}\geq 0} the gap of arm a {\displaystyle a} . An arm a {\displaystyle a} with μ a = μ ∗ {\displaystyle \mu _{a}=\mu ^{}} is called an optimal arm; otherwise it is a suboptimal arm. The goal is to minimize the regret at horizon T {\displaystyle T} , defined by R T := ∑ a = 1 K Δ a E [ N a ( T ) ] . {\displaystyle R_{T}:=\sum _{a=1}^{K}\Delta _{a}\,\mathbb {E} [N_{a}(T)].} Intuitively, the regret is the (expected) total loss compared to always playing an optimal arm: regret = ∑ a ( cost of playing a ) × ( times a is played ) . {\displaystyle {\text{regret}}=\sum _{a}\ ({\text{cost of playing }}a)\times ({\text{times }}a{\text{ is played}}).} An MAB algorithm is a (possibly randomized) policy that, at each round t {\displaystyle t} , choose an arm a_t by using the observations received from previous turns. === Intuitive example === Suppose a farmer must choose, each year, one of K {\displaystyle K} seed varieties to plant. Each variety k {\displaystyle k} has an unknown average yield μ k {\displaystyle \mu _{k}} . If the farmer knew the best variety (with mean μ ∗ {\displaystyle \mu ^{}} ) he would plant it every year; in reality he must try varieties to learn which is best. The cumulative regret after T {\displaystyle T} years measures the total expected loss in yield due to imperfect knowledge. Remarks The model above is the stochastic MAB; there also exist adversarial variants. One may consider a fixed-horizon setting (known T {\displaystyle T} ) or an anytime setting (unknown T {\displaystyle T} ). == Lai–Robbins lower bound == The theorem gives the right amount of time we should pull a suboptimal arm k {\displaystyle k} to distinguish whether we are in the instance with ν k {\displaystyle \nu _{k}} or with ν ~ k {\displaystyle {\tilde {\nu }}_{k}} where ν ~ k {\displaystyle {\tilde {\nu }}_{k}} is such that μ ~ k > μ ∗ {\displaystyle {\tilde {\mu }}_{k}>\mu ^{}} . Knowning a lower bound on the number of pull of every suboptimal arm gives a lower bound on the regret as only suboptimal arms contribute to the regret. Before stating the formal theorem we need to define what is a consistent algorithm. === Consistency (uniformly good algorithms) === Let D {\displaystyle {\mathcal {D}}} be a class of probability distributions and consider K {\displaystyle K} arms with reward distributions ν = ( ν 1 , … , ν K ) ∈ D K {\displaystyle \nu =(\nu _{1},\dots ,\nu _{K})\in {\mathcal {D}}^{K}} . An algorithm is said to be consistent (also called uniformly good) on D K {\displaystyle {\mathcal {D}}^{K}} if, for every instance ν ∈ D K {\displaystyle \nu \in {\mathcal {D}}^{K}} , the expected regret R T ( ν ) {\displaystyle R_{T}(\nu )} grows subpolynomially: ∀ α > 0 , R T ( ν ) = o ( T α ) as T → ∞ {\displaystyle \forall \alpha >0,\qquad R_{T}(\nu )=o(T^{\alpha })\quad {\text{as }}T\to \infty } This assumption excludes algorithms that perform well on some instances but incur linear regret on others. === Formal lower bound === For any suboptimal arm a {\displaystyle a} . For a distribution ν a ∈ D {\displaystyle \nu _{a}\in {\mathcal {D}}} and a threshold x {\displaystyle x} , define K inf ( ν a , x , D ) := inf { KL ( ν a , ν ′ ) : ν ′ ∈ D , μ ′ > x } {\displaystyle {\mathcal {K}}_{\inf }(\nu _{a},x,{\mathcal {D}}):=\inf {\Bigl \{}\operatorname {KL} (\nu _{a},\nu '):\nu '\in {\mathcal {D}},\ \mu '>x{\Bigr \}}} where KL ( ⋅ , ⋅ ) {\displaystyle \operatorname {KL} (\cdot ,\cdot )} denotes the Kullback-Leibler divergence. Then, for any algorithm consistent on D K {\displaystyle {\mathcal {D}}^{K}} and for every instance ν ∈ D K {\displaystyle \nu \in {\mathcal {D}}^{K}} , every suboptimal arm a {\displaystyle a} satisfies E ν [ N a ( T ) ] ≥ ln T K inf ( ν a , μ ∗ , D ) + o ( ln T ) {\displaystyle \mathbb {E} _{\nu }[N_{a}(T)]\geq {\frac {\ln T}{{\mathcal {K}}_{\inf }(\nu _{a},\mu ^{},{\mathcal {D}})}}+o(\ln T)} Consequently, the regret satisfies R T ( ν ) ≥ ( ∑ a : μ a < μ ∗ Δ a K inf ( ν a , μ ∗ , D ) ) ln T + o ( ln T ) {\displaystyle R_{T}(\nu )\geq \left(\sum _{a:\,\mu _{a}<\mu ^{}}{\frac {\Delta _{a}}{{\mathcal {K}}_{\inf }(\nu _{a},\mu ^{},{\mathcal {D}})}}\right)\ln T+o(\ln T)} The original 1985 paper established this result for exponential families; later work showed that the bound holds under much weaker assumptions on D {\displaystyle {\mathcal {D}}} . === Intuition === Consistency imposes that, for every ν {\displaystyle \nu } , the number of pulls of an optimal arm must be large. This means that μ ∗ {\displaystyle \mu ^{}} is estimated very accurately. The goal is to determine, for a suboptimal arm k {\displaystyle k} , how many samples are needed to be confident, with the appropriate level of confidence, that μ k < μ ∗ {\displaystyle \mu _{k}<\mu ^{}} . To do so, we use what is called the most confusing instance: an instance close to ν {\displaystyle \nu } such that arm k {\displaystyle k} is optimal. We define it as ν ~ {\displaystyle {\tilde {\nu }}} such that, for all a ≠ k {\displaystyle a\neq k} , ν ~ a = ν a {\displaystyle {\tilde {\nu }}_{a}=\nu _{a}} , and ν ~ k {\displaystyle {\tilde {\nu }}_{k}} is chosen so that μ ~ k > μ ∗ {\displaystyle {\tilde {\mu }}_{k}>\mu ^{}} . The objective is to determine how many samples of arm k {\displaystyle k} are required to distinguish whether we are in the instance with ν k {\displaystyle \nu _{k}} or with ν ~ k {\displaystyle {\tilde {\nu }}_{k}} in terms of KL {\displaystyle \operatorname {KL} } distance. == Algorithms achieving the Lai–Robbins lower bound == Several algorithms are known to achieve the Lai–Robbins asymptotic lower bound under specific assumptions on the reward distribution class D {\displaystyle {\mathcal {D}}} . The following list summarizes a non-exhaustive list of algorithms matching the lower bound. == Extension to other problems == === Structured bandit === A more complexe is structured bandit where we know that the mean of each arm is in a set with some restriction. In this case we can prove a smaller lower bound that use the knowledge of this set. === Best arm identification (BAI) === A similar result has been proved for best arm identification, which is the same game except that, instead of minimizing the regret, the goal is to identify the best arm with probability 1 − δ {\displaystyle 1-\delta } using as few rounds as possible. === Reinforcement Learning (RL) === Similar results have been proved for regret minimization in average-reward reinforcement learning. The order is also ln T {\displaystyle \ln T} , with a constant that depends on the problem.