Statistical learning theory

Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. Statistical learning theory deals with the statistical inference problem of finding a predictive function based on data. Statistical learning theory has led to successful applications in fields such as computer vision, speech recognition, and bioinformatics. == Introduction == The goals of learning are understanding and prediction. Learning falls into many categories, including supervised learning, unsupervised learning, online learning, and reinforcement learning. From the perspective of statistical learning theory, supervised learning is best understood. Supervised learning involves learning from a training set of data. Every point in the training is an input–output pair, where the input maps to an output. The learning problem consists of inferring the function that maps between the input and the output, such that the learned function can be used to predict the output from future input. Depending on the type of output, supervised learning problems are either problems of regression or problems of classification. If the output takes a continuous range of values, it is a regression problem. Using Ohm's law as an example, a regression could be performed with voltage as input and current as an output. The regression would find the functional relationship between voltage and current to be R {\displaystyle R} , such that V = I R {\displaystyle V=IR} Classification problems are those for which the output will be an element from a discrete set of labels. Classification is very common for machine learning applications. In facial recognition, for instance, a picture of a person's face would be the input, and the output label would be that person's name. The input would be represented by a large multidimensional vector whose elements represent pixels in the picture. After learning a function based on the training set data, that function is validated on a test set of data, data that did not appear in the training set. == Formal description == Take X {\displaystyle X} to be the vector space of all possible inputs, and Y {\displaystyle Y} to be the vector space of all possible outputs. Statistical learning theory takes the perspective that there is some unknown probability distribution over the product space Z = X × Y {\displaystyle Z=X\times Y} , i.e. there exists some unknown p ( z ) = p ( x , y ) {\displaystyle p(z)=p(\mathbf {x} ,y)} . The training set is made up of n {\displaystyle n} samples from this probability distribution, and is notated S = { ( x 1 , y 1 ) , … , ( x n , y n ) } = { z 1 , … , z n } {\displaystyle S=\{(\mathbf {x} _{1},y_{1}),\dots ,(\mathbf {x} _{n},y_{n})\}=\{\mathbf {z} _{1},\dots ,\mathbf {z} _{n}\}} Every x i {\displaystyle \mathbf {x} _{i}} is an input vector from the training data, and y i {\displaystyle y_{i}} is the output that corresponds to it. In this formalism, the inference problem consists of finding a function f : X → Y {\displaystyle f:X\to Y} such that f ( x ) ∼ y {\displaystyle f(\mathbf {x} )\sim y} . Let H {\displaystyle {\mathcal {H}}} be a space of functions f : X → Y {\displaystyle f:X\to Y} called the hypothesis space. The hypothesis space is the space of functions the algorithm will search through. Let V ( f ( x ) , y ) {\displaystyle V(f(\mathbf {x} ),y)} be the loss function, a metric for the difference between the predicted value f ( x ) {\displaystyle f(\mathbf {x} )} and the actual value y {\displaystyle y} . The expected risk is defined to be I [ f ] = ∫ X × Y V ( f ( x ) , y ) p ( x , y ) d x d y {\displaystyle I[f]=\int _{X\times Y}V(f(\mathbf {x} ),y)\,p(\mathbf {x} ,y)\,d\mathbf {x} \,dy} The target function, the best possible function f {\displaystyle f} that can be chosen, is given by the f {\displaystyle f} that satisfies f = argmin h ∈ H ⁡ I [ h ] {\displaystyle f=\mathop {\operatorname {argmin} } _{h\in {\mathcal {H}}}I[h]} Because the probability distribution p ( x , y ) {\displaystyle p(\mathbf {x} ,y)} is unknown, a proxy measure for the expected risk must be used. This measure is based on the training set, a sample from this unknown probability distribution. It is called the empirical risk I S [ f ] = 1 n ∑ i = 1 n V ( f ( x i ) , y i ) {\displaystyle I_{S}[f]={\frac {1}{n}}\sum _{i=1}^{n}V(f(\mathbf {x} _{i}),y_{i})} A learning algorithm that chooses the function f S {\displaystyle f_{S}} that minimizes the empirical risk is called empirical risk minimization. == Loss functions == The choice of loss function is a determining factor on the function f S {\displaystyle f_{S}} that will be chosen by the learning algorithm. The loss function also affects the convergence rate for an algorithm. It is important for the loss function to be convex. Different loss functions are used depending on whether the problem is one of regression or one of classification. === Regression === The most common loss function for regression is the square loss function (also known as the L2-norm). This familiar loss function is used in Ordinary Least Squares regression. The form is: V ( f ( x ) , y ) = ( y − f ( x ) ) 2 {\displaystyle V(f(\mathbf {x} ),y)=(y-f(\mathbf {x} ))^{2}} The absolute value loss (also known as the L1-norm) is also sometimes used: V ( f ( x ) , y ) = | y − f ( x ) | {\displaystyle V(f(\mathbf {x} ),y)=|y-f(\mathbf {x} )|} === Classification === In some sense the 0-1 indicator function is the most natural loss function for classification. It takes the value 0 if the predicted output is the same as the actual output, and it takes the value 1 if the predicted output is different from the actual output. For binary classification with Y = { − 1 , 1 } {\displaystyle Y=\{-1,1\}} , this is: V ( f ( x ) , y ) = θ ( − y f ( x ) ) {\displaystyle V(f(\mathbf {x} ),y)=\theta (-yf(\mathbf {x} ))} where θ {\displaystyle \theta } is the Heaviside step function. == Regularization == In machine learning problems, a major problem that arises is that of overfitting. Because learning is a prediction problem, the goal is not to find a function that most closely fits the (previously observed) data, but to find one that will most accurately predict output from future input. Empirical risk minimization runs this risk of overfitting: finding a function that matches the data exactly but does not predict future output well. Overfitting is symptomatic of unstable solutions; a small perturbation in the training set data would cause a large variation in the learned function. It can be shown that if the stability for the solution can be guaranteed, generalization and consistency are guaranteed as well. Regularization can solve the overfitting problem and give the problem stability. Regularization can be accomplished by restricting the hypothesis space H {\displaystyle {\mathcal {H}}} . A common example would be restricting H {\displaystyle {\mathcal {H}}} to linear functions: this can be seen as a reduction to the standard problem of linear regression. H {\displaystyle {\mathcal {H}}} could also be restricted to polynomial of degree p {\displaystyle p} , exponentials, or bounded functions on L1. Restriction of the hypothesis space avoids overfitting because the form of the potential functions are limited, and so does not allow for the choice of a function that gives empirical risk arbitrarily close to zero. One example of regularization is Tikhonov regularization. This consists of minimizing 1 n ∑ i = 1 n V ( f ( x i ) , y i ) + γ ‖ f ‖ H 2 {\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}V(f(\mathbf {x} _{i}),y_{i})+\gamma \left\|f\right\|_{\mathcal {H}}^{2}} where γ {\displaystyle \gamma } is a fixed and positive parameter, the regularization parameter. Tikhonov regularization ensures existence, uniqueness, and stability of the solution. == Bounding empirical risk == Consider a binary classifier f : X → { 0 , 1 } {\displaystyle f:{\mathcal {X}}\to \{0,1\}} . We can apply Hoeffding's inequality to bound the probability that the empirical risk deviates from the true risk to be a Sub-Gaussian distribution. P ( | R ^ ( f ) − R ( f ) | ≥ ϵ ) ≤ 2 e − 2 n ϵ 2 {\displaystyle \mathbb {P} (|{\hat {R}}(f)-R(f)|\geq \epsilon )\leq 2e^{-2n\epsilon ^{2}}} But generally, when we do empirical risk minimization, we are not given a classifier; we must choose it. Therefore, a more useful result is to bound the probability of the supremum of the difference over the whole class. P ( sup f ∈ F | R ^ ( f ) − R ( f ) | ≥ ϵ ) ≤ 2 S ( F , n ) e − n ϵ 2 / 8 ≈ n d e − n ϵ 2 / 8 {\displaystyle \mathbb {P} {\bigg (}\sup _{f\in {\mathcal {F}}}|{\hat {R}}(f)-R(f)|\geq \epsilon {\bigg )}\leq 2S({\mathcal {F}},n)e^{-n\epsilon ^{2}/8}\approx n^{d}e^{-n\epsilon ^{2}/8}} where S ( F , n ) {\displaystyle S({\mathcal {F}},n)} is the shattering number and n {\displaystyle n} is the number of samples in your dataset. The exponential term comes from Hoeffding but there is an extra cost of taking the supremum over the whole cla

Google Cloud Dataflow

Google Cloud Dataflow is a fully managed service for executing Apache Beam pipelines within the Google Cloud Platform ecosystem. Dataflow provides a fully managed service for executing Apache Beam pipelines, offering features like autoscaling, dynamic work rebalancing, and a managed execution environment. Dataflow is suitable for large-scale, continuous data processing jobs, and is one of the major components of Google's big data architecture on the Google Cloud Platform. At its core, Dataflow's architecture is designed to abstract away infrastructure management, allowing developers to focus purely on the logic of their data processing tasks. When a pipeline written using the Apache Beam SDK is submitted, Dataflow translates this high-level definition into an optimized job graph. The service then provisions and manages a fleet of Google Compute Engine workers to execute this graph in a highly parallelized and fault-tolerant manner. This serverless approach, combined with intelligent autoscaling of both the number of workers (horizontal) and the resources per worker (vertical), ensures that jobs have the precise amount of computational power needed at any given time, optimizing both performance and cost. The service's deep integration with the Google Cloud ecosystem makes it a powerful tool for a variety of use cases beyond simple data movement. For real-time analytics, Dataflow can ingest unbounded streams of data from Cloud Pub/Sub, perform complex transformations, and load results into BigQuery for immediate querying. In machine learning workflows, it is commonly used to preprocess and transform massive datasets stored in Cloud Storage, preparing them for training models in Vertex AI. This versatility makes it the central processing engine for modern ETL (Extract, Transform, Load) operations, streaming analytics, and large-scale data preparation within the cloud. == History == Google Cloud Dataflow was announced in June, 2014 and released to the general public as an open beta in April, 2015. In January, 2016 Google donated the underlying SDK, the implementation of a local runner, and a set of IOs (data connectors) to access Google Cloud Platform data services to the Apache Software Foundation. The donated code formed the original basis for Apache Beam. In August 2022, there was an incident where user timers were broken for certain Dataflow streaming pipelines in multiple regions, which was later resolved. Throughout 2023 and 2024, there have been various other updates and incidents affecting Google Cloud Dataflow, as documented in the release notes and service health history. The donation of the Dataflow SDK to the Apache Software Foundation was a pivotal moment, establishing Apache Beam as a unified, open-source programming model for defining both batch and streaming data pipelines. This strategic move decoupled the pipeline definition from the execution engine. As a result, developers could write portable data processing logic that was not locked into Google's ecosystem. A Beam pipeline can be executed on various runners, including Apache Flink, Apache Spark, and, of course, the highly optimized Google Cloud Dataflow service, providing flexibility and future-proofing data processing investments. == Features == Google Cloud Dataflow supports both batch and streaming data processing pipelines. It automatically handles resource provisioning, data sharding, and scaling according to workload, reducing manual configuration needed for large-scale data operations. == Use cases == Dataflow is used for ETL (Extract, Transform, Load) data pipelines, real-time analytics, and event stream processing for companies in industries such as finance, advertising, and IoT.

The Best Free AI Art Generator for Beginners

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Is an AI Image Generator Worth It in 2026?

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Iterative Viterbi decoding

Iterative Viterbi decoding is an algorithm that spots the subsequence S of an observation O = {o1, ..., on} having the highest average probability (i.e., probability scaled by the length of S) of being generated by a given hidden Markov model M with m states. The algorithm uses a modified Viterbi algorithm as an internal step. The scaled probability measure was first proposed by John S. Bridle. An early algorithm to solve this problem, sliding window, was proposed by Jay G. Wilpon et al., 1989, with constant cost T = mn2/2. A faster algorithm consists of an iteration of calls to the Viterbi algorithm, reestimating a filler score until convergence. == The algorithm == A basic (non-optimized) version, finding the sequence s with the smallest normalized distance from some subsequence of t is: // input is placed in observation s[1..n], template t[1..m], // and [[distance matrix]] d[1..n,1..m] // remaining elements in matrices are solely for internal computations (int, int, int) AverageSubmatchDistance(char s[0..(n+1)], char t[0..(m+1)], int d[1..n,0..(m+1)]) { // score, subsequence start, subsequence end declare int e, B, E t'[0] := t'[m+1] := s'[0] := s'[n+1] := 'e' e := random() do e' := e for i := 1 to n do d'[i,0] := d'[i,m+1] := e (e, B, E) := ViterbiDistance(s', t', d') e := e/(E-B+1) until (e == e') return (e, B, E) } The ViterbiDistance() procedure returns the tuple (e, B, E), i.e., the Viterbi score "e" for the match of t and the selected entry (B) and exit (E) points from it. "B" and "E" have to be recorded using a simple modification to Viterbi. A modification that can be applied to CYK tables, proposed by Antoine Rozenknop, consists in subtracting e from all elements of the initial matrix d.

Microsoft Azure

Microsoft Azure, sometimes stylized Azure, and formerly Windows Azure, is the cloud computing platform developed by Microsoft. It offers management, access and development of applications and services to individuals, companies, and governments through its global infrastructure. Microsoft Azure supports many programming languages, tools, and frameworks, including Microsoft-specific and third-party software and systems. Azure was first introduced at the Professional Developers Conference (PDC) in October 2008 under the codename "Project Red Dog". It was officially launched as Windows Azure in February 2010 and later renamed to Microsoft Azure on March 25, 2014. == Services == Microsoft Azure uses large-scale virtualization at Microsoft data centers worldwide and offers more than 600 services. Microsoft Azure offers a service level agreement (SLA) that guarantees 99.9% availability for applications and data hosted on its platform, subject to specific terms and conditions outlined in the SLA documentation. === Computer services === Virtual machines, infrastructure as a service (IaaS), allowing users to launch general-purpose Microsoft Windows and Linux virtual machines, software as a service (SaaS), as well as preconfigured machine images for popular software packages. Starting in 2022, these virtual machines are now powered by Ampere Cloud-native processors. Most users run Linux on Azure, some of the many Linux distributions offered, including Microsoft's own Linux-based Azure Sphere. App services, platform as a service (PaaS) environment, letting developers easily publish and manage websites. Azure Web Sites allows developers to build sites using ASP.NET, PHP, Node.js, Java, or Python, which can be deployed using FTP, Git, Mercurial, Azure DevOps, or uploaded through the user portal. This feature was announced in preview form in June 2012 at the Meet Microsoft Azure event. Customers can create websites in PHP, ASP.NET, Node.js, or Python, or select from several open-source applications from a gallery to deploy. This comprises one aspect of the platform as a service (PaaS) offerings for the Microsoft Azure Platform. It was renamed Web Apps in April 2015. Web Jobs are applications that can be deployed to an App Service environment to implement background processing that can be invoked on a schedule, on-demand, or run continuously. The Blob, Table, and Queue services can be used to communicate between Web Apps and Web Jobs and to provide state. Azure Kubernetes Service (AKS) provides the capability to deploy production-ready Kubernetes clusters in Azure. In July 2023, watermarking support on Azure Virtual Desktop was announced as an optional feature of Screen Capture to provide additional security against data leakage. === Identity === Entra ID connect is used to synchronize on-premises directories and enable SSO (Single Sign On). Entra ID B2C allows the use of consumer identity and access management in the cloud. Entra Domain Services is used to join Azure virtual machines to a domain without domain controllers. Azure information protection can be used to protect sensitive information. Entra ID External Identities is a set of capabilities that allow organizations to collaborate with external users, including customers and partners. On July 11, 2023, Microsoft announced the renaming of Azure AD to Microsoft Entra ID. The name change took place four days later. === Mobile services === Mobile Engagement collects real-time analytics that highlight users' behavior. It also provides push notifications to mobile devices. HockeyApp can be used to develop, distribute, and beta-test mobile apps. === Storage services === Storage Services provides REST and SDK APIs for storing and accessing data on the cloud. Table Service lets programs store structured text in partitioned collections of entities that are accessed by the partition key and primary key. Azure Table Service is a NoSQL non-relational database. Blob Service allows programs to store unstructured text and binary data as object storage blobs that can be accessed by an HTTP(S) path. Blob service also provides security mechanisms to control access to data. Queue Service lets programs communicate asynchronously by message using queues. File Service allows storing and access of data on the cloud using the REST APIs or the SMB protocol. === Communication services === Azure Communication Services offers an SDK for creating web and mobile communications applications that include SMS, video calling, VOIP and PSTN calling, and web-based chat. === Data management === Azure Data Explorer provides big data analytics and data-exploration capabilities. Azure Search provides text search and a subset of OData's structured filters using REST or SDK APIs. Cosmos DB is a NoSQL database service that implements a subset of the SQL SELECT statement on JSON documents. Azure Cache for Redis is a managed implementation of Redis. StorSimple manages storage tasks between on-premises devices and cloud storage. Azure SQL Database works to create, scale, and extend applications into the cloud using Microsoft SQL Server technology. It also integrates with Active Directory, Microsoft System Center, and Hadoop. Azure Synapse Analytics is a fully managed cloud data warehouse. Azure Data Factory is a data integration service that allows creation of data-driven workflows in the cloud for orchestrating and automating data movement and data transformation. Azure Data Lake is a scalable data storage and analytic service for big data analytics workloads that require developers to run massively parallel queries. Azure HDInsight is a big data-relevant service that deploys Hortonworks Hadoop on Microsoft Azure and supports the creation of Hadoop clusters using Linux with Ubuntu. Azure Stream Analytics is a Serverless scalable event-processing engine that enables users to develop and run real-time analytics on multiple streams of data from sources such as devices, sensors, websites, social media, and other applications. === Messaging === The Microsoft Azure Service Bus allows applications running on Azure premises or off-premises devices to communicate with Azure. This helps to build scalable and reliable applications in a service-oriented architecture (SOA). The Azure service bus supports four different types of communication mechanisms: Event Hubs, which provides event and telemetry ingress to the cloud at a massive scale, with low latency and high reliability. For example, an event hub can be used to track data from cell phones such as coordinating with a GPS in real time. Queues, which allows one-directional communication. A sender application would send the message to the service bus queue and a receiver would read from the queue. Though there can be multiple readers for the queue, only one would process a single message. Topics, which provides one-directional communication using a subscriber pattern. It is similar to a queue; however, each subscriber will receive a copy of the message sent to a Topic. Optionally, the subscriber can filter out messages based on specific criteria defined by the subscriber. Relays, which provides bi-directional communication. Unlike queues and topics, a relay does not store in-flight messages in its memory; instead, it just passes them on to the destination application. === Media services === A PaaS offering that can be used for encoding, content protection, streaming, or analytics. === CDN === Azure has a worldwide content delivery network (CDN) designed to efficiently deliver audio, video, applications, images, and other static files. It improves the performance of websites by caching static files closer to users, based on their geographic location. Users can manage the network using a REST-based HTTP API. Azure has 118 point-of-presence locations across 100 cities worldwide (also known as Edge locations) as of January 2023. === Developer === Application Insights Azure DevOps === Management === With Azure Automation, users can easily automate repetitive and time-consuming tasks, often prone to cloud or enterprise setting errors. They can accomplish it using runbooks or desired state configurations for process automation. Microsoft SMA === Azure AI === Microsoft Azure Machine Learning (Azure ML) provides tools and frameworks for developers to create their own machine learning and artificial intelligence (AI) services. Azure AI Services by Microsoft comprises prebuilt APIs, SDKs, and services developers can customize. These services encompass perceptual and cognitive intelligence features such as speech recognition, speaker recognition, neural speech synthesis, face recognition, computer vision, OCR/form understanding, natural language processing, machine translation, and business decision services. Many AI characteristics in Microsoft's products and services, namely Bing, Office, Teams, Xbox, and Windows, are driven by Azure AI Services. Microsoft Foundry (formerly known as Azure AI Studio)

How to Choose an AI Code-review Tool

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