In computational learning theory, probably approximately correct (PAC) learning is a framework for mathematical analysis of machine learning. It was proposed in 1984 by Leslie Valiant. In this framework, the learner receives samples and must select a generalization function (called the hypothesis) from a certain class of possible functions. The goal is that, with high probability (the "probably" part), the selected function will have low generalization error (the "approximately correct" part). The learner must be able to learn the concept given any arbitrary approximation ratio, probability of success, or distribution of the samples. The model was later extended to treat noise (misclassified samples). An important innovation of the PAC framework is the introduction of computational complexity theory concepts to machine learning. In particular, the learner is expected to find efficient functions (time and space requirements bounded to a polynomial of the example size), and the learner itself must implement an efficient procedure (requiring an example count bounded to a polynomial of the concept size, modified by the approximation and likelihood bounds). == Definitions and terminology == In order to give the definition for something that is PAC-learnable, we first have to introduce some terminology. For the following definitions, two examples will be used. The first is the problem of character recognition given an array of n {\displaystyle n} bits encoding a binary-valued image. The other example is the problem of finding an interval that will correctly classify points within the interval as positive and the points outside of the range as negative. Let X {\displaystyle X} be a set called the instance space or the encoding of all the samples. In the character recognition problem, the instance space is X = { 0 , 1 } n {\displaystyle X=\{0,1\}^{n}} . In the interval problem the instance space, X {\displaystyle X} , is the set of all bounded intervals in R {\displaystyle \mathbb {R} } , where R {\displaystyle \mathbb {R} } denotes the set of all real numbers. A concept is a subset c ⊂ X {\displaystyle c\subset X} . One concept is the set of all patterns of bits in X = { 0 , 1 } n {\displaystyle X=\{0,1\}^{n}} that encode a picture of the letter "P". An example concept from the second example is the set of open intervals, { ( a , b ) ∣ 0 ≤ a ≤ π / 2 , π ≤ b ≤ 13 } {\displaystyle \{(a,b)\mid 0\leq a\leq \pi /2,\pi \leq b\leq {\sqrt {13}}\}} , each of which contains only the positive points. A concept class C {\displaystyle C} is a collection of concepts over X {\displaystyle X} . This could be the set of all subsets of the array of bits that are skeletonized 4-connected (width of the font is 1). Let EX ( c , D ) {\displaystyle \operatorname {EX} (c,D)} be a procedure that draws an example, x {\displaystyle x} , using a probability distribution D {\displaystyle D} and gives the correct label c ( x ) {\displaystyle c(x)} , that is 1 if x ∈ c {\displaystyle x\in c} and 0 otherwise. Now, given 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} , assume there is an algorithm A {\displaystyle A} and a polynomial p {\displaystyle p} in 1 / ϵ , 1 / δ {\displaystyle 1/\epsilon ,1/\delta } (and other relevant parameters of the class C {\displaystyle C} ) such that, given a sample of size p {\displaystyle p} drawn according to EX ( c , D ) {\displaystyle \operatorname {EX} (c,D)} , then, with probability of at least 1 − δ {\displaystyle 1-\delta } , A {\displaystyle A} outputs a hypothesis h ∈ C {\displaystyle h\in C} that has an average error less than or equal to ϵ {\displaystyle \epsilon } on X {\displaystyle X} with the same distribution D {\displaystyle D} . Further if the above statement for algorithm A {\displaystyle A} is true for every concept c ∈ C {\displaystyle c\in C} and for every distribution D {\displaystyle D} over X {\displaystyle X} , and for all 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} then C {\displaystyle C} is (efficiently) PAC learnable (or distribution-free PAC learnable). We can also say that A {\displaystyle A} is a PAC learning algorithm for C {\displaystyle C} . == Equivalence == Under some regularity conditions these conditions are equivalent: The concept class C is PAC learnable. The VC dimension of C is finite. C is a uniformly Glivenko-Cantelli class. C is compressible in the sense of Littlestone and Warmuth
Data cube
In computer programming, a data cube (or datacube) is a multi-dimensional array of values. Typically, the term "data cube" is applied in contexts where these arrays are massively larger than the hosting computer's main memory; examples include multi-terabyte/petabyte data warehouses and time series of image data. Even though it is called a cube, a data cube generally is a multi-dimensional concept which can be 1-dimensional, 2-dimensional, 3-dimensional, or higher-dimensional. The data cube is used to represent data (sometimes called facts) along some dimensions of interest. In satellite image timeseries, dimensions would be latitude and longitude coordinates and time; a fact (sometimes called measure) would be a pixel at a given space and time as taken by the satellite. For example, in online analytical processing, an OLAP cube about a company would have dimensions that could be the company subsidiaries, the company products, and time; in this setup, a fact would be a sales event where a particular product has been sold in a particular subsidiary at a particular time. In any case, every dimension divides data into groups of cells whereas each cell in the cube represents a single measure of interest. Sometimes cubes hold only a few values with the rest being empty, i.e. undefined, while sometimes most or all cube coordinates hold a cell value. In the first case such data are called sparse, and in the second case they are called dense, although there is no hard delineation between the two. Data cubes may be stored in database management systems (DBMS) as part of array DBMS. Spatio-temporal databases and geospatial databases may also be represented as coverage data. == History == Multi-dimensional arrays have long been familiar in programming languages. Fortran offers arbitrarily-indexed 1-D arrays and arrays of arrays, which allows the construction of higher-dimensional arrays, up to 15 dimensions. APL supports n-D arrays with a rich set of operations. All these have in common that arrays must fit into the main memory and are available only while the particular program maintaining them (such as image processing software) is running. A series of data exchange formats support storage and transmission of data cube-like data, often tailored towards particular application domains. Examples include MDX for statistical (in particular, business) data, Zarr and Hierarchical Data Format for general scientific data, and TIFF for imagery. In 1992, Peter Baumann introduced management of massive data cubes with high-level user functionality combined with an efficient software architecture. Datacube operations include subset extraction, processing, fusion, and in general queries in the spirit of data manipulation languages like SQL. Some years after, the data cube concept was applied to describe time-varying business data as data cubes by Jim Gray, et al., and by Venky Harinarayan, Anand Rajaraman and Jeff Ullman. Around that time, a working group on Multi-Dimensional Databases ("Arbeitskreis Multi-Dimensionale Datenbanken") was established at German Gesellschaft für Informatik. Datacube Inc. was an image processing company selling hardware and software applications for the PC market in 1996, however without addressing data cubes as such. The EarthServer initiative has established geo data cube service requirements. == Standardization == In 2018, the ISO SQL database language was extended with data cube functionality as "SQL – Part 15: Multi-dimensional arrays (SQL/MDA)". Web Coverage Processing Service is a geo data cube analytics language issued by the Open Geospatial Consortium in 2008. In addition to the common data cube operations, the language knows about the semantics of space and time and supports both regular and irregular grid data cubes, based on the concept of coverage data. An industry standard for querying business data cubes, originally developed by Microsoft, is MultiDimensional eXpressions. == Implementation == Many high-level computer languages treat data cubes and other large arrays as single entities distinct from their contents. These languages, of which Fortran, APL, IDL, NumPy, PDL, and S-Lang are examples, allow the programmer to manipulate complete film clips and other data en masse with simple expressions derived from linear algebra and vector mathematics. Some languages (such as PDL) distinguish between a list of images and a data cube, while many (such as IDL) do not. Array DBMSs (Database Management Systems) offer a data model which generically supports definition, management, retrieval, and manipulation of n-dimensional data cubes. This database category has been pioneered by the rasdaman system since 1994. == Applications == Multi-dimensional arrays can meaningfully represent spatio-temporal sensor, image, and simulation data, but also statistics data where the semantics of dimensions is not necessarily of spatial or temporal nature. Generally, any kind of axis can be combined with any other into a data cube. === Mathematics === In mathematics, a one-dimensional array corresponds to a vector, a two-dimensional array resembles a matrix; more generally, a tensor may be represented as an n-dimensional data cube. === Science and engineering === For a time sequence of color images, the array is generally four-dimensional, with the dimensions representing image X and Y coordinates, time, and RGB (or other color space) color plane. For example, the EarthServer initiative unites data centers from different continents offering 3-D x/y/t satellite image timeseries and 4-D x/y/z/t weather data for retrieval and server-side processing through the Open Geospatial Consortium WCPS geo data cube query language standard. A data cube is also used in the field of imaging spectroscopy, since a spectrally-resolved image is represented as a three-dimensional volume. Earth observation data cubes combine satellite imagery such as Landsat 8 and Sentinel-2 with Geographic information system analytics. === Business intelligence === In online analytical processing (OLAP), data cubes are a common arrangement of business data suitable for analysis from different perspectives through operations like slicing, dicing, pivoting, and aggregation.
Ilya Sutskever
Ilya Sutskever (Hebrew: איליה סוצקבר; born 1986) is a computer scientist who specializes in machine learning. He has made several major contributions to the field of deep learning, including sequence-to-sequence learning, reasoning models, GPT models, and contributions to CLIP, DALL-E, and AlphaGo. With Alex Krizhevsky and Geoffrey Hinton, he co-created AlexNet, a convolutional neural network. One of the most highly cited computer scientists in history, he has won the NeurIPS Test of Time Award for his lasting impact on AI research three times in a row (2022–2024) and received the National Academy of Sciences Award for the Industrial Application of Science in 2026. Sutskever co-founded and was chief scientist at OpenAI, where he oversaw the research breakthroughs that led to large language models and to the launch of ChatGPT. He also led the research that led to reasoning models such as o1. In 2023, he was one of the members of OpenAI's board that ousted Sam Altman as its CEO; Altman was reinstated a week later, and Sutskever stepped down from the board. In June 2024, Sutskever co-founded the company Safe Superintelligence Inc., alongside Daniel Gross and Daniel Levy. Within a year, the company was valued at more than $30 billion. == Early life and education == Sutskever was born in 1986 into a Jewish family in Nizhny Novgorod, Russia (then Gorky, Russian SFSR, Soviet Union). At the age of 5, he immigrated to Israel with his family and grew up in Jerusalem. Sutskever proved to be a good student in school, and in eighth grade started taking classes at the Open University of Israel. At 16, he moved with his family to Canada, where he attended high school for a month before being admitted to the University of Toronto in Ontario as a third-year undergraduate student. At the University of Toronto, Sutskever received a bachelor's degree in mathematics in 2005, a master's degree in computer science in 2007, and a PhD in computer science in 2013. His doctoral advisor was Geoffrey Hinton. In 2012, Sutskever built AlexNet in collaboration with Geoffrey Hinton and Alex Krizhevsky. == Career and research == In 2012, Sutskever spent about two months as a postdoc with Andrew Ng at Stanford University. He then returned to the University of Toronto and joined Hinton's new research company DNNResearch, a spinoff of Hinton's research group. In 2013, Google acquired DNNResearch and hired Sutskever as a research scientist at Google Brain. At Google Brain, Sutskever worked with Oriol Vinyals and Quoc Viet Le to create the sequence-to-sequence learning algorithm, and worked on TensorFlow. He is also one of the AlphaGo paper's many co-authors. At the end of 2015, Sutskever left Google to become cofounder and chief scientist of the newly founded organization OpenAI. In 2022, Sutskever tweeted, "it may be that today's large neural networks are slightly conscious", which triggered debates about AI consciousness. He is considered to have played a key role in the development of ChatGPT, and later in leading the research that led to reasoning models. He is credited with establishing OpenAI’s scaling ethos. In 2023, he announced that he would co-lead OpenAI's new "Superalignment" project, which was trying to solve the alignment of superintelligences within four years. He wrote that even if superintelligence seems far off, it could happen this decade. Sutskever was formerly one of the six board members of the nonprofit entity that controlled OpenAI. In November 2023, the board fired Sam Altman, saying that "he was not consistently candid in his communications with the board". He authored a 52-page memo that relied heavily on information from Mira Murati, accusing Altman of lying, manipulating executives, and fostering internal division. Sutskever submitted the memo to the board after months of tension and dissatisfaction with Altman's leadership style, and ultimately joined the board in voting for Altman's termination. In an all-hands company meeting shortly after the board meeting, Sutskever said that firing Altman was "the board doing its duty", but the next week, he expressed regret at having participated in Altman's ouster. Altman's firing and OpenAI's co-founder Greg Brockman's resignation led three senior researchers to resign from OpenAI. After that, Sutskever stepped down from the OpenAI board and was absent from OpenAI's office. Some sources suggested he was leading the team remotely, while others said he no longer had access to the team's work. In May 2024, Sutskever announced his departure from OpenAI to focus on a new project that was "very personally meaningful" to him. His decision followed a turbulent period at OpenAI marked by leadership crises and internal debates about the direction of AI development and alignment protocols. Jan Leike, the other leader of the superalignment project, announced his departure hours later, citing an erosion of safety and trust in OpenAI's leadership. In June 2024, Sutskever announced Safe Superintelligence Inc., a new company he founded with Daniel Gross and Daniel Levy with offices in Palo Alto and Tel Aviv. In contrast to OpenAI, which releases revenue-generating products, Sutskever said the new company's "first product will be the safe superintelligence, and it will not do anything else up until then". In September 2024, the company announced that it had raised $1 billion from venture capital firms including Andreessen Horowitz, Sequoia Capital, DST Global, and SV Angel. In March 2025, Safe Superintelligence Inc. raised $2 billion more and reportedly reached a $32 billion valuation, notably due to Sutskever's reputation. In June 2025, SSI rejected an offer from Meta Platforms to buy the company. Sutskever became CEO of SSI shortly thereafter, after co-founder and CEO Gross left for Meta. In an October 2024 interview after winning the Nobel Prize in Physics, Geoffrey Hinton expressed support for Sutskever's decision to fire Altman, emphasizing concerns about AI safety. During the Musk v. Altman trial in 2026, Sutskever confirmed he had a $7 billion stake in OpenAI. === Awards and honors === In 2015, Sutskever was named in MIT Technology Review's 35 Innovators Under 35. In 2018, he was the keynote speaker at Nvidia Ntech 2018 and AI Frontiers Conference 2018. In 2022, he was elected a Fellow of the Royal Society (FRS). In 2023 and 2024, included in Time's list of the 100 most influential people in AI In 2022, 2023, and 2024, he won Neural Information Processing Systems’ Test of Time award, which recognizes papers that significantly shaped the AI field over at least ten years. In 2025, he received an honorary doctorate from his alma mater, the University of Toronto In 2026, he received the National Academy of Sciences Award for the Industrial Application of Science, presented for the first time in artificial intelligence.
John F. Sowa
John Florian Sowa (born 1940) is an American computer scientist, an expert in artificial intelligence and computer design, and the inventor of conceptual graphs. == Biography == Sowa received a BS in mathematics from Massachusetts Institute of Technology in 1962, an MA in applied mathematics from Harvard University in 1966, and a PhD in computer science from the Vrije Universiteit Brussel in 1999 with a dissertation titled "Knowledge Representation: Logical, Philosophical, and Computational Foundations". Sowa spent most of his professional career at IBM, starting in 1962 at IBM's applied mathematics group. Over the decades he has researched and developed emerging fields of computer science from compilers, programming languages, and system architecture to artificial intelligence and knowledge representation. In the 1990s Sowa was associated with the IBM Educational Center in New York. Over the years he taught courses at the IBM Systems Research Institute, Binghamton University, Stanford University, the Linguistic Society of America and the Université du Québec à Montréal. He is a fellow of the Association for the Advancement of Artificial Intelligence. After early retirement at IBM, Sowa in 2001 cofounded VivoMind Intelligence, Inc. with Arun K. Majumdar. With this company he was developing data-mining and database technology, more specifically high-level "ontologies" for artificial intelligence and automated natural language understanding. Currently Sowa is working with Kyndi Inc., also founded by Majumdar. John Sowa is married to the philologist Cora Angier Sowa, and they live in Croton-on-Hudson, New York. == Work == Sowa's research interests since the 1970s were in the field of artificial intelligence, expert systems and database query linked to natural languages. In his work he combines ideas from numerous disciplines and eras modern and ancient, for example, applying ideas from Aristotle, the medieval scholastics to Alfred North Whitehead and including database schema theory, and incorporating the model of analogy of Islamic scholar Ibn Taymiyyah in his works. === Conceptual graph === Sowa invented conceptual graphs, a graphic notation for logic and natural language, based on the structures in semantic networks and on the existential graphs of Charles S. Peirce. He introduced the concept in the 1976 article "Conceptual graphs for a data base interface" in the IBM Journal of Research and Development. He elaborated upon it in the 1983 book Conceptual structures: information processing in mind and machine. In the 1980s, this theory had "been adopted by a number of research and development groups throughout the world. International conferences on conceptual structures (ICCS) have been held since 1993, following a series of conceptual graph workshops that began in 1986. === Sowa's law of standards === In 1991, Sowa first stated his Law of Standards: "Whenever a major organization develops a new system as an official standard for X, the primary result is the widespread adoption of some simpler system as a de facto standard for X." Like Gall's law, The Law of Standards is essentially an argument in favour of underspecification. Examples include: The introduction of PL/I resulting in COBOL and FORTRAN becoming the de facto standards for business and scientific programming respectively The introduction of Algol-68 resulting in Pascal becoming the de facto standard for academic programming The introduction of the Ada language resulting in C becoming the de facto standard for US Department of Defense programming The introduction of OS/2 resulting in Windows becoming the de facto standard for desktop OS The introduction of X.400 resulting in SMTP becoming the de facto standard for electronic mail The introduction of X.500 resulting in LDAP becoming the de facto standard for directory services == Publications == 1984. Conceptual Structures - Information Processing in Mind and Machine. The Systems Programming Series, Addison-Wesley 1991. Principles of Semantic Networks. Morgan Kaufmann. Mineau, Guy W; Moulin, Bernard; Sowa, John F, eds. (1993). Conceptual Graphs for Knowledge Representation. LNCS. Vol. 699. doi:10.1007/3-540-56979-0. ISBN 978-3-540-56979-4. S2CID 32275791. 1994. International Conference on Conceptual Structures (2nd : 1994 : College Park, Md.) Conceptual structures, current practices : Second International Conference on Conceptual Structures, ICCS'94, College Park, Maryland, USA, August 16–20, 1994 : proceedings. William M. Tepfenhart, Judith P. Dick, John F. Sowa, eds. Ellis, Gerard; Levinson, Robert; Rich, William; Sowa, John F, eds. (1995). Conceptual Structures: Applications, Implementation and Theory. LNCS. Vol. 954. doi:10.1007/3-540-60161-9. ISBN 978-3-540-60161-6. S2CID 27300281. Lukose, Dickson; Delugach, Harry; Keeler, Mary; Searle, Leroy; Sowa, John, eds. (1997). Conceptual Structures: Fulfilling Peirce's Dream. LNCS. Vol. 1257. doi:10.1007/BFb0027865. ISBN 3-540-63308-1. S2CID 1934069. 2000. Knowledge representation : logical, philosophical, and computational foundations, Brooks Cole Publishing Co., Pacific Grove Articles, a selection Sowa, J. F. (July 1976). "Conceptual Graphs for a Data Base Interface". IBM Journal of Research and Development. 20 (4): 336–357. doi:10.1147/rd.204.0336. Sowa, J. F.; Zachman, J. A. (1992). "Extending and formalizing the framework for information systems architecture". IBM Systems Journal. 31 (3): 590–616. doi:10.1147/sj.313.0590. 1992. "Conceptual Graph Summary"; In: T.E. Nagle et al. (Eds.). Conceptual Structures: Current Research and Practice. Chichester: Ellis Horwood. 1995. "Top-level ontological categories." in: International journal of human-computer studies. Vol. 43, Iss. 5–6, Nov. 1995, pp. 669–685 2006. "Semantic Networks". In: Encyclopedia of Cognitive Science.. John Wiley & Sons.
Tim Houlne
Tim Houlne is an American business executive, entrepreneur, and author known for his work in outsourcing and homeshoring, remote working, and artificial intelligence (AI) in customer service. He is the founder and CEO of Humach, a company that uses human agents and AI in customer experience solutions. Previously, he was co-founder and CEO of Working Solutions, a virtual contact center company in the United States. == Early life and education == Houlne graduated from Missouri Western State University (MWSU) in 1986 with a bachelor's degree in business administration and from the University of Texas in Dallas with an MBA. In 2024, MWSU and North Central Missouri College renamed the Convergent Technology Alliance Center to the Houlne Center for Convergent Technology. The 20,000 square-foot learning laboratory provides training and applied education experiences in industries such as AI, cybersecurity, manufacturing and construction, and service technologies. == Career == In 1998, Houlne co-founded Working Solutions, a Plano, Texas-based U.S. outsourcing company that provides customer service using remote, home-based agents. As CEO, he oversaw the development of a virtual workforce model that routes service calls to either domestic or offshore agents, according to client needs and service requirements. In 2015, Houlne founded Humach, a customer experience outsourcing provider that uses human service agents with AI-based digital agents. The company derives its name from the combination of services provided by humans and machines. Its clients include Amazon, Carfax and McDonald's. The company acquired InfiniteAI in 2020, and Markets EQ in 2025. In 2013, Houlne was named a finalist for the Ernst & Young Entrepreneur of the Year Award (Southwest Region).He is the co-author of several books focused on the evolution of work, the gig economy, and the influence of AI in customer-facing roles. == Works == The New World of Work: From the Cube to the Cloud (2013) ISBN 0982562276 OCLC 813933360 The New World of Work, Second Edition: The Cube, the Cloud and What's Next (2023) ISBN 9781642258318 OCLC 1389815847 The Intelligent Workforce: How Humans & Machines Will Co-Create a Better Future (2024) ISBN 9798887501604 OCLC 1439598569
Cloem
Cloem is a company based in Cannes, France, which applies natural language processing (NLP) technologies to assist patent applicants in creating variants of patent claims, called "cloems". According to the company, these "computer-generated claims can be published to keep potential competitors from attempting to file adjacent patent claims." == Technology == According to Cloem, dictionaries, ontologies and proprietary claim-drafting algorithms are used to draft alternative claims based on a client's original set of claims. In particular, the original set of claims is subject to various permutations and linguistic manipulations "by considering alternative definitions for terms as well as “synonyms, hyponyms, hyperonyms, meronyms, holonyms, and antonyms.”" == Possible uses == Cloem can optionally publish one or more created texts, as electronic publications or as paper-printed publications. These can potentially serve – through a defensive publication – as prior art to prevent another party for obtaining a patent on the subject-matter at stake. In other words, after an initial patent filing, an "improvement" patent (adjacent invention) can be applied for by another party, such as a competitor. By publishing variants of a patent claim, the risk of adverse patenting may potentially be decreased (improvement inventions may no longer be patentable). Cloems may also be potentially patentable. One of the issues of patentability, however, is that only a natural person can be a listed as an inventor on a patent. Since cloems are produced by a computer based on a person's input, it is not clear if the computer or the person is the inventor. The inventorship of Cloem texts is an open question.
Infer.NET
Infer.NET is a free and open source .NET software library for machine learning. It supports running Bayesian inference in graphical models and can also be used for probabilistic programming. == Overview == Infer.NET follows a model-based approach and is used to solve different kinds of machine learning problems including standard problems like classification, recommendation or clustering, customized solutions and domain-specific problems. The framework is used in various different domains such as bioinformatics, epidemiology, computer vision, and information retrieval. Development of the framework was started by a team at Microsoft's research centre in Cambridge, UK in 2004. It was first released for academic use in 2008 and later open sourced in 2018. In 2013, Microsoft was awarded the USPTO's Patents for Humanity Award in Information Technology category for Infer.NET and the work in advanced machine learning techniques. Infer.NET is used internally at Microsoft as the machine learning engine in some of their products such as Office, Azure, and Xbox. The source code is licensed under MIT License and available on GitHub. It is also available as NuGet package.