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Agent verification

Agent verification is activity to gain assurances that purposeful artificial constructs act in accordance with their specifications. While primitive forms of inorganic agents have been used in manufacturing for centuries, the study of artificial agents did not begin until the mid 20th century. Foundational work on such agents was closely bound with the emergence of artificial intelligence as an academic discipline. Early agents deployed for industrial control systems and in computing were often controlled by quite simple logic however, not involving artificial intelligence as such. When deployed as part of a multi-agent system, even such simple agents could require special agent orientated testing methods, as their collective behaviour was challenging to verify with traditional testing techniques. Difficulties in providing assurances that agents will not behave in dangerous ways became more prevalent after the introduction of LLM agents, especially after the rapid acceleration of their deployment in 2025. The verification of agent behaviour can be conducted by formal or informal methods. Informal verification requires less mathematical skill. But when agents are part of systems where errors have significant risks — such as danger to human life, environmental damage or major financial loss — formal verification is preferred. Both regulators and system designers themselves like formal verification as it provides a high degree of mathematical certainty. It is not however always possible to formally test all aspects of an agent based system's behaviour, especially where newer LLM based agents are concerned, due in part to their high degree of autonomy. Accordingly, agent verification for low impact deployments might be carried out only with informal methods, while for high impact deployments, it may be performed with a mix of formal and informal techniques. == Terminology == In academia, the term agent verification is often defined to mean activity concerned with gaining assurance that the agent behaves in accordance with its specification - whether by processes such as testing or simulation. 'Verification' is typically contrasted with 'validation', the latter meaning activity concerned with checking that the specification itself meets user or real world needs. Such definitions are not universally adhered to however - for example, in some workplaces and documents, the words 'verification' and 'validation' can be used synonymously. Efforts to gain confidence in Agents have intensified sharply since 2025 due to the rapid roll out of LLM agents; different terms are sometimes used in the commercial sector. Here the term 'agent verification' can be used in the same sense as it is in academia, but sometimes the same activity can be covered by more ambiguous and wider ranging terms such as 'Agent governance' , 'Agent observability' or 'AI agent policing'. == History == === Classical agents === The theoretical underpinnings for artificial (inorganic) agents emerged in the mid 20th century, with establishment of cybernetics and artificial intelligence. Oliver Selfridge's 1958 Pandemonium - A Paradigm for Learning paper was an important early theoretical contribution in establishing agent oriented architecture. Practical implementations of agents for real world applications began to become widespread in the 1990s, after the introduction of the belief–desire–intention software model (BDI), and agent-oriented programming. Pure digital agents were deployed in computer infrastructure for purposes such as monitoring, while agents connected to real-world sensors and actuators were increasingly used in industrial control systems. While the concept of artificial agents was interwoven with early artificial intelligence studies right from the start, early agents lacked general purpose reasoning capabilities, often only having simple if then logic. Even a device as simple as a thermostat, which has a sensor and a means of acting, can be considered a proto agent in this sense. Verifying the behaviours of a simple single agent system is not generally especially difficult, but it can be a different matter when several simple agents coexist in the same system. Craig Reynolds's work on boids showed that relatively complex, "intelligent" behaviour can emerge from a number of such simple agents working together in a Multi-agent system (MAS). By the 1990s, even the behaviour of a single agent system could sometimes be quite complex; in accordance with the Belief–desire–intention software model, agents could have believes that might evolve over time. Agents were increasingly introduced that were controlled by quite large decision tree models, which had new vulnerabilities to adversarial attack. It was becoming increasingly apparent that traditional software verification methods had limitations for testing such agents, or even for the more primitive type of agents when they were deployed as part of a MAS. It was the use of agents for industrial control systems, sometimes associated with robotics, that lent urgency to the practice of agent verification. Informal testing might be acceptable for digital agents used say to monitor whether each of an organisation's computers are properly licensed. But with an increasing potential for faulty agents to result in a failure that might cause a large fire to break out at a chemical manufacturing plant, a botched medical operation, or even a crashed aircraft, the need to develop reliable means of verifying behaviour of such agents was considered urgent. The Foundation for Intelligent Physical Agents was established in 1996. From the late 90s, a growing number of industry and university based scientists began working on the problem, with researchers publishing papers on the verification of both single and multi agent systems. Much of this work showed how formal verification techniques like model checking could be used to gain a high level of assurance that agent based systems would conform with their specification. A 2018 systematic review covering 231 studies found that model checking was the most common technique for agent verification, with theorem proving the second most commonly used formal verification method. In the first two decades of the 20th century, agents run by AI became more common, with Siri and Alexa being well known examples. But such agents still lacked general reasoning capabilities and did not pose new pressing problems for agent verification. === General purpose reasoning agents === The advent of LLMs created huge potential for further use of artificial agents, as agents based on them could have general purpose cognitive abilities. Agents run by LLMs (and occasionally non-LLM foundation models) have similar vulnerability to adversarial attack as those run by decision tree models. The wider scope of actions for LLM agents has created new challenges for their verification, over and above those present for classical agents. For example, the LLM's neural network endows it with infinite domains, an especial challenge for traditional formal verification techniques. Academics began to study the problems involved in verifying LLM agents from 2018. Deployment of such agents began to accelerate in late 2023 after OpenAI's "function-calling" API was made available, and especially after Anthropic's late 2024 introduction of Model Context Protocol (MCP), a standardised way for LLM agents to gain contextual awareness, and to act on the world by calling various external tools. The rapid rollout of LLM agents following MCP's release has seen the task of agent verification receive increased attention within academia, and also from the private sector. In 2024 and 2025 several startups focusing on LLM agent verification have been founded in both Europe and the US to meet growing demand. == Approaches == === Formal verification === Formal verification involves proving the correctness of some or all aspects of a system using mathematical methods. Such methods can range from manual formal proof, to verification assisted with automated theorem provers like Isabelle. For agent verification, model checking is by far the most frequently used formal verification method; for pre-LLM models it was often complemented with techniques using computation tree logic. Another common method is theorem proving. Formal verification provides a higher degree of confidence than informal methods, but it is not always used, even when it is possible. Sometimes a person or organisation developing software agents won't have the necessary skills, or may not see it as worth the effort if the agent(s) will not have the ability to cause much harm even if they malfunction. When agents are deployed in systems where errors could have serious consequences, the ability of formal verification methods to provide mathematical certainty tends to be strongly preferred by both regulators and designers themselves. But even for high impact systems, formal verificatio
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Latent Dirichlet allocation

In natural language processing, latent Dirichlet allocation (LDA) is a generative statistical model that explains how a collection of text documents can be described by a set of unobserved "topics." For example, given a set of news articles, LDA might discover that one topic is characterized by words like "president", "government", and "election", while another is characterized by "team", "game", and "score". It is one of the most common topic models. The LDA model was first presented as a graphical model for population genetics by J. K. Pritchard, M. Stephens and P. Donnelly in 2000. The model was subsequently applied to machine learning by David Blei, Andrew Ng, and Michael I. Jordan in 2003. Although its most frequent application is in modeling text corpora, it has also been used for other problems, such as in clinical psychology, social science, and computational musicology. The core assumption of LDA is that documents are represented as a random mixture of latent topics, and each topic is characterized by a probability distribution over words. The model is a generalization of probabilistic latent semantic analysis (pLSA), differing primarily in that LDA treats the topic mixture as a Dirichlet prior, leading to more reasonable mixtures and less susceptibility to overfitting. Learning the latent topics and their associated probabilities from a corpus is typically done using Bayesian inference, often with methods like Gibbs sampling or variational Bayes. == History == In the context of population genetics, LDA was proposed by J. K. Pritchard, M. Stephens and P. Donnelly in 2000. LDA was applied in machine learning by David Blei, Andrew Ng and Michael I. Jordan in 2003. == Overview == === Population genetics === In population genetics, the model is used to detect the presence of structured genetic variation in a group of individuals. The model assumes that alleles carried by individuals under study have origin in various extant or past populations. The model and various inference algorithms allow scientists to estimate the allele frequencies in those source populations and the origin of alleles carried by individuals under study. The source populations can be interpreted ex-post in terms of various evolutionary scenarios. In association studies, detecting the presence of genetic structure is considered a necessary preliminary step to avoid confounding. === Clinical psychology, mental health, and social science === In clinical psychology research, LDA has been used to identify common themes of self-images experienced by young people in social situations. Other social scientists have used LDA to examine large sets of topical data from discussions on social media (e.g., tweets about prescription drugs). Additionally, supervised Latent Dirichlet Allocation with covariates (SLDAX) has been specifically developed to combine latent topics identified in texts with other manifest variables. This approach allows for the integration of text data as predictors in statistical regression analyses, improving the accuracy of mental health predictions. One of the main advantages of SLDAX over traditional two-stage approaches is its ability to avoid biased estimates and incorrect standard errors, allowing for a more accurate analysis of psychological texts. In the field of social sciences, LDA has proven to be useful for analyzing large datasets, such as social media discussions. For instance, researchers have used LDA to investigate tweets discussing socially relevant topics, like the use of prescription drugs and cultural differences in China. By analyzing these large text corpora, it is possible to uncover patterns and themes that might otherwise go unnoticed, offering valuable insights into public discourse and perception in real time. === Musicology === In the context of computational musicology, LDA has been used to discover tonal structures in different corpora. === Machine learning === One application of LDA in machine learning – specifically, topic discovery, a subproblem in natural language processing – is to discover topics in a collection of documents, and then automatically classify any individual document within the collection in terms of how "relevant" it is to each of the discovered topics. A topic is considered to be a set of terms (i.e., individual words or phrases) that, taken together, suggest a shared theme. For example, in a document collection related to pet animals, the terms dog, spaniel, beagle, golden retriever, puppy, bark, and woof would suggest a DOG_related theme, while the terms cat, siamese, Maine coon, tabby, manx, meow, purr, and kitten would suggest a CAT_related theme. There may be many more topics in the collection – e.g., related to diet, grooming, healthcare, behavior, etc. that we do not discuss for simplicity's sake. (Very common, so called stop words in a language – e.g., "the", "an", "that", "are", "is", etc., – would not discriminate between topics and are usually filtered out by pre-processing before LDA is performed. Pre-processing also converts terms to their "root" lexical forms – e.g., "barks", "barking", and "barked" would be converted to "bark".) If the document collection is sufficiently large, LDA will discover such sets of terms (i.e., topics) based upon the co-occurrence of individual terms, though the task of assigning a meaningful label to an individual topic (i.e., that all the terms are DOG_related) is up to the user, and often requires specialized knowledge (e.g., for collection of technical documents). The LDA approach assumes that: The semantic content of a document is composed by combining one or more terms from one or more topics. Certain terms are ambiguous, belonging to more than one topic, with different probability. (For example, the term training can apply to both dogs and cats, but are more likely to refer to dogs, which are used as work animals or participate in obedience or skill competitions.) However, in a document, the accompanying presence of specific neighboring terms (which belong to only one topic) will disambiguate their usage. Most documents will contain only a relatively small number of topics. In the collection, e.g., individual topics will occur with differing frequencies. That is, they have a probability distribution, so that a given document is more likely to contain some topics than others. Within a topic, certain terms will be used much more frequently than others. In other words, the terms within a topic will also have their own probability distribution. When LDA machine learning is employed, both sets of probabilities are computed during the training phase, using Bayesian methods and an expectation–maximization algorithm. LDA is a generalization of older approach of probabilistic latent semantic analysis (pLSA), The pLSA model is equivalent to LDA under a uniform Dirichlet prior distribution. pLSA relies on only the first two assumptions above and does not care about the remainder. While both methods are similar in principle and require the user to specify the number of topics to be discovered before the start of training (as with k-means clustering) LDA has the following advantages over pLSA: LDA yields better disambiguation of words and a more precise assignment of documents to topics. Computing probabilities allows a "generative" process by which a collection of new "synthetic documents" can be generated that would closely reflect the statistical characteristics of the original collection. Unlike LDA, pLSA is vulnerable to overfitting especially when the size of corpus increases. The LDA algorithm is more readily amenable to scaling up for large data sets using the MapReduce approach on a computing cluster. == Model == With plate notation, which is often used to represent probabilistic graphical models (PGMs), the dependencies among the many variables can be captured concisely. The boxes are "plates" representing replicates, which are repeated entities. The outer plate represents documents, while the inner plate represents the repeated word positions in a given document; each position is associated with a choice of topic and word. The variable names are defined as follows: M denotes the number of documents N is number of words in a given document (document i has N i {\displaystyle N_{i}} words) α is the parameter of the Dirichlet prior on the per-document topic distributions β is the parameter of the Dirichlet prior on the per-topic word distribution θ i {\displaystyle \theta _{i}} is the topic distribution for document i φ k {\displaystyle \varphi _{k}} is the word distribution for topic k z i j {\displaystyle z_{ij}} is the topic for the j-th word in document i w i j {\displaystyle w_{ij}} is the specific word. The fact that W is grayed out means that words w i j {\displaystyle w_{ij}} are the only observable variables, and the other variables are latent variables. As proposed in the original paper, a sparse Dirichlet prior can be used to model the to
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Targeted maximum likelihood estimation

Targeted Maximum Likelihood Estimation (TMLE) (also more accurately referred to as Targeted Minimum Loss-Based Estimation) is a general statistical estimation framework for causal inference and semiparametric models. TMLE combines ideas from maximum likelihood estimation, semiparametric efficiency theory, and machine learning. It was introduced by Mark J. van der Laan and colleagues in the mid-2000s as a method that yields asymptotically efficient plug-in estimators while allowing the use of flexible, data-adaptive algorithms such as ensemble machine learning for nuisance parameter estimation. TMLE is used in epidemiology, biostatistics, and the social sciences to estimate causal effects in observational and experimental studies. Applications of TMLE include Longitudinal TMLE (LTMLE) for time-varying treatments and confounders. Variations in how the targeting step in TMLE is carried out have resulted in various versions of TMLE such as Collaborative TMLE (CTMLE) and Adaptive TMLE for improved finite-sample performance and automated variable selection. == History == The TMLE framework was first described by van der Laan and Rubin (2006) as a general approach for the construction of efficient plug-in estimators of smooth features of the data density. It was demonstrated in the context of causal inference and missing data problems. It was developed to address limitations of traditional doubly robust methods, such as Augmented Inverse Probability Weighting (AIPW), by respecting the plug-in principle in the sense that it respects that the target parameter is a function of the data density that is an element of the statistical model. TMLE estimates the data density or relevant parts of it with machine learning and targets these machine learning fits before it is plugged in the target parameter mapping. In this manner, a TMLE always respects global knowledge and satisfies known bounds such as that the target parameter is a probability . Since its introduction, TMLE has been developed in a series of theoretical and applied papers, culminating in book-length treatments of the method and its applications to survival analysis, adaptive designs, and longitudinal data. == Methodology == At its core, TMLE is a two-step estimation procedure: Initial estimation: Machine learning methods (such as the Super Learner ensemble) are used to obtain flexible estimates of nuisance parameters, such as outcome regressions and propensity scores. Targeting step: The initial estimate is updated by solving a score equation (the efficient influence function) so that the final estimator is consistent, asymptotically normal, and efficient under mild regularity conditions. The targeted machine learning fit is then mapped into the corresponding estimator of the target parameter by simply plugging it in the target parameter mapping. This approach balances the bias–variance trade-off by combining data-adaptive estimation with semiparametric efficiency theory. TMLE is doubly robust, meaning it remains consistent if either the outcome model or the treatment model is consistently estimated. === Formula === Here we explain the TMLE of the average treatment effect of a binary treatment on an outcome adjusting for baseline covariates. Consider i.i.d. observations O i = ( W i , A i , Y i ) {\displaystyle O_{i}=(W_{i},A_{i},Y_{i})} from a distribution P 0 {\displaystyle P_{0}} , where W {\displaystyle W} are baseline covariates, A {\displaystyle A} is a binary treatment, and Y {\displaystyle Y} is an outcome. Let Q ¯ ( a , w ) = E [ Y ∣ A = a , W = w ] {\displaystyle {\bar {Q}}(a,w)=\mathbb {E} [Y\mid A=a,W=w]} represent the outcome model and g ( a ∣ w ) = P ( A = a ∣ W = w ) {\displaystyle g(a\mid w)=P(A=a\mid W=w)} represent the propensity score. The average treatment effect (ATE) is given by ψ 0 = E { Q ¯ ( 1 , W ) − Q ¯ ( 0 , W ) } . {\displaystyle \psi _{0}=\mathbb {E} \{{\bar {Q}}(1,W)-{\bar {Q}}(0,W)\}.} A basic TMLE for the ATE proceeds as follows: Step 1: Estimate initial models. Obtain estimates Q ¯ ^ ( a , w ) {\displaystyle {\hat {\bar {Q}}}(a,w)} and g ^ ( a ∣ w ) {\displaystyle {\hat {g}}(a\mid w)} , often using flexible methods such as Super Learner. Step 2: Compute the clever covariate. Define: H ( A , W ) = A g ^ ( 1 ∣ W ) − 1 − A g ^ ( 0 ∣ W ) . {\displaystyle H(A,W)={\frac {A}{{\hat {g}}(1\mid W)}}-{\frac {1-A}{{\hat {g}}(0\mid W)}}.} Step 3: Estimate the fluctuation parameter. Fit a logistic regression of Y {\displaystyle Y} on H ( A , W ) {\displaystyle H(A,W)} with logit ⁡ ( Q ¯ ^ ( A , W ) ) {\displaystyle \operatorname {logit} ({\hat {\bar {Q}}}(A,W))} as offset. This yields ε ^ {\displaystyle {\hat {\varepsilon }}} , the MLE that solves the score equation: 1 n ∑ i = 1 n H ( A i , W i ) { Y i − Q ¯ ^ ε ( A i , W i ) } = 0. {\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}H(A_{i},W_{i}){\big \{}Y_{i}-{\hat {\bar {Q}}}^{\varepsilon }(A_{i},W_{i}){\big \}}=0.} Step 4: Update the initial estimate. Apply the "blip" to obtain the targeted estimate: Q ¯ ^ ∗ ( A , W ) = expit ⁡ ( logit ⁡ ( Q ¯ ^ ( A , W ) ) + ε ^ H ( A , W ) ) . {\displaystyle {\hat {\bar {Q}}}^{}(A,W)=\operatorname {expit} {\Big (}\operatorname {logit} {\big (}{\hat {\bar {Q}}}(A,W){\big )}+{\hat {\varepsilon }}\,H(A,W){\Big )}.} Step 5: Compute the TMLE. The ATE estimate is: ψ ^ TMLE = 1 n ∑ i = 1 n [ Q ¯ ^ ∗ ( 1 , W i ) − Q ¯ ^ ∗ ( 0 , W i ) ] . {\displaystyle {\hat {\psi }}_{\text{TMLE}}={\frac {1}{n}}\sum _{i=1}^{n}{\big [}{\hat {\bar {Q}}}^{}(1,W_{i})-{\hat {\bar {Q}}}^{}(0,W_{i}){\big ]}.} Inference. The efficient influence function (EIF) for the ATE is: D ∗ ( O ) = H ( A , W ) { Y − Q ¯ ∗ ( A , W ) } + Q ¯ ∗ ( 1 , W ) − Q ¯ ∗ ( 0 , W ) − ψ . {\displaystyle D^{}(O)=H(A,W)\{Y-{\bar {Q}}^{}(A,W)\}+{\bar {Q}}^{}(1,W)-{\bar {Q}}^{}(0,W)-\psi .} The variance is estimated by σ ^ 2 = n − 1 ∑ i = 1 n ( D ∗ ( O i ) ) 2 {\displaystyle {\hat {\sigma }}^{2}=n^{-1}\sum _{i=1}^{n}{\big (}D^{}(O_{i}){\big )}^{2}} , yielding Wald-type confidence intervals ψ ^ TMLE ± z 1 − α / 2 σ ^ / n {\displaystyle {\hat {\psi }}_{\text{TMLE}}\pm z_{1-\alpha /2}\,{\hat {\sigma }}/{\sqrt {n}}} . Remark. For continuous outcomes, a linear fluctuation Q ¯ ^ ∗ = Q ¯ ^ + ε ^ H {\displaystyle {\hat {\bar {Q}}}^{}={\hat {\bar {Q}}}+{\hat {\varepsilon }}\,H} may be used instead. For bounded continuous outcomes, the logistic fluctuation (after rescaling Y {\displaystyle Y} to [ 0 , 1 ] {\displaystyle [0,1]} ) is often preferred for improved finite-sample performance. == Applications == TMLE has been applied in: Epidemiology: Estimating causal effects of exposures and interventions in observational cohort studies. Clinical trials and real-world evidence: The Targeted Learning roadmap provides a structured framework for generating and validating real-world evidence (RWE), bridging randomized trials and observational data using TMLE and related estimation techniques. This approach enables transparency, sensitivity analysis, and stronger causal inference for regulatory and clinical trial contexts. High-dimensional settings: Integration with ensemble methods for causal effect estimation. TMLE has been successfully applied in pharmacoepidemiology where a large number of covariates are automatically selected to adjust for confounding. In a study of post–myocardial infarction statin use and 1-year mortality, TMLE demonstrated robust performance relative to inverse probability weighting in scenarios with hundreds of potential confounders. == Derivatives and extensions == Longitudinal TMLE (LTMLE): A methodological extension of TMLE for longitudinal data with time-varying treatments, confounders, and censoring. It allows the estimation of dynamic treatment regimes and intervention-specific causal effects over time. This framework was originally introduced by van der Laan & Gruber (2012). Collaborative TMLE (CTMLE): Enhances finite-sample performance and variable selection by collaboratively fitting the treatment mechanism in conjunction with the target parameter. == Software == Several R packages implement TMLE and related methods: tmle: Functions for binary, categorical, and continuous outcomes. ltmle: Implementation for longitudinal data with time-varying treatments and outcomes. ctmle: Algorithms for collaborative TMLE and adaptive variable selection. SuperLearner: A theoretically grounded, cross-validated ensemble learning method that combines predictions from multiple algorithms to minimize predictive risk. Widely used in TMLE for estimating nuisance parameters. The original implementation is available as the R package SuperLearner. Recent machine learning platforms like H2O AutoML implement similar ensemble strategies, combining diverse learners in parallel and leveraging stacking and blending techniques, effectively functioning as a large-scale Super Learner.
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Rule-based machine learning

Rule-based machine learning (RBML) is a term in computer science intended to encompass any machine learning method that identifies, learns, or evolves 'rules' to store, manipulate or apply. The defining characteristic of a rule-based machine learner is the identification and utilization of a set of relational rules that collectively represent the knowledge captured by the system. Rule-based machine learning approaches include learning classifier systems, association rule learning, artificial immune systems, and any other method that relies on a set of rules, each covering contextual knowledge. While rule-based machine learning is conceptually a type of rule-based system, it is distinct from traditional rule-based systems, which are often hand-crafted, and other rule-based decision makers. This is because rule-based machine learning applies some form of learning algorithm such as Rough sets theory to identify and minimise the set of features and to automatically identify useful rules, rather than a human needing to apply prior domain knowledge to manually construct rules and curate a rule set. == Rules == Rules typically take the form of an '{IF:THEN} expression', (e.g. {IF 'condition' THEN 'result'}, or as a more specific example, {IF 'red' AND 'octagon' THEN 'stop-sign}). An individual rule is not in itself a model, since the rule is only applicable when its condition is satisfied. Therefore rule-based machine learning methods typically comprise a set of rules, or knowledge base, that collectively make up the prediction model usually known as decision algorithm. Rules can also be interpreted in various ways depending on the domain knowledge, data types(discrete or continuous) and in combinations. == RIPPER == Repeated incremental pruning to produce error reduction (RIPPER) is a propositional rule learner proposed by William W. Cohen as an optimized version of IREP.
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Learnable function class

In statistical learning theory, a learnable function class is a set of functions for which an algorithm can be devised to asymptotically minimize the expected risk, uniformly over all probability distributions. The concept of learnable classes are closely related to regularization in machine learning, and provides large sample justifications for certain learning algorithms. == Definition == === Background === Let Ω = X × Y = { ( x , y ) } {\displaystyle \Omega ={\mathcal {X}}\times {\mathcal {Y}}=\{(x,y)\}} be the sample space, where y {\displaystyle y} are the labels and x {\displaystyle x} are the covariates (predictors). F = { f : X ↦ Y } {\displaystyle {\mathcal {F}}=\{f:{\mathcal {X}}\mapsto {\mathcal {Y}}\}} is a collection of mappings (functions) under consideration to link x {\displaystyle x} to y {\displaystyle y} . L : Y × Y ↦ R {\displaystyle L:{\mathcal {Y}}\times {\mathcal {Y}}\mapsto \mathbb {R} } is a pre-given loss function (usually non-negative). Given a probability distribution P ( x , y ) {\displaystyle P(x,y)} on Ω {\displaystyle \Omega } , define the expected risk I P ( f ) {\displaystyle I_{P}(f)} to be: I P ( f ) = ∫ L ( f ( x ) , y ) d P ( x , y ) {\displaystyle I_{P}(f)=\int L(f(x),y)dP(x,y)} The general goal in statistical learning is to find the function in F {\displaystyle {\mathcal {F}}} that minimizes the expected risk. That is, to find solutions to the following problem: f ^ = arg ⁡ min f ∈ F I P ( f ) {\displaystyle {\hat {f}}=\arg \min _{f\in {\mathcal {F}}}I_{P}(f)} But in practice the distribution P {\displaystyle P} is unknown, and any learning task can only be based on finite samples. Thus we seek instead to find an algorithm that asymptotically minimizes the empirical risk, i.e., to find a sequence of functions { f ^ n } n = 1 ∞ {\displaystyle \{{\hat {f}}_{n}\}_{n=1}^{\infty }} that satisfies lim n → ∞ P ( I P ( f ^ n ) − inf f ∈ F I P ( f ) > ϵ ) = 0 {\displaystyle \lim _{n\rightarrow \infty }\mathbb {P} (I_{P}({\hat {f}}_{n})-\inf _{f\in {\mathcal {F}}}I_{P}(f)>\epsilon )=0} One usual algorithm to find such a sequence is through empirical risk minimization. === Learnable function class === We can make the condition given in the above equation stronger by requiring that the convergence is uniform for all probability distributions. That is: The intuition behind the more strict requirement is as such: the rate at which sequence { f ^ n } {\displaystyle \{{\hat {f}}_{n}\}} converges to the minimizer of the expected risk can be very different for different P ( x , y ) {\displaystyle P(x,y)} . Because in real world the true distribution P {\displaystyle P} is always unknown, we would want to select a sequence that performs well under all cases. However, by the no free lunch theorem, such a sequence that satisfies (1) does not exist if F {\displaystyle {\mathcal {F}}} is too complex. This means we need to be careful and not allow too "many" functions in F {\displaystyle {\mathcal {F}}} if we want (1) to be a meaningful requirement. Specifically, function classes that ensure the existence of a sequence { f ^ n } {\displaystyle \{{\hat {f}}_{n}\}} that satisfies (1) are known as learnable classes. It is worth noting that at least for supervised classification and regression problems, if a function class is learnable, then the empirical risk minimization automatically satisfies (1). Thus in these settings not only do we know that the problem posed by (1) is solvable, we also immediately have an algorithm that gives the solution. == Interpretations == If the true relationship between y {\displaystyle y} and x {\displaystyle x} is y ∼ f ∗ ( x ) {\displaystyle y\sim f^{}(x)} , then by selecting the appropriate loss function, f ∗ {\displaystyle f^{}} can always be expressed as the minimizer of the expected loss across all possible functions. That is, f ∗ = arg ⁡ min f ∈ F ∗ I P ( f ) {\displaystyle f^{}=\arg \min _{f\in {\mathcal {F}}^{}}I_{P}(f)} Here we let F ∗ {\displaystyle {\mathcal {F}}^{}} be the collection of all possible functions mapping X {\displaystyle {\mathcal {X}}} onto Y {\displaystyle {\mathcal {Y}}} . f ∗ {\displaystyle f^{}} can be interpreted as the actual data generating mechanism. However, the no free lunch theorem tells us that in practice, with finite samples we cannot hope to search for the expected risk minimizer over F ∗ {\displaystyle {\mathcal {F}}^{}} . Thus we often consider a subset of F ∗ {\displaystyle {\mathcal {F}}^{}} , F {\displaystyle {\mathcal {F}}} , to carry out searches on. By doing so, we risk that f ∗ {\displaystyle f^{}} might not be an element of F {\displaystyle {\mathcal {F}}} . This tradeoff can be mathematically expressed as In the above decomposition, part ( b ) {\displaystyle (b)} does not depend on the data and is non-stochastic. It describes how far away our assumptions ( F {\displaystyle {\mathcal {F}}} ) are from the truth ( F ∗ {\displaystyle {\mathcal {F}}^{}} ). ( b ) {\displaystyle (b)} will be strictly greater than 0 if we make assumptions that are too strong ( F {\displaystyle {\mathcal {F}}} too small). On the other hand, failing to put enough restrictions on F {\displaystyle {\mathcal {F}}} will cause it to be not learnable, and part ( a ) {\displaystyle (a)} will not stochastically converge to 0. This is the well-known overfitting problem in statistics and machine learning literature. == Example: Tikhonov regularization == A good example where learnable classes are used is the so-called Tikhonov regularization in reproducing kernel Hilbert space (RKHS). Specifically, let F ∗ {\displaystyle {\mathcal {F^{}}}} be an RKHS, and | | ⋅ | | 2 {\displaystyle ||\cdot ||_{2}} be the norm on F ∗ {\displaystyle {\mathcal {F^{}}}} given by its inner product. It is shown in that F = { f : | | f | | 2 ≤ γ } {\displaystyle {\mathcal {F}}=\{f:||f||_{2}\leq \gamma \}} is a learnable class for any finite, positive γ {\displaystyle \gamma } . The empirical minimization algorithm to the dual form of this problem is arg ⁡ min f ∈ F ∗ { ∑ i = 1 n L ( f ( x i ) , y i ) + λ | | f | | 2 } {\displaystyle \arg \min _{f\in {\mathcal {F}}^{}}\left\{\sum _{i=1}^{n}L(f(x_{i}),y_{i})+\lambda ||f||_{2}\right\}} This was first introduced by Tikhonov to solve ill-posed problems. Many statistical learning algorithms can be expressed in such a form (for example, the well-known ridge regression). The tradeoff between ( a ) {\displaystyle (a)} and ( b ) {\displaystyle (b)} in (2) is geometrically more intuitive with Tikhonov regularization in RKHS. We can consider a sequence of { F γ } {\displaystyle \{{\mathcal {F}}_{\gamma }\}} , which are essentially balls in F ∗ {\displaystyle {\mathcal {F^{}}}} with centers at 0. As γ {\displaystyle \gamma } gets larger, F γ {\displaystyle {\mathcal {F}}_{\gamma }} gets closer to the entire space, and ( b ) {\displaystyle (b)} is likely to become smaller. However we will also suffer smaller convergence rates in ( a ) {\displaystyle (a)} . The way to choose an optimal γ {\displaystyle \gamma } in finite sample settings is usually through cross-validation. == Relationship to empirical process theory == Part ( a ) {\displaystyle (a)} in (2) is closely linked to empirical process theory in statistics, where the empirical risk { ∑ i = 1 n L ( y i , f ( x i ) ) , f ∈ F } {\displaystyle \{\sum _{i=1}^{n}L(y_{i},f(x_{i})),f\in {\mathcal {F}}\}} are known as empirical processes. In this field, the function class F {\displaystyle {\mathcal {F}}} that satisfies the stochastic convergence are known as uniform Glivenko–Cantelli classes. It has been shown that under certain regularity conditions, learnable classes and uniformly Glivenko-Cantelli classes are equivalent. Interplay between ( a ) {\displaystyle (a)} and ( b ) {\displaystyle (b)} in statistics literature is often known as the bias-variance tradeoff. However, note that in the authors gave an example of stochastic convex optimization for General Setting of Learning where learnability is not equivalent with uniform convergence.
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Probably approximately correct learning

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
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Abess

abess (Adaptive Best Subset Selection, also ABESS) is a machine learning method designed to address the problem of best subset selection. It aims to determine which features or variables are crucial for optimal model performance when provided with a dataset and a prediction task. abess was introduced by Zhu in 2020 and it dynamically selects the appropriate model size adaptively, eliminating the need for selecting regularization parameters. abess is applicable in various statistical and machine learning tasks, including linear regression, the Single-index model, and other common predictive models. abess can also be applied in biostatistics. == Basic Form == The basic form of abess is employed to address the optimal subset selection problem in general linear regression. abess is an l 0 {\displaystyle l_{0}} method, it is characterized by its polynomial time complexity and the property of providing both unbiased and consistent estimates. In the context of linear regression, assuming we have knowledge of n {\displaystyle n} independent samples ( x i , y i ) , i = 1 , … , n {\displaystyle (x_{i},y_{i}),i=1,\ldots ,n} , where x i ∈ R p × 1 {\displaystyle x_{i}\in \mathbb {R} ^{p\times 1}} and y i ∈ R {\displaystyle y_{i}\in \mathbb {R} } , we define X = ( x 1 , … , x n ) ⊤ {\displaystyle X=(x_{1},\ldots ,x_{n})^{\top }} and y = ( y 1 , … , y n ) ⊤ {\displaystyle y=(y_{1},\ldots ,y_{n})^{\top }} . The following equation represents the general linear regression model: y = X β + ε . {\displaystyle y=X\beta +\varepsilon .} To obtain appropriate parameters β {\displaystyle \beta } , one can consider the loss function for linear regression: L n LR ( β ; X , y ) = 1 2 n ‖ y − X β ‖ 2 2 . {\displaystyle {\mathcal {L}}_{n}^{\text{LR}}(\beta ;X,y)={\frac {1}{2n}}\|y-X\beta \|_{2}^{2}.} In abess, the initial focus is on optimizing the loss function under the l 0 {\displaystyle l_{0}} constraint. That is, we consider the following problem: min β ∈ R p × 1 L n LR ( β ; X , y ) , subject to ‖ β ‖ 0 ≤ s , {\displaystyle \min _{\beta \in \mathbb {R} ^{p\times 1}}{\mathcal {L}}_{n}^{\text{LR}}(\beta ;X,y),{\text{ subject to }}\|\beta \|_{0}\leq s,} where s {\displaystyle s} represents the desired size of the support set, and ‖ β ‖ 0 = ∑ i = 1 p I ( β i ≠ 0 ) {\displaystyle \|\beta \|_{0}=\sum _{i=1}^{p}{\mathcal {I}}_{(\beta _{i}\neq 0)}} is the l 0 {\displaystyle l_{0}} norm of the vector. To address the optimization problem described above, abess iteratively exchanges an equal number of variables between the active set and the inactive set. In each iteration, the concept of sacrifice is introduced as follows: For j in the active set ( j ∈ A ^ {\displaystyle j\in {\hat {\mathcal {A}}}} ): ξ j = L n LR ( β ^ A ∖ { j } ) − L n LR ( β ^ A ) = X j ⊤ X j 2 n ( β ^ j ) 2 {\displaystyle \xi _{j}={\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{{\mathcal {A}}\backslash \{j\}}\right)-{\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{\mathcal {A}}\right)={\frac {{\boldsymbol {X}}_{j}^{\top }{\boldsymbol {X}}_{j}}{2n}}\left({\hat {\beta }}_{j}\right)^{2}} For j in the inactive set ( j ∉ A ^ {\displaystyle j\notin {\hat {\mathcal {A}}}} ): ξ j = L n LR ( β ^ A ) − L n LR ( β ^ A + t ^ { j } ) = X j ⊤ X j 2 n ( d ^ j X j ⊤ X j / n ) 2 {\displaystyle \xi _{j}={\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{\mathcal {A}}\right)-{\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{\mathcal {A}}+{\hat {\boldsymbol {t}}}^{\{j\}}\right)={\frac {{\boldsymbol {X}}_{j}^{\top }{\boldsymbol {X}}_{j}}{2n}}\left({\frac {{\hat {\mathrm {d} }}_{j}}{{\boldsymbol {X}}_{j}^{\top }{\boldsymbol {X}}_{j}/n}}\right)^{2}} Here are the key elements in the above equations: β ^ A {\displaystyle {\hat {\beta }}^{\mathcal {A}}} : This represents the estimate of β {\displaystyle \beta } obtained in the previous iteration. A ^ {\displaystyle {\hat {\mathcal {A}}}} : It denotes the estimated active set from the previous iteration. β ^ A ∖ { j } {\displaystyle {\hat {\boldsymbol {\beta }}}^{{\mathcal {A}}\backslash \{j\}}} : This is a vector where the j-th element is set to 0, while the other elements are the same as β ^ A {\displaystyle {\hat {\beta }}^{\mathcal {A}}} . t ^ { j } = arg ⁡ min t L n LR ( β ^ A + t { j } ) {\displaystyle {\hat {\boldsymbol {t}}}^{\{j\}}=\arg \min _{t}{\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{\mathcal {A}}+{\boldsymbol {t}}^{\{j\}}\right)} : Here, t { j } {\displaystyle t^{\{j\}}} represents a vector where all elements are 0 except the j-th element. d ^ j = X j ⊤ ( y − X β ^ ) / n {\displaystyle {\hat {d}}_{j}={\boldsymbol {X}}_{j}^{\top }({\boldsymbol {y}}-{\boldsymbol {X}}{\hat {\boldsymbol {\beta }}})/n} : This is calculated based on the equation mentioned. The iterative process involves exchanging variables, with the aim of minimizing the sacrifices in the active set while maximizing the sacrifices in the inactive set during each iteration. This approach allows abess to efficiently search for the optimal feature subset. In abess, select an appropriate s max {\displaystyle s_{\max }} and optimize the above problem for active sets size s = 1 , … , s max {\displaystyle s=1,\ldots ,s_{\max }} using the information criterion GIC = n log ⁡ L n LR + s log ⁡ p log ⁡ log ⁡ n , {\displaystyle {\text{GIC}}=n\log {\mathcal {L}}_{n}^{\text{LR}}+s\log p\log \log n,} to adaptively choose the appropriate active set size s {\displaystyle s} and obtain its corresponding abess estimator. == Generalizations == The splicing algorithm in abess can be employed for subset selection in other models. === Distribution-Free Location-Scale Regression === In 2023, Siegfried extends abess to the case of Distribution-Free and Location-Scale. Specifically, it considers the optimization problem max ϑ ∈ R P , β ∈ R J , γ ∈ R J ∑ i = 1 N ℓ i ( ϑ , x i ⊤ β , exp ⁡ ( x i ⊤ γ ) − 1 ) , {\displaystyle \max _{{\boldsymbol {\vartheta }}\in \mathbb {R} ^{P},{\boldsymbol {\beta }}\in \mathbb {R} ^{J},{\boldsymbol {\gamma }}\in \mathbb {R} ^{J}}\sum _{i=1}^{N}\ell _{i}\left({\boldsymbol {\vartheta }},{\boldsymbol {x}}_{i}^{\top }{\boldsymbol {\beta }},{\sqrt {\exp \left({\boldsymbol {x}}_{i}^{\top }{\boldsymbol {\gamma }}\right)}}^{-1}\right),} subject to ‖ ( β ⊤ , γ ⊤ ) ⊤ ‖ 0 ≤ s , {\displaystyle \left\|\left({\boldsymbol {\beta }}^{\top },{\boldsymbol {\gamma }}^{\top }\right)^{\top }\right\|_{0}\leq s,} where ℓ i {\displaystyle \ell _{i}} is a loss function, ϑ {\displaystyle {\boldsymbol {\vartheta }}} is a parameter vector, β {\displaystyle {\boldsymbol {\beta }}} and γ {\displaystyle {\boldsymbol {\gamma }}} are vectors, and x i {\displaystyle {\boldsymbol {x}}_{i}} is a data vector. This approach, demonstrated across various applications, enables parsimonious regression modeling for arbitrary outcomes while maintaining interpretability through innovative subset selection procedures. === Groups Selection === In 2023, Zhang applied the splicing algorithm to group selection, optimizing the following model: min β ∈ R p L n LR ( β ; X , y ) subject to ∑ j = 1 J I ( ‖ β G j ‖ 2 ≠ 0 ) ≤ s {\displaystyle \min _{{\boldsymbol {\beta }}\in \mathbb {R} ^{p}}{\mathcal {L}}_{n}^{\text{LR}}(\beta ;X,y){\text{ subject to }}\sum _{j=1}^{J}I\left(\|{\boldsymbol {\beta }}_{G_{j}}\|_{2}\neq 0\right)\leq s} Here are the symbols involved: J {\displaystyle J} : Total number of feature groups, representing the existence of J {\displaystyle J} non-overlapping feature groups in the dataset. G j {\displaystyle G_{j}} : Index set for the j {\displaystyle j} -th feature group, where j {\displaystyle j} ranges from 1 to J {\displaystyle J} , representing the feature grouping structure in the data. s {\displaystyle s} : Model size, a positive integer determined from the data, limiting the number of selected feature groups. === Regression with Corrupted Data === Zhang applied the splicing algorithm to handle corrupted data. Corrupted data refers to information that has been disrupted or contains errors during the data collection or recording process. This interference may include sensor inaccuracies, recording errors, communication issues, or other external disturbances, leading to inaccurate or distorted observations within the dataset. === Single Index Models === In 2023, Tang applied the splicing algorithm to optimal subset selection in the Single-index model. The form of the Single Index Model (SIM) is given by y i = g ( b ⊤ x i , e i ) , i = 1 , … , n , {\displaystyle y_{i}=g({\boldsymbol {b}}^{\top }{\boldsymbol {x}}_{i},e_{i}),\quad i=1,\ldots ,n,} where b {\displaystyle {\boldsymbol {b}}} is the parameter vector, e i {\displaystyle e_{i}} is the error term. The corresponding loss function is defined as l n ( β ) = ∑ i = 1 n ( r i n − 1 2 − x i ⊤ β ) 2 , {\displaystyle l_{n}({\boldsymbol {\beta }})=\sum _{i=1}^{n}\left({\frac {r_{i}}{n}}-{\frac {1}{2}}-{\boldsymbol {x}}_{i}^{\top }{\boldsymbol {\beta }}\right)^{2},} where r {\disp
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Multi expression programming

Multi Expression Programming (MEP) is an evolutionary algorithm for generating mathematical functions describing a given set of data. MEP is a Genetic Programming variant encoding multiple solutions in the same chromosome. MEP representation is not specific (multiple representations have been tested). In the simplest variant, MEP chromosomes are linear strings of instructions. This representation was inspired by Three-address code. MEP strength consists in the ability to encode multiple solutions, of a problem, in the same chromosome. In this way, one can explore larger zones of the search space. For most of the problems this advantage comes with no running-time penalty compared with genetic programming variants encoding a single solution in a chromosome. == Representation == MEP chromosomes are arrays of instructions represented in Three-address code format. Each instruction contains a variable, a constant, or a function. If the instruction is a function, then the arguments (given as instruction's addresses) are also present. === Example of MEP program === Here is a simple MEP chromosome (labels on the left side are not a part of the chromosome): 1: a 2: b 3: + 1, 2 4: c 5: d 6: + 4, 5 7: 3, 5 == Fitness computation == When the chromosome is evaluated it is unclear which instruction will provide the output of the program. In many cases, a set of programs is obtained, some of them being completely unrelated (they do not have common instructions). For the above chromosome, here is the list of possible programs obtained during decoding: E1 = a, E2 = b, E4 = c, E5 = d, E3 = a + b. E6 = c + d. E7 = (a + b) d. Each instruction is evaluated as a possible output of the program. The fitness (or error) is computed in a standard manner. For instance, in the case of symbolic regression, the fitness is the sum of differences (in absolute value) between the expected output (called target) and the actual output. == Fitness assignment process == Which expression will represent the chromosome? Which one will give the fitness of the chromosome? In MEP, the best of them (which has the lowest error) will represent the chromosome. This is different from other GP techniques: In Linear genetic programming the last instruction will give the output. In Cartesian Genetic Programming the gene providing the output is evolved like all other genes. Note that, for many problems, this evaluation has the same complexity as in the case of encoding a single solution in each chromosome. Thus, there is no penalty in running time compared to other techniques. == Software == === MEPX === MEPX is a cross-platform (Windows, macOS, and Linux Ubuntu) free software for the automatic generation of computer programs. It can be used for data analysis, particularly for solving symbolic regression, statistical classification and time-series problems. === libmep === Libmep is a free and open source library implementing Multi Expression Programming technique. It is written in C++. === hmep === hmep is a new open source library implementing Multi Expression Programming technique in Haskell programming language.
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Fatsecret

Fatsecret, commonly styled as fatsecret, is a mobile application, website and API that helps people achieve their weight loss goals and find accurate nutrition information. It also offers a weight loss clinic with coaching and medically supported programs. The platform powers global health apps. == History == Fatsecret was founded in 2006 in Melbourne, Australia by Lenny Moses and Rodney Moses. As of 2019, Lenny serves as the company's CEO. The company is known for its calorie counting and meal tracking app, and by April 2016, the company claimed to have 45 million users of its services. In August 2018, a premium version of its app was released. Since August 2009, the company has operated the Fatsecret Platform API, which allows access to its global food and nutrition database. Fatsecret reportedly had 900,000 downloads of its app in January 2020. In an analysis of several Health & Fitness app subcategories for the United States in January 2021, Fatsecret was reported to have the highest 30 day user retention rate of top Calorie Counter + Meal Planner for Weight Loss apps.
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Time-aware long short-term memory

Time-aware LSTM (T-LSTM) is a long short-term memory (LSTM) unit capable of handling irregular time intervals in longitudinal patient records. T-LSTM was developed by researchers from Michigan State University, IBM Research, and Cornell University and was first presented in the Knowledge Discovery and Data Mining (KDD) conference. Experiments using real and synthetic data proved that T-LSTM auto-encoder outperformed widely used frameworks including LSTM and MF1-LSTM auto-encoders.
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Exploratory blockmodeling

Exploratory blockmodeling is an (inductive) approach (or a group of approaches) in blockmodeling regarding the specification of an ideal blockmodel. This approach, also known as hypotheses-generating, is the simplest approach, as it "merely involves the definition of the block types permitted as well as of the number of clusters." With this approach, researcher usually defines the best possible blockmodel, which then represent the base for the analysis of the whole network. This approach is usually based on: previous analyses and theoretical considerations, using stricker blockmodel and block types, where the structural equivalence is stricker than the regular equivalence and using smaller number of classes. The opposite approach is called a confirmatory blockmodeling.
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FastICA

FastICA is an efficient and popular algorithm for independent component analysis invented by Aapo Hyvärinen at Helsinki University of Technology. Like most ICA algorithms, FastICA seeks an orthogonal rotation of prewhitened data, through a fixed-point iteration scheme, that maximizes a measure of non-Gaussianity of the rotated components. Non-gaussianity serves as a proxy for statistical independence, which is a very strong condition and requires infinite data to verify. FastICA can also be alternatively derived as an approximative Newton iteration. == Algorithm == === Prewhitening the data === Let the X := ( x i j ) ∈ R N × M {\displaystyle \mathbf {X} :=(x_{ij})\in \mathbb {R} ^{N\times M}} denote the input data matrix, M {\displaystyle M} the number of columns corresponding with the number of samples of mixed signals and N {\displaystyle N} the number of rows corresponding with the number of independent source signals. The input data matrix X {\displaystyle \mathbf {X} } must be prewhitened, or centered and whitened, before applying the FastICA algorithm to it. Centering the data entails demeaning each component of the input data X {\displaystyle \mathbf {X} } , that is, for each i = 1 , … , N {\displaystyle i=1,\ldots ,N} and j = 1 , … , M {\displaystyle j=1,\ldots ,M} . After centering, each row of X {\displaystyle \mathbf {X} } has an expected value of 0 {\displaystyle 0} . Whitening the data requires a linear transformation L : R N × M → R N × M {\displaystyle \mathbf {L} :\mathbb {R} ^{N\times M}\to \mathbb {R} ^{N\times M}} of the centered data so that the components of L ( X ) {\displaystyle \mathbf {L} (\mathbf {X} )} are uncorrelated and have variance one. More precisely, if X {\displaystyle \mathbf {X} } is a centered data matrix, the covariance of L x := L ( X ) {\displaystyle \mathbf {L} _{\mathbf {x} }:=\mathbf {L} (\mathbf {X} )} is the ( N × N ) {\displaystyle (N\times N)} -dimensional identity matrix, that is, A common method for whitening is by performing an eigenvalue decomposition on the covariance matrix of the centered data X {\displaystyle \mathbf {X} } , E { X X T } = E D E T {\displaystyle E\left\{\mathbf {X} \mathbf {X} ^{T}\right\}=\mathbf {E} \mathbf {D} \mathbf {E} ^{T}} , where E {\displaystyle \mathbf {E} } is the matrix of eigenvectors and D {\displaystyle \mathbf {D} } is the diagonal matrix of eigenvalues. The whitened data matrix is defined thus by === Single component extraction === The iterative algorithm finds the direction for the weight vector w ∈ R N {\displaystyle \mathbf {w} \in \mathbb {R} ^{N}} that maximizes a measure of non-Gaussianity of the projection w T X {\displaystyle \mathbf {w} ^{T}\mathbf {X} } , with X ∈ R N × M {\displaystyle \mathbf {X} \in \mathbb {R} ^{N\times M}} denoting a prewhitened data matrix as described above. Note that w {\displaystyle \mathbf {w} } is a column vector. To measure non-Gaussianity, FastICA relies on a nonquadratic nonlinear function f ( u ) {\displaystyle f(u)} , its first derivative g ( u ) {\displaystyle g(u)} , and its second derivative g ′ ( u ) {\displaystyle g^{\prime }(u)} . Hyvärinen states that the functions are useful for general purposes, while may be highly robust. The steps for extracting the weight vector w {\displaystyle \mathbf {w} } for single component in FastICA are the following: Randomize the initial weight vector w {\displaystyle \mathbf {w} } Let w + ← E { X g ( w T X ) T } − E { g ′ ( w T X ) } w {\displaystyle \mathbf {w} ^{+}\leftarrow E\left\{\mathbf {X} g(\mathbf {w} ^{T}\mathbf {X} )^{T}\right\}-E\left\{g'(\mathbf {w} ^{T}\mathbf {X} )\right\}\mathbf {w} } , where E { . . . } {\displaystyle E\left\{...\right\}} means averaging over all column-vectors of matrix X {\displaystyle \mathbf {X} } Let w ← w + / ‖ w + ‖ {\displaystyle \mathbf {w} \leftarrow \mathbf {w} ^{+}/\|\mathbf {w} ^{+}\|} If not converged, go back to 2 === Multiple component extraction === The single unit iterative algorithm estimates only one weight vector which extracts a single component. Estimating additional components that are mutually "independent" requires repeating the algorithm to obtain linearly independent projection vectors - note that the notion of independence here refers to maximizing non-Gaussianity in the estimated components. Hyvärinen provides several ways of extracting multiple components with the simplest being the following. Here, 1 M {\displaystyle \mathbf {1_{M}} } is a column vector of 1's of dimension M {\displaystyle M} . Algorithm FastICA Input: C {\displaystyle C} Number of desired components Input: X ∈ R N × M {\displaystyle \mathbf {X} \in \mathbb {R} ^{N\times M}} Prewhitened matrix, where each column represents an N {\displaystyle N} -dimensional sample, where C <= N {\displaystyle C<=N} Output: W ∈ R N × C {\displaystyle \mathbf {W} \in \mathbb {R} ^{N\times C}} Un-mixing matrix where each column projects X {\displaystyle \mathbf {X} } onto independent component. Output: S ∈ R C × M {\displaystyle \mathbf {S} \in \mathbb {R} ^{C\times M}} Independent components matrix, with M {\displaystyle M} columns representing a sample with C {\displaystyle C} dimensions. for p in 1 to C: w p ← {\displaystyle \mathbf {w_{p}} \leftarrow } Random vector of length N while w p {\displaystyle \mathbf {w_{p}} } changes w p ← 1 M X g ( w p T X ) T − 1 M g ′ ( w p T X ) 1 M w p {\displaystyle \mathbf {w_{p}} \leftarrow {\frac {1}{M}}\mathbf {X} g(\mathbf {w_{p}} ^{T}\mathbf {X} )^{T}-{\frac {1}{M}}g'(\mathbf {w_{p}} ^{T}\mathbf {X} )\mathbf {1_{M}} \mathbf {w_{p}} } w p ← w p − ∑ j = 1 p − 1 ( w p T w j ) w j {\displaystyle \mathbf {w_{p}} \leftarrow \mathbf {w_{p}} -\sum _{j=1}^{p-1}(\mathbf {w_{p}} ^{T}\mathbf {w_{j}} )\mathbf {w_{j}} } w p ← w p ‖ w p ‖ {\displaystyle \mathbf {w_{p}} \leftarrow {\frac {\mathbf {w_{p}} }{\|\mathbf {w_{p}} \|}}} output W ← [ w 1 , … , w C ] {\displaystyle \mathbf {W} \leftarrow {\begin{bmatrix}\mathbf {w_{1}} ,\dots ,\mathbf {w_{C}} \end{bmatrix}}} output S ← W T X {\displaystyle \mathbf {S} \leftarrow \mathbf {W^{T}} \mathbf {X} }
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Tinybop

Tinybop is a Brooklyn based publisher of apps for children. == History == Tinybop is a Brooklyn-based children's media company established in 2011 by Raul Gutierrez. App titles are released in two series: the Explorer's Library - a series of science apps and Digital Toys - series of open-ended construction apps. == Published apps == Explorer's Library Titles: The Human Body – An anatomy app for children. Released 2013. The company's first app was illustrated by Kelli Anderson and has been downloaded millions of times. Selected for the American Library Association's Notable Children's Media List in 2022. Named Apple App Store's Best of 2013. Winner of the Digital Ehon Yuichi Kimura Prize for Children's Digital Media. Plants – An app about biomes around the world. Homes – An app about houses around with world. Illustrated by Tuesday Bassen. Winner of the Parents Gold Choice Award for children's apps. Simple Machines – A children's physics app about simple machines. The Earth – An app for children about the geologic Earth illustrated by Sarah Jacoby. Weather – A children's weather app. Skyscrapers – A children's app about building tall buildings. Space – An interactive solar system. Mammals – A children's app about mammals illustrated by Wenjia Tang. Winner of the Digital Ehon Award for Children's Educational media. Coral Reef – An app about marine ecosystems. Winner of an Excellence in Early Learning Digital Media Honor from the American Library Association. State of Matter – An app covering solids, liquids, and gases. Winner of Excellence in Early Learning Digital Media Honor from the American Library Association. Light and Color – An app about light and color. Selected for The American Library Association's Notable Children's Media List 2023. Winner of the 2022 Yoichi Sakakihara Prize for Children's Media. Digital Toys Titles: The Robot Factory – A robot building app for children illustrated by Owen Davey. Apple named The Robot Factory as iPad App of the Year in 2015. The Everything Machine – A visual coding app for children. The Everything Machine was named Apple's Best of 2015. Monsters – A monster creation app illustrated by Tianhua Mao. The Infinite Arcade – An arcade game building app. Me: A Kids Diary – A digital journal for children. Selected for The American Library Association's Notable Children's Media List 2020. The Creature Garden – An app that allows children to create fantastical animals illustrated by Natasha Durley. Selected for The American Library Association's Notable Children's Media List 2021. Things that Go Bump – A multiplayer game set in an enchanted Japanese house, released on Apple Arcade in 2018.
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Correlation clustering

Clustering is the problem of partitioning data points into groups based on similarity or dissimilarity. Correlation clustering is a clustering framework in which a set of objects is partitioned into clusters based on pairwise similarity and dissimilarity information, without requiring the number of clusters to be specified in advance. == Description of the problem == In machine learning, correlation clustering (also known as cluster editing) considers settings in which pairwise similarity or dissimilarity relationships between objects are known. A standard formulation models the input as an unweighted complete graph G = ( V , E ) {\displaystyle G=(V,E)} , where each edge is labeled either + {\displaystyle +} or − {\displaystyle -} (that is, the graph is a signed graph), indicating whether the corresponding endpoints are similar or dissimilar. The goal is to find a clustering (that is, a partition of V {\displaystyle V} ) that either maximizes the number of agreements—the sum of positive edges whose endpoints lie in the same cluster and negative edges whose endpoints lie in different clusters—or minimizes the number of disagreements—the sum of positive edges whose endpoints are separated and negative edges whose endpoints lie in the same cluster. Unlike other clustering methods such as k-means, correlation clustering does not require choosing the number of clusters k {\displaystyle k} in advance. It is not always possible to find a clustering with zero disagreements. For example, consider a triangle graph containing two positive edges and one negative edge. In this case, every clustering incurs at least one disagreement. Such configurations are referred to in the literature as bad triangles. From a computational perspective, optimizing the correlation clustering objective is challenging. The (decision version of the) problem is NP-complete. A large body of subsequent work has developed approximation algorithms for correlation clustering under various assumptions, including complete or general graphs and unweighted or weighted graphs, for both minimization and maximization objectives. This problem is considered one of the fundamental combinatorial optimization problems, and many algorithmic techniques have been developed to address it. The problem has also been studied extensively across multiple disciplines. A comprehensive literature review of early correlation clustering research is provided by Wahid and Hassini. == Formal Definitions == Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph with nodes V {\displaystyle V} and edges E {\displaystyle E} . A clustering of G {\displaystyle G} is a partition of its node set Π = { π 1 , … , π k } {\displaystyle \Pi =\{\pi _{1},\dots ,\pi _{k}\}} with V = π 1 ∪ ⋯ ∪ π k {\displaystyle V=\pi _{1}\cup \dots \cup \pi _{k}} and π i ∩ π j = ∅ {\displaystyle \pi _{i}\cap \pi _{j}=\emptyset } for i ≠ j {\displaystyle i\neq j} . For a given clustering Π {\displaystyle \Pi } , let δ ( Π ) = { { u , v } ∈ E ∣ { u , v } ⊈ π ∀ π ∈ Π } {\displaystyle \delta (\Pi )=\{\{u,v\}\in E\mid \{u,v\}\not \subseteq \pi \;\forall \pi \in \Pi \}} denote the subset of edges of G {\displaystyle G} whose endpoints are in different subsets of the clustering Π {\displaystyle \Pi } . Now, let w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} be a function that assigns a non-negative weight to each edge of the graph and let E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} be a partition of the edges into attractive ( E + {\displaystyle E^{+}} ) and repulsive ( E − {\displaystyle E^{-}} ) edges; that is, the edges are signed. The minimum disagreement correlation clustering problem is the following optimization problem: minimize Π ∑ e ∈ E + ∩ δ ( Π ) w e + ∑ e ∈ E − ∖ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in E^{+}\cap \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\setminus \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∩ δ ( Π ) {\displaystyle E^{+}\cap \delta (\Pi )} contains the attractive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } and the set E − ∖ δ ( Π ) {\displaystyle E^{-}\setminus \delta (\Pi )} contains the repulsive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that disagree with the clustering Π {\displaystyle \Pi } . Similarly to the minimum disagreement correlation clustering problem, the maximum agreement correlation clustering problem is defined as maximize Π ∑ e ∈ E + ∖ δ ( Π ) w e + ∑ e ∈ E − ∩ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E^{+}\setminus \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\cap \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∖ δ ( Π ) {\displaystyle E^{+}\setminus \delta (\Pi )} contains the attractive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } and the set E − ∩ δ ( Π ) {\displaystyle E^{-}\cap \delta (\Pi )} contains the repulsive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that agree with the clustering Π {\displaystyle \Pi } . Instead of formulating the correlation clustering problem in terms of non-negative edge weights and a partition of the edges into attractive and repulsive edges the problem is also formulated in terms of positive and negative edge costs without partitioning the set of edges explicitly. For given weights w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} and a given partition E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} of the edges into attractive and repulsive edges, the edge costs can be defined by c e = { w e if e ∈ E + − w e if e ∈ E − {\displaystyle {\begin{aligned}c_{e}={\begin{cases}\;\;w_{e}&{\text{if }}e\in E^{+}\\-w_{e}&{\text{if }}e\in E^{-}\end{cases}}\end{aligned}}} for all e ∈ E {\displaystyle e\in E} . An edge whose endpoints are in different clusters is said to be cut. The set δ ( Π ) {\displaystyle \delta (\Pi )} of all edges that are cut is often called a multicut of G {\displaystyle G} . The minimum cost multicut problem is the problem of finding a clustering Π {\displaystyle \Pi } of G {\displaystyle G} such that the sum of the costs of the edges whose endpoints are in different clusters is minimal: minimize Π ∑ e ∈ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in \delta (\Pi )}c_{e}\;.\end{aligned}}} Similar to the minimum cost multicut problem, coalition structure generation in weighted graph games is the problem of finding a clustering such that the sum of the costs of the edges that are not cut is maximal: maximize Π ∑ e ∈ E ∖ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E\setminus \delta (\Pi )}c_{e}\;.\end{aligned}}} This formulation is also known as the clique partitioning problem. It can be shown that all four problems that are formulated above are equivalent. This means that a clustering that is optimal with respect to any of the four objectives is optimal for all of the four objectives. == Algorithms == If the graph admits a clustering with zero disagreements, then deleting all negative edges and computing the connected components of the remaining graph yields an optimal clustering. A necessary and sufficient condition for the existence of such a clustering was given by Davis: no cycle in the graph may contain exactly one negative edge. Bansal et al. discuss the NP-completeness proof and also present both a constant factor approximation algorithm and polynomial-time approximation scheme to find the clusters in this setting. Ailon et al. propose a randomized 3-approximation algorithm for the same problem. CC-Pivot(G=(V,E+,E−)) Pick random pivot i ∈ V Set C = { i } {\displaystyle C=\{i\}} , V'=Ø For all j ∈ V, j ≠ i; If (i,j) ∈ E+ then Add j to C Else (If (i,j) ∈ E−) Add j to V' Let G' be the subgraph induced by V' Return clustering C,CC-Pivot(G') The authors show that the above algorithm is a 3-approximation algorithm for correlation clustering. The best polynomial-time approximation algorithm known at the moment for this problem achieves a ~2.06 approximation by rounding a linear program, as shown by Chawla, Makarychev, Schramm, and Yaroslavtsev. Karpinski and Schudy proved existence of a polynomial time approximation scheme (PTAS) for that problem on complete graphs and fixed number of clusters. == Optimal number of clusters == In 2011, it was shown by Bagon and Galun that the optimization of the correlation clustering functional is closely related to well known discrete optimization methods. In their work they proposed a probabilistic analysis of the underlying implicit model that allows the correlation clustering functional to estimate the
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Logic learning machine

Logic learning machine (LLM) is a machine learning method based on the generation of intelligible rules. LLM is an efficient implementation of the Switching Neural Network (SNN) paradigm, developed by Marco Muselli, Senior Researcher at the Italian National Research Council CNR-IEIIT in Genoa. LLM has been employed in many different sectors, including the field of medicine (orthopedic patient classification, DNA micro-array analysis and Clinical Decision Support Systems), financial services and supply chain management. == History == The Switching Neural Network approach was developed in the 1990s to overcome the drawbacks of the most commonly used machine learning methods. In particular, black box methods, such as multilayer perceptron and support vector machine, had good accuracy but could not provide deep insight into the studied phenomenon. On the other hand, decision trees were able to describe the phenomenon but often lacked accuracy. Switching Neural Networks made use of Boolean algebra to build sets of intelligible rules able to obtain very good performance. In 2014, an efficient version of Switching Neural Network was developed and implemented in the Rulex suite with the name Logic Learning Machine. Also, an LLM version devoted to regression problems was developed. == General == Like other machine learning methods, LLM uses data to build a model able to perform a good forecast about future behaviors. LLM starts from a table including a target variable (output) and some inputs and generates a set of rules that return the output value y {\displaystyle y} corresponding to a given configuration of inputs. A rule is written in the form: if premise then consequence where consequence contains the output value whereas premise includes one or more conditions on the inputs. According to the input type, conditions can have different forms: for categorical variables the input value must be in a given subset: x 1 ∈ { A , B , C , . . . } {\displaystyle x_{1}\in \{A,B,C,...\}} . for ordered variables the condition is written as an inequality or an interval: x 2 ≤ α {\displaystyle x_{2}\leq \alpha } or β ≤ x 3 ≤ γ {\displaystyle \beta \leq x_{3}\leq \gamma } A possible rule is therefore in the form if x 1 ∈ { A , B , C , . . . } {\displaystyle x_{1}\in \{A,B,C,...\}} AND x 2 ≤ α {\displaystyle x_{2}\leq \alpha } AND β ≤ x 3 ≤ γ {\displaystyle \beta \leq x_{3}\leq \gamma } then y = y ¯ {\displaystyle y={\bar {y}}} == Types == According to the output type, different versions of the Logic Learning Machine have been developed: Logic Learning Machine for classification, when the output is a categorical variable, which can assume values in a finite set Logic Learning Machine for regression, when the output is an integer or real number.
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