The universal portfolio algorithm is a portfolio selection algorithm from the field of machine learning and information theory. The algorithm learns adaptively from historical data and maximizes log-optimal growth rate in the long run, per the Kelly criterion. It was introduced by the late Stanford University information theorist Thomas M. Cover. The algorithm rebalances the portfolio at the beginning of each trading period. At the beginning of the first trading period it starts with a naive diversification. In the following trading periods the portfolio composition depends on the historical total return of all possible constant-rebalanced portfolios. The universal portfolio algorithm is the predecessor of the various online portfolio selection methodologies.
IT operations analytics
In the fields of information technology (IT) and systems management, IT operations analytics (ITOA) is an approach or method to retrieve, analyze, and report data for IT operations. ITOA may apply big data analytics to large datasets to produce business insights. In 2014, Gartner predicted its use might increase revenue or reduce costs. By 2017, it predicted that 15% of enterprises will use IT operations analytics technologies. == Definition == IT operations analytics (ITOA) (also known as advanced operational analytics, or IT data analytics) technologies are primarily used to discover complex patterns in high volumes of often "noisy" IT system availability and performance data. Forrester Research defined IT analytics as "The use of mathematical algorithms and other innovations to extract meaningful information from the sea of raw data collected by management and monitoring technologies." Note, ITOA is different than AIOps, which focuses on applying artificial intelligence and machine learning to the applications of ITOA. == History == Operations research as a discipline emerged from the Second World War to improve military efficiency and decision-making on the battlefield. However, only with the emergence of machine learning tech in the early 2000s could an artificially intelligent operational analytics platform actually begin to engage in the high-level pattern recognition that could adequately serve business needs. A critical catalyst towards ITOA development was the rise of Google, which pioneered a predictive analytics model that represented the first attempt to read into patterns of human behavior on the Internet. IT specialists then applied predictive analytics to the IT Industry, coming forward with platforms that can sift through data to generate insights without the need for human intervention. Due to the mainstream embrace of cloud computing and the increasing desire for businesses to adopt more big data practices, the ITOA industry has grown significantly since 2010. A 2016 ExtraHop survey of large and mid-size corporations indicates that 65 percent of the businesses surveyed will seek to integrate their data silos either this year or the next. The current goals of ITOA platforms are to improve the accuracy of their APM services, facilitate better integration with the data, and to enhance their predictive analytics capabilities. == Applications == ITOA systems tend to be used by IT operations teams, and Gartner describes seven applications of ITOA systems: Root cause analysis: The models, structures and pattern descriptions of IT infrastructure or application stack being monitored can help users pinpoint fine-grained and previously unknown root causes of overall system behavior pathologies. Proactive control of service performance and availability: Predicts future system states and the impact of those states on performance. Problem assignment: Determines how problems may be resolved or, at least, direct the results of inferences to the most appropriate individuals, or communities in the enterprise for problem resolution. Service impact analysis: When multiple root causes are known, the analytics system's output is used to determine and rank the relative impact, so that resources can be devoted to correcting the fault in the most timely and cost-effective way possible. Complement best-of-breed technology: The models, structures and pattern descriptions of IT infrastructure or application stack being monitored are used to correct or extend the outputs of other discovery-oriented tools to improve the fidelity of information used in operational tasks (e.g., service dependency maps, application runtime architecture topologies, network topologies). Real time application behavior learning: Learns & correlates the behavior of Application based on user pattern and underlying Infrastructure on various application patterns, create metrics of such correlated patterns and store it for further analysis. Dynamically baselines threshold: Learns behavior of Infrastructure on various application user patterns and determines the Optimal behavior of the Infra and technological components, bench marks and baselines the low and high water mark for the specific environments and dynamically changes the bench mark baselines with the changing infra and user patterns without any manual intervention. == Types == In their Data Growth Demands a Single, Architected IT Operations Analytics Platform, Gartner Research describes five types of analytics technologies: Log analysis Unstructured text indexing, search and inference (UTISI) Topological analysis (TA) Multidimensional database search and analysis (MDSA) Complex operations event processing (COEP) Statistical pattern discovery and recognition (SPDR) == Tools and ITOA platforms == A number of vendors operate in the ITOA space:
XGBoost
XGBoost (eXtreme Gradient Boosting) is an open-source software library which provides a regularizing gradient boosting framework for C++, Java, Python, R, Julia, Perl, and Scala. It works on Linux, Microsoft Windows, and macOS. From the project description, it aims to provide a "Scalable, Portable and Distributed Gradient Boosting (GBM, GBRT, GBDT) Library". It runs on a single machine, as well as the distributed processing frameworks Apache Hadoop, Apache Spark, Apache Flink, and Dask. XGBoost gained much popularity and attention in the mid-2010s as the algorithm of choice for many winning teams of machine learning competitions. == History == XGBoost initially started as a research project by Tianqi Chen as part of the Distributed (Deep) Machine Learning Community (DMLC) group at the University of Washington. Initially, it began as a terminal application which could be configured using a libsvm configuration file. It became well known in the ML competition circles after its use in the winning solution of the Higgs Machine Learning Challenge. Soon after, the Python and R packages were built, and XGBoost now has package implementations for Java, Scala, Julia, Perl, and other languages. This brought the library to more developers and contributed to its popularity among the Kaggle community, where it has been used for a large number of competitions. It was soon integrated with a number of other packages making it easier to use in their respective communities. It has now been integrated with scikit-learn for Python users and with the caret package for R users. It can also be integrated into Data Flow frameworks like Apache Spark, Apache Hadoop, and Apache Flink using the abstracted Rabit and XGBoost4J. XGBoost is also available on OpenCL for FPGAs. An efficient, scalable implementation of XGBoost has been published by Tianqi Chen and Carlos Guestrin. While the XGBoost model often achieves higher accuracy than a single decision tree, it sacrifices the intrinsic interpretability of decision trees. For example, following the path that a decision tree takes to make its decision is trivial and self-explained, but following the paths of hundreds or thousands of trees is much harder. == Features == Salient features of XGBoost which make it different from other gradient boosting algorithms include: Clever penalization of trees A proportional shrinking of leaf nodes Newton Boosting Extra randomization parameter Implementation on single, distributed systems and out-of-core computation Automatic feature selection Theoretically justified weighted quantile sketching for efficient computation Parallel tree structure boosting with sparsity Efficient cacheable block structure for decision tree training == The algorithm == XGBoost works as Newton–Raphson in function space unlike gradient boosting that works as gradient descent in function space, a second order Taylor approximation is used in the loss function to make the connection to Newton–Raphson method. A generic unregularized XGBoost algorithm is: == Parameters == XGBoost has parameters which can be specified to affect how it functions and performs. Some parameters include: Learning rate (also known as the "step size" or “"shrinkage"), it is a number between 0 and 1 (default is 0.3), which determines the rate the algorithm learns from each iteration. n_estimators, sets the number of trees to be built in the ensemble, where more trees generally increases the complexity of the model, but can lead to overfitting with too many trees. Gamma (also known as Lagrange multiplier or the minimum loss reduction parameter), controls the minimum amount of loss reduction required to make a further split on a leaf node of the tree. The default in XGBoost is 0 . max_depth, represents how deeply each tree in the boosting process can grow during training, where the default is 6. == Awards == John Chambers Award (2016) High Energy Physics meets Machine Learning award (HEP meets ML) (2016)
Oja's rule
Oja's learning rule, or simply Oja's rule, named after Finnish computer scientist Erkki Oja (Finnish pronunciation: [ˈojɑ], AW-yuh), is a model of how neurons in the brain or in artificial neural networks change connection strength, or learn, over time. It is a modification of the standard Hebb's Rule that, through multiplicative normalization, solves all stability problems and generates an algorithm for principal components analysis. This is a computational form of an effect which is believed to happen in biological neurons. == Theory == Oja's rule requires a number of simplifications to derive, but in its final form it is demonstrably stable, unlike Hebb's rule. It is a single-neuron special case of the Generalized Hebbian Algorithm. However, Oja's rule can also be generalized in other ways to varying degrees of stability and success. === Formula === Consider a simplified model of a neuron y {\displaystyle y} that returns a linear combination of its inputs x using presynaptic weights w: y ( x ) = ∑ j = 1 m x j w j {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}} Oja's rule defines the change in presynaptic weights w given the output response y {\displaystyle y} of a neuron to its inputs x to be Δ w = w n + 1 − w n = η y n ( x n − y n w n ) , {\displaystyle \,\Delta \mathbf {w} ~=~\mathbf {w} _{n+1}-\mathbf {w} _{n}~=~\eta \,y_{n}(\mathbf {x} _{n}-y_{n}\mathbf {w} _{n}),} where η is the learning rate which can also change with time. Note that the bold symbols are vectors and n defines a discrete time iteration. The rule can also be made for continuous iterations as d w d t = η y ( t ) ( x ( t ) − y ( t ) w ( t ) ) . {\displaystyle \,{\frac {d\mathbf {w} }{dt}}~=~\eta \,y(t)(\mathbf {x} (t)-y(t)\mathbf {w} (t)).} === Derivation === The simplest learning rule known is Hebb's rule, which states in conceptual terms that neurons that fire together, wire together. In component form as a difference equation, it is written Δ w = η y ( x n ) x n {\displaystyle \,\Delta \mathbf {w} ~=~\eta \,y(\mathbf {x} _{n})\mathbf {x} _{n}} , or in scalar form with implicit n-dependence, w i ( n + 1 ) = w i ( n ) + η y ( x ) x i {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}} , where y(xn) is again the output, this time explicitly dependent on its input vector x. Hebb's rule has synaptic weights approaching infinity with a positive learning rate. We can stop this by normalizing the weights so that each weight's magnitude is restricted between 0, corresponding to no weight, and 1, corresponding to being the only input neuron with any weight. We do this by normalizing the weight vector to be of length one: w i ( n + 1 ) = w i ( n ) + η y ( x ) x i ( ∑ j = 1 m [ w j ( n ) + η y ( x ) x j ] p ) 1 / p {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}}{\left(\sum _{j=1}^{m}[w_{j}(n)+\eta \,y(\mathbf {x} )x_{j}]^{p}\right)^{1/p}}}} . Note that in Oja's original paper, p=2, corresponding to quadrature (root sum of squares), which is the familiar Cartesian normalization rule. However, any type of normalization, even linear, will give the same result without loss of generality. For a small learning rate | η | ≪ 1 {\displaystyle |\eta |\ll 1} the equation can be expanded as a Power series in η {\displaystyle \eta } . w i ( n + 1 ) = w i ( n ) ( ∑ j w j p ( n ) ) 1 / p + η ( y x i ( ∑ j w j p ( n ) ) 1 / p − w i ( n ) ∑ j y x j w j p − 1 ( n ) ( ∑ j w j p ( n ) ) ( 1 + 1 / p ) ) + O ( η 2 ) {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}~+~\eta \left({\frac {yx_{i}}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}-{\frac {w_{i}(n)\sum _{j}yx_{j}w_{j}^{p-1}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{(1+1/p)}}}\right)~+~O(\eta ^{2})} . For small η, our higher-order terms O(η2) go to zero. We again make the specification of a linear neuron, that is, the output of the neuron is equal to the sum of the product of each input and its synaptic weight to the power of p-1, which in the case of p=2 is synaptic weight itself, or y ( x ) = ∑ j = 1 m x j w j p − 1 {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}^{p-1}} . We also specify that our weights normalize to 1, which will be a necessary condition for stability, so | w | = ( ∑ j = 1 m w j p ) 1 / p = 1 {\displaystyle \,|\mathbf {w} |~=~\left(\sum _{j=1}^{m}w_{j}^{p}\right)^{1/p}~=~1} , which, when substituted into our expansion, gives Oja's rule, or w i ( n + 1 ) = w i ( n ) + η y ( x i − w i ( n ) y ) {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(x_{i}-w_{i}(n)y)} . === Stability and PCA === In analyzing the convergence of a single neuron evolving by Oja's rule, one extracts the first principal component, or feature, of a data set. Furthermore, with extensions using the Generalized Hebbian Algorithm, one can create a multi-Oja neural network that can extract as many features as desired, allowing for principal components analysis. A principal component aj is extracted from a dataset x through some associated vector qj, or aj = qj⋅x, and we can restore our original dataset by taking x = ∑ j a j q j {\displaystyle \mathbf {x} ~=~\sum _{j}a_{j}\mathbf {q} _{j}} . In the case of a single neuron trained by Oja's rule, we find the weight vector converges to q1, or the first principal component, as time or number of iterations approaches infinity. We can also define, given a set of input vectors Xi, that its correlation matrix Rij = XiXj has an associated eigenvector given by qj with eigenvalue λj. The variance of outputs of our Oja neuron σ2(n) = ⟨y2(n)⟩ then converges with time iterations to the principal eigenvalue, or lim n → ∞ σ 2 ( n ) = λ 1 {\displaystyle \lim _{n\rightarrow \infty }\sigma ^{2}(n)~=~\lambda _{1}} . These results are derived using Lyapunov function analysis, and they show that Oja's neuron necessarily converges on strictly the first principal component if certain conditions are met in our original learning rule. Most importantly, our learning rate η is allowed to vary with time, but only such that its sum is divergent but its power sum is convergent, that is ∑ n = 1 ∞ η ( n ) = ∞ , ∑ n = 1 ∞ η ( n ) p < ∞ , p > 1 {\displaystyle \sum _{n=1}^{\infty }\eta (n)=\infty ,~~~\sum _{n=1}^{\infty }\eta (n)^{p}<\infty ,~~~p>1} . Our output activation function y(x(n)) is also allowed to be nonlinear and nonstatic, but it must be continuously differentiable in both x and w and have derivatives bounded in time. == Applications == Oja's rule was originally described in Oja's 1982 paper, but the principle of self-organization to which it is applied is first attributed to Alan Turing in 1952. PCA has also had a long history of use before Oja's rule formalized its use in network computation in 1989. The model can thus be applied to any problem of self-organizing mapping, in particular those in which feature extraction is of primary interest. Therefore, Oja's rule has an important place in image and speech processing. It is also useful as it expands easily to higher dimensions of processing, thus being able to integrate multiple outputs quickly. A canonical example is its use in binocular vision. === Biology and Oja's subspace rule === There is clear evidence for both long-term potentiation and long-term depression in biological neural networks, along with a normalization effect in both input weights and neuron outputs. However, while there is no direct experimental evidence yet of Oja's rule active in a biological neural network, a biophysical derivation of a generalization of the rule is possible. Such a derivation requires retrograde signalling from the postsynaptic neuron, which is biologically plausible (see neural backpropagation), and takes the form of Δ w i j ∝ ⟨ x i y j ⟩ − ϵ ⟨ ( c p r e ∗ ∑ k w i k y k ) ⋅ ( c p o s t ∗ y j ) ⟩ , {\displaystyle \Delta w_{ij}~\propto ~\langle x_{i}y_{j}\rangle -\epsilon \left\langle \left(c_{\mathrm {pre} }\sum _{k}w_{ik}y_{k}\right)\cdot \left(c_{\mathrm {post} }y_{j}\right)\right\rangle ,} where as before wij is the synaptic weight between the ith input and jth output neurons, x is the input, y is the postsynaptic output, and we define ε to be a constant analogous the learning rate, and cpre and cpost are presynaptic and postsynaptic functions that model the weakening of signals over time. Note that the angle brackets denote the average and the ∗ operator is a convolution. By taking the pre- and post-synaptic functions into frequency space and combining integration terms with the convolution, we find that this gives an arbitrary-dimensional generalization of Oja's rule known as Oja's Subspace, namely Δ w = C x ⋅ w − w ⋅ C y . {\displaystyle \Delta w~=~Cx\cdot w-w\cdot Cy.}
Algorithmic learning theory
Algorithmic learning theory is a mathematical framework for analyzing machine learning problems and algorithms. Synonyms include formal learning theory and algorithmic inductive inference. Algorithmic learning theory is different from statistical learning theory in that it does not make use of statistical assumptions and analysis. Both algorithmic and statistical learning theory are concerned with machine learning and can thus be viewed as branches of computational learning theory. == Distinguishing characteristics == Unlike statistical learning theory and most statistical theory in general, algorithmic learning theory does not assume that data are random samples, that is, that data points are independent of each other. This makes the theory suitable for domains where observations are (relatively) noise-free but not random, such as language learning and automated scientific discovery. The fundamental concept of algorithmic learning theory is learning in the limit: as the number of data points increases, a learning algorithm should converge to a correct hypothesis on every possible data sequence consistent with the problem space. This is a non-probabilistic version of statistical consistency, which also requires convergence to a correct model in the limit, but allows a learner to fail on data sequences with probability measure 0 . Algorithmic learning theory investigates the learning power of Turing machines. Other frameworks consider a much more restricted class of learning algorithms than Turing machines, for example, learners that compute hypotheses more quickly, for instance in polynomial time. An example of such a framework is probably approximately correct learning . == Learning in the limit == The concept was introduced in E. Mark Gold's seminal paper "Language identification in the limit". The objective of language identification is for a machine running one program to be capable of developing another program by which any given sentence can be tested to determine whether it is "grammatical" or "ungrammatical". The language being learned need not be English or any other natural language - in fact the definition of "grammatical" can be absolutely anything known to the tester. In Gold's learning model, the tester gives the learner an example sentence at each step, and the learner responds with a hypothesis, which is a suggested program to determine grammatical correctness. It is required of the tester that every possible sentence (grammatical or not) appears in the list eventually, but no particular order is required. It is required of the learner that at each step the hypothesis must be correct for all the sentences so far. A particular learner is said to be able to "learn a language in the limit" if there is a certain number of steps beyond which its hypothesis no longer changes. At this point it has indeed learned the language, because every possible sentence appears somewhere in the sequence of inputs (past or future), and the hypothesis is correct for all inputs (past or future), so the hypothesis is correct for every sentence. The learner is not required to be able to tell when it has reached a correct hypothesis, all that is required is that it be true. Gold showed that any language which is defined by a Turing machine program can be learned in the limit by another Turing-complete machine using enumeration. This is done by the learner testing all possible Turing machine programs in turn until one is found which is correct so far - this forms the hypothesis for the current step. Eventually, the correct program will be reached, after which the hypothesis will never change again (but note that the learner does not know that it won't need to change). Gold also showed that if the learner is given only positive examples (that is, only grammatical sentences appear in the input, not ungrammatical sentences), then the language can only be guaranteed to be learned in the limit if there are only a finite number of possible sentences in the language (this is possible if, for example, sentences are known to be of limited length). Language identification in the limit is a highly abstract model. It does not allow for limits of runtime or computer memory which can occur in practice, and the enumeration method may fail if there are errors in the input. However the framework is very powerful, because if these strict conditions are maintained, it allows the learning of any program known to be computable. This is because a Turing machine program can be written to mimic any program in any conventional programming language. See Church-Turing thesis. == Other identification criteria == Learning theorists have investigated other learning criteria, such as the following. Efficiency: minimizing the number of data points required before convergence to a correct hypothesis. Mind Changes: minimizing the number of hypothesis changes that occur before convergence. Mind change bounds are closely related to mistake bounds that are studied in statistical learning theory. Kevin Kelly has suggested that minimizing mind changes is closely related to choosing maximally simple hypotheses in the sense of Occam’s Razor. == Annual conference == Since 1990, there is an International Conference on Algorithmic Learning Theory (ALT), called Workshop in its first years (1990–1997). Between 1992 and 2016, proceedings were published in the LNCS series. Starting from 2017, they are published by the Proceedings of Machine Learning Research. The 34th conference will be held in Singapore in Feb 2023. The topics of the conference cover all of theoretical machine learning, including statistical and computational learning theory, online learning, active learning, reinforcement learning, and deep learning.
Tiimo
Tiimo is an app designed to help neurodivergent individuals with planning their life. In August 2024 the company raised €1.4 million, bringing their total funding to €4.3 million. At that point they had over 500,000 users, including 50,000 paid users. The app has Apple Watch support and a learning platform that includes courses on well-being and neurodiversity. The app was founded by Helene Lassen Nørlem and Melissa Würtz Azari in 2015. After being a finalist in 2024, in December 2025 Tiimo was won Apple’s iPhone App of the Year. The premium version is $10/mo and features an AI chatbot alongside the daily planner.
Apache Giraph
Apache Giraph is an Apache project to perform graph processing on big data. Giraph utilizes Apache Hadoop's MapReduce implementation to process graphs. Facebook used Giraph with some performance improvements to analyze one trillion edges using 200 machines in 4 minutes. Giraph is based on a paper published by Google about its own graph processing system called Pregel. It can be compared to other Big Graph processing libraries such as Cassovary. As of September 2023, it is no longer actively developed.