Site reliability engineering (SRE) is a discipline in the field of software engineering and IT infrastructure support that monitors and improves the availability and performance of deployed software systems and large software services (which are expected to deliver reliable response times across events such as new software deployments, hardware failures, and cybersecurity attacks). There is typically a focus on automation and an infrastructure as code methodology. SRE uses elements of software engineering, IT infrastructure, web development, and operations to assist with reliability. It is similar to DevOps as they both aim to improve the reliability and availability of deployed software systems. == History == Site Reliability Engineering originated at Google with Benjamin Treynor Sloss, who founded SRE team in 2003. The concept expanded within the software development industry, leading various companies to employ site reliability engineers. By March 2016, Google had more than 1,000 site reliability engineers on staff. Dedicated SRE teams are common at larger web development companies. In middle-sized and smaller companies, DevOps teams sometimes perform SRE, as well. Organizations that have adopted the concept include Airbnb, Dropbox, IBM, LinkedIn, Netflix, and Wikimedia. == Definition == Site reliability engineers (SREs) are responsible for a combination of system availability, latency, performance, efficiency, change management, monitoring, emergency response, and capacity planning. SREs often have backgrounds in software engineering, systems engineering, and/or system administration. The focuses of SRE include automation, system design, and improvements to system resilience. SRE is considered a specific implementation of DevOps; focusing specifically on building reliable systems, whereas DevOps covers a broader scope of operations. Despite having different focuses, some companies have rebranded their operations teams to SRE teams. == Principles and practices == Common definitions of the practices include (but are not limited to): Automation of repetitive tasks for cost-effectiveness. Defining reliability goals to prevent endless effort. Design of systems with a goal to reduce risks to availability, latency, and efficiency. Observability, the ability to ask arbitrary questions about a system without having to know ahead of time what to ask. Common definitions of the principles include (but are not limited to): Toil management, the implementation of the first principle outlined above. Defining and measuring reliability goals—SLIs, SLOs, and error budgets. Non-Abstract Large Scale Systems Design (NALSD) with a focus on reliability. Designing for and implementing observability. Defining, testing, and running an incident management process. Capacity planning. Change and release management, including CI/CD. Chaos engineering. == Deployment == SRE teams collaborate with other departments within organizations to guide the implementation of the mentioned principles. Below is an overview of common practices: === Kitchen Sink === Kitchen Sink refers to the expansive and often unbounded scope of services and workflows that SRE teams oversee. Unlike traditional roles with clearly defined boundaries, SREs are tasked with various responsibilities, including system performance optimization, incident management, and automation. This approach allows SREs to address multiple challenges, ensuring that systems run efficiently and evolve in response to changing demands and complexities. === Infrastructure === Infrastructure SRE teams focus on maintaining and improving the reliability of systems that support other teams' workflows. While they sometimes collaborate with platform engineering teams, their primary responsibility is ensuring up-time, performance, and efficiency. Platform teams, on the other hand, primarily develop the software and systems used across the organization. While reliability is a goal for both, platform teams prioritize creating and maintaining the tools and services used by internal stakeholders, whereas Infrastructure SRE teams are tasked with ensuring those systems run smoothly and meet reliability standards. === Tools === SRE teams utilize a variety of tools with the aim of measuring, maintaining, and enhancing system reliability. These tools play a role in monitoring performance, identifying issues, and facilitating proactive maintenance. For instance, Nagios Core is commonly employed for system monitoring and alerting, while Prometheus (software) is frequently used for collecting and querying metrics in cloud-native environments. === Product or Application === SRE teams dedicated to specific products or applications are common in large organizations. These teams are responsible for ensuring the reliability, scalability, and performance of key services. In larger companies, it's typical to have multiple SRE teams, each focusing on different products or applications, ensuring that each area receives specialized attention to meet performance and availability targets. === Embedded === In an embedded model, individual SREs or small SRE pairs are integrated within software engineering teams. These SREs collaborate with developers, applying core SRE principles—such as automation, monitoring, and incident response—directly to the software development lifecycle. This approach aims to enhance reliability, performance, and collaboration between SREs and developers. === Consulting === Consulting SRE teams specialize in advising organizations on the implementation of SRE principles and practices. Typically composed of seasoned SREs with a history across various implementations, these teams provide insights and guidance for specific organizational needs. When working directly with clients, these SREs are often referred to as 'Customer Reliability Engineers.' In large organizations that have adopted SRE, a hybrid model is common. This model includes various implementations, such as multiple Product/Application SRE teams dedicated to addressing the specific reliability needs of different products. An Infrastructure SRE team may collaborate with a Platform engineering group to achieve shared reliability goals for a unified platform that supports all products and applications. == Industry == Since 2014, the USENIX organization has hosted the annual SREcon conference, bringing together site reliability engineers from various industries. This conference is a platform for professionals to share knowledge, explore effective practices, and discuss trends in site reliability engineering.
Convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions f {\displaystyle f} and g {\displaystyle g} that produces a third function f ∗ g {\displaystyle fg} , as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term convolution refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution f ∗ g {\displaystyle fg} differs from cross-correlation f ⋆ g {\displaystyle f\star g} only in that either f ( x ) {\displaystyle f(x)} or g ( x ) {\displaystyle g(x)} is reflected about the y-axis in convolution; thus it is a cross-correlation of g ( − x ) {\displaystyle g(-x)} and f ( x ) {\displaystyle f(x)} , or f ( − x ) {\displaystyle f(-x)} and g ( x ) {\displaystyle g(x)} . For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, computer vision and human vision, geophysics, engineering, physics, and differential equations. The convolution can be defined for functions on Euclidean space and other groups (as algebraic structures). For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing. Computing the inverse of the convolution operation is known as deconvolution. == Definition == The convolution of f {\displaystyle f} and g {\displaystyle g} is written f ∗ g {\displaystyle fg} , denoting the operator with the symbol ∗ {\displaystyle } . It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. As such, it is a particular kind of integral transform: ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .} An equivalent definition is (see commutativity): ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( t − τ ) g ( τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau .} While the symbol t {\displaystyle t} is used above, it need not represent the time domain. At each t {\displaystyle t} , the convolution formula can be described as the area under the function f ( τ ) {\displaystyle f(\tau )} weighted by the function g ( − τ ) {\displaystyle g(-\tau )} shifted by the amount t {\displaystyle t} . As t {\displaystyle t} changes, the weighting function g ( t − τ ) {\displaystyle g(t-\tau )} emphasizes different parts of the input function f ( τ ) {\displaystyle f(\tau )} ; If t {\displaystyle t} is a positive value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted along the τ {\displaystyle \tau } -axis toward the right (toward + ∞ {\displaystyle +\infty } ) by the amount of t {\displaystyle t} , while if t {\displaystyle t} is a negative value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted toward the left (toward − ∞ {\displaystyle -\infty } ) by the amount of | t | {\displaystyle |t|} . For functions f {\displaystyle f} , g {\displaystyle g} supported on only [ 0 , ∞ ) {\displaystyle [0,\infty )} (i.e., zero for negative arguments), the integration limits can be truncated, resulting in: ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ for f , g : [ 0 , ∞ ) → R . {\displaystyle (fg)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau \quad \ {\text{for }}f,g:[0,\infty )\to \mathbb {R} .} For the multi-dimensional formulation of convolution, see domain of definition (below). === Notation === A common engineering notational convention is: f ( t ) ∗ g ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ ⏟ ( f ∗ g ) ( t ) , {\displaystyle f(t)g(t)\mathrel {:=} \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(fg)(t)},} which has to be interpreted carefully to avoid confusion. For instance, f ( t ) ∗ g ( t − t 0 ) {\displaystyle f(t)g(t-t_{0})} is equivalent to ( f ∗ g ) ( t − t 0 ) {\displaystyle (fg)(t-t_{0})} , but f ( t − t 0 ) ∗ g ( t − t 0 ) {\displaystyle f(t-t_{0})g(t-t_{0})} is in fact equivalent to ( f ∗ g ) ( t − 2 t 0 ) {\displaystyle (fg)(t-2t_{0})} . === Relations with other transforms === Given two functions f ( t ) {\displaystyle f(t)} and g ( t ) {\displaystyle g(t)} with bilateral Laplace transforms (two-sided Laplace transform) F ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u {\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u} and G ( s ) = ∫ − ∞ ∞ e − s v g ( v ) d v {\displaystyle G(s)=\int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v} respectively, the convolution operation ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} can be defined as the inverse Laplace transform of the product of F ( s ) {\displaystyle F(s)} and G ( s ) {\displaystyle G(s)} . More precisely, F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u ⋅ ∫ − ∞ ∞ e − s v g ( v ) d v = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s ( u + v ) f ( u ) g ( v ) d u d v {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u\cdot \int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-s(u+v)}\ f(u)\ g(v)\ {\text{d}}u\ {\text{d}}v\end{aligned}}} Let t = u + v {\displaystyle t=u+v} , then F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s t f ( u ) g ( t − u ) d u d t = ∫ − ∞ ∞ e − s t ∫ − ∞ ∞ f ( u ) g ( t − u ) d u ⏟ ( f ∗ g ) ( t ) d t = ∫ − ∞ ∞ e − s t ( f ∗ g ) ( t ) d t . {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-st}\ f(u)\ g(t-u)\ {\text{d}}u\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}\underbrace {\int _{-\infty }^{\infty }f(u)\ g(t-u)\ {\text{d}}u} _{(fg)(t)}\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}(fg)(t)\ {\text{d}}t.\end{aligned}}} Note that F ( s ) ⋅ G ( s ) {\displaystyle F(s)\cdot G(s)} is the bilateral Laplace transform of ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} . A similar derivation can be done using the unilateral Laplace transform (one-sided Laplace transform). The convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms. == Visual explanation == == Historical developments == One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754. Also, an expression of the type: ∫ f ( u ) ⋅ g ( x − u ) d u {\displaystyle \int f(u)\cdot g(x-u)\,du} is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, which is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800. Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace, Jean-Baptiste Joseph Fourier, Siméon Denis Poisson, and others. The term itself did not come into wide use until the 1950s or 1960s. Prior to that it was sometimes known as Faltung (which means folding in German), composition product, superposition integral, and Carson's integral. Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses. The operation: ∫ 0 t φ ( s ) ψ ( t − s ) d s , 0 ≤ t < ∞ , {\displaystyle \int _{0}^{t}\varphi (s)\psi (t-s)\,ds,\quad 0\leq t<\infty ,} is a particular case of composition products considered by the Italian mathematician Vito Volterra in 1913. == Circular c
Magic Quadrant
Magic Quadrant (MQ) is a series of market research reports published by research and advisory firm Gartner that rely on proprietary qualitative data analysis methods to demonstrate market trends, such as direction, maturity, and participants. Their analyses are conducted for several specific technology industries and are updated every 1–2 years: once an updated report has been published, its predecessor is "retired". == Rating == Gartner rates vendors upon two criteria: completeness of vision and ability to execute. Completeness of vision – Reflects the vendor's innovation, and whether the vendor drives or follows the market. Ability to execute – Summarizes factors such as the vendor's financial viability, market responsiveness, product development, sales channels and customer base. The two component scores lead to a vendor position in one of four quadrants: === Leaders === Vendors in the "Leaders" quadrant have the highest composite scores for their completeness of vision and ability to execute. A vendor in the Leaders quadrant has the market share, credibility, and marketing & sales capabilities needed to drive the acceptance of new technologies. These vendors demonstrate a clear understanding of market needs, they are innovators and thought leaders, and they have well-articulated plans that customers and prospects can use when designing their infrastructures and strategies. In addition, they have a presence in the five major geographical regions, consistent financial performance, and broad platform support. === Challengers === Vendors in the "Challengers" quadrant have high scores mainly for their ability to execute. They both participate in the market and execute well enough to be a serious threat to vendors in the "Leaders" quadrant. They have strong products, as well as sufficiently credible market position and resources to sustain continued growth. Financial viability is not an issue for vendors in the "Challengers" quadrant, but they lack the size and influence of vendors in the "Leaders" quadrant due to their relative lack of vision. === Visionaries === Vendors in the "Visionaries" quadrant have high scores mainly for their completeness of vision. They deliver innovative products that address operationally or financially important end-user problems at a broad scale, but have not yet demonstrated the ability to capture market share or maintain sustainable levels of profitability. Visionary vendors are frequently privately held companies and acquisition targets for larger, established companies. The likelihood of acquisition often reduces the risks associated with installing their systems. === Niche Players === Vendors in the "Niche Players" quadrant have relatively low scores for both their ability to execute and their completeness of vision. They are often narrowly focused on specific market or vertical segments. This quadrant often also includes vendors that are adapting their existing products to enter the market under consideration, or larger vendors having difficulty developing and executing on their vision. == Gartner Critical Capabilities == Gartner Critical Capabilities complement Magic Quadrant analysis to offer deeper insight into the products and services offered by multiple vendors by a comparative analysis that scores competing products or services against a set of critical differentiators identified by Gartner. Gartner has periodically ended Magic Quadrant listings for IT Service Management, Web Content Management, and other industries as those markets have fully matured or other factors rendered the analytic framework inapplicable. == Criticism == The Magic Quadrant, and analysts in general, skew the market: according to research, by applying their methodologies to describe a market, they change that marketplace to fit their tools. Another criticism is that open source vendors are not considered sufficiently by analysts like Gartner, as has been published in an online discussion between a VP from Talend and a German Research VP from Gartner. On May 29, 2009 (2009-05-29), software vendor ZL Technologies filed a federal lawsuit against Gartner that challenged the "legitimacy" of Gartner's Magic Quadrant rating system. Gartner filed a motion to dismiss by claiming First Amendment protection since it contends that its MQ reports contain "pure opinion", which legally means opinions that are not based on fact. The court threw out the ZL case because it lacked a specific complaint. The decision was upheld on appeal.
Generalized distributive law
The generalized distributive law (GDL) is a generalization of the distributive property which gives rise to a general message passing algorithm. It is a synthesis of the work of many authors in the information theory, digital communications, signal processing, statistics, and artificial intelligence communities. The law and algorithm were introduced in a semi-tutorial by Srinivas M. Aji and Robert J. McEliece with the same title. == Introduction == "The distributive law in mathematics is the law relating the operations of multiplication and addition, stated symbolically, a ∗ ( b + c ) = a ∗ b + a ∗ c {\displaystyle a(b+c)=ab+ac} ; that is, the monomial factor a {\displaystyle a} is distributed, or separately applied, to each term of the binomial factor b + c {\displaystyle b+c} , resulting in the product a ∗ b + a ∗ c {\displaystyle ab+ac} " – Britannica. As it can be observed from the definition, application of distributive law to an arithmetic expression reduces the number of operations in it. In the previous example the total number of operations reduced from three (two multiplications and an addition in a ∗ b + a ∗ c {\displaystyle ab+ac} ) to two (one multiplication and one addition in a ∗ ( b + c ) {\displaystyle a(b+c)} ). Generalization of distributive law leads to a large family of fast algorithms. This includes the FFT and Viterbi algorithm. This is explained in a more formal way in the example below: α ( a , b ) = d e f ∑ c , d , e ∈ A f ( a , c , b ) g ( a , d , e ) {\displaystyle \alpha (a,\,b){\stackrel {\mathrm {def} }{=}}\displaystyle \sum \limits _{c,d,e\in A}f(a,\,c,\,b)\,g(a,\,d,\,e)} where f ( ⋅ ) {\displaystyle f(\cdot )} and g ( ⋅ ) {\displaystyle g(\cdot )} are real-valued functions, a , b , c , d , e ∈ A {\displaystyle a,b,c,d,e\in A} and | A | = q {\displaystyle |A|=q} (say) Here we are "marginalizing out" the independent variables ( c {\displaystyle c} , d {\displaystyle d} , and e {\displaystyle e} ) to obtain the result. When we are calculating the computational complexity, we can see that for each q 2 {\displaystyle q^{2}} pairs of ( a , b ) {\displaystyle (a,b)} , there are q 3 {\displaystyle q^{3}} terms due to the triplet ( c , d , e ) {\displaystyle (c,d,e)} which needs to take part in the evaluation of α ( a , b ) {\displaystyle \alpha (a,\,b)} with each step having one addition and one multiplication. Therefore, the total number of computations needed is 2 ⋅ q 2 ⋅ q 3 = 2 q 5 {\displaystyle 2\cdot q^{2}\cdot q^{3}=2q^{5}} . Hence the asymptotic complexity of the above function is O ( n 5 ) {\displaystyle O(n^{5})} . If we apply the distributive law to the RHS of the equation, we get the following: α ( a , b ) = d e f ∑ c ∈ A f ( a , c , b ) ⋅ ∑ d , e ∈ A g ( a , d , e ) {\displaystyle \alpha (a,\,b){\stackrel {\mathrm {def} }{=}}\displaystyle \sum \limits _{c\in A}f(a,\,c,\,b)\cdot \sum _{d,\,e\in A}g(a,\,d,\,e)} This implies that α ( a , b ) {\displaystyle \alpha (a,\,b)} can be described as a product α 1 ( a , b ) ⋅ α 2 ( a ) {\displaystyle \alpha _{1}(a,\,b)\cdot \alpha _{2}(a)} where α 1 ( a , b ) = d e f ∑ c ∈ A f ( a , c , b ) {\displaystyle \alpha _{1}(a,b){\stackrel {\mathrm {def} }{=}}\displaystyle \sum \limits _{c\in A}f(a,\,c,\,b)} and α 2 ( a ) = d e f ∑ d , e ∈ A g ( a , d , e ) {\displaystyle \alpha _{2}(a){\stackrel {\mathrm {def} }{=}}\displaystyle \sum \limits _{d,\,e\in A}g(a,\,d,\,e)} Now, when we are calculating the computational complexity, we can see that there are q 3 {\displaystyle q^{3}} additions in α 1 ( a , b ) {\displaystyle \alpha _{1}(a,\,b)} and α 2 ( a ) {\displaystyle \alpha _{2}(a)} each and there are q 2 {\displaystyle q^{2}} multiplications when we are using the product α 1 ( a , b ) ⋅ α 2 ( a ) {\displaystyle \alpha _{1}(a,\,b)\cdot \alpha _{2}(a)} to evaluate α ( a , b ) {\displaystyle \alpha (a,\,b)} . Therefore, the total number of computations needed is q 3 + q 3 + q 2 = 2 q 3 + q 2 {\displaystyle q^{3}+q^{3}+q^{2}=2q^{3}+q^{2}} . Hence the asymptotic complexity of calculating α ( a , b ) {\displaystyle \alpha (a,b)} reduces to O ( n 3 ) {\displaystyle O(n^{3})} from O ( n 5 ) {\displaystyle O(n^{5})} . This shows by an example that applying distributive law reduces the computational complexity which is one of the good features of a "fast algorithm". == History == Some of the problems that used distributive law to solve can be grouped as follows: Decoding algorithms: A GDL like algorithm was used by Gallager's for decoding low density parity-check codes. Based on Gallager's work Tanner introduced the Tanner graph and expressed Gallagers work in message passing form. The tanners graph also helped explain the Viterbi algorithm. It is observed by Forney that Viterbi's maximum likelihood decoding of convolutional codes also used algorithms of GDL-like generality. Forward–backward algorithm: The forward backward algorithm helped as an algorithm for tracking the states in the Markov chain. And this also was used the algorithm of GDL like generality Artificial intelligence: The notion of junction trees has been used to solve many problems in AI. Also the concept of bucket elimination used many of the concepts. == The MPF problem == MPF or marginalize a product function is a general computational problem which as special case includes many classical problems such as computation of discrete Hadamard transform, maximum likelihood decoding of a linear code over a memory-less channel, and matrix chain multiplication. The power of the GDL lies in the fact that it applies to situations in which additions and multiplications are generalized. A commutative semiring is a good framework for explaining this behavior. It is defined over a set K {\displaystyle K} with operators " + {\displaystyle +} " and " . {\displaystyle .} " where ( K , + ) {\displaystyle (K,\,+)} and ( K , . ) {\displaystyle (K,\,.)} are a commutative monoids and the distributive law holds. Let p 1 , … , p n {\displaystyle p_{1},\ldots ,p_{n}} be variables such that p 1 ∈ A 1 , … , p n ∈ A n {\displaystyle p_{1}\in A_{1},\ldots ,p_{n}\in A_{n}} where A {\displaystyle A} is a finite set and | A i | = q i {\displaystyle |A_{i}|=q_{i}} . Here i = 1 , … , n {\displaystyle i=1,\ldots ,n} . If S = { i 1 , … , i r } {\displaystyle S=\{i_{1},\ldots ,i_{r}\}} and S ⊂ { 1 , … , n } {\displaystyle S\,\subset \{1,\ldots ,n\}} , let A S = A i 1 × ⋯ × A i r {\displaystyle A_{S}=A_{i_{1}}\times \cdots \times A_{i_{r}}} , p S = ( p i 1 , … , p i r ) {\displaystyle p_{S}=(p_{i_{1}},\ldots ,p_{i_{r}})} , q S = | A S | {\displaystyle q_{S}=|A_{S}|} , A = A 1 × ⋯ × A n {\displaystyle \mathbf {A} =A_{1}\times \cdots \times A_{n}} , and p = { p 1 , … , p n } {\displaystyle \mathbf {p} =\{p_{1},\ldots ,p_{n}\}} Let S = { S j } j = 1 M {\displaystyle S=\{S_{j}\}_{j=1}^{M}} where S j ⊂ { 1 , . . . , n } {\displaystyle S_{j}\subset \{1,...\,,n\}} . Suppose a function is defined as α i : A S i → R {\displaystyle \alpha _{i}:A_{S_{i}}\rightarrow R} , where R {\displaystyle R} is a commutative semiring. Also, p S i {\displaystyle p_{S_{i}}} are named the local domains and α i {\displaystyle \alpha _{i}} as the local kernels. Now the global kernel β : A → R {\displaystyle \beta :\mathbf {A} \rightarrow R} is defined as: β ( p 1 , . . . , p n ) = ∏ i = 1 M α ( p S i ) {\displaystyle \beta (p_{1},...\,,p_{n})=\prod _{i=1}^{M}\alpha (p_{S_{i}})} Definition of MPF problem: For one or more indices i = 1 , . . . , M {\displaystyle i=1,...\,,M} , compute a table of the values of S i {\displaystyle S_{i}} -marginalization of the global kernel β {\displaystyle \beta } , which is the function β i : A S i → R {\displaystyle \beta _{i}:A_{S_{i}}\rightarrow R} defined as β i ( p S i ) = ∑ p S i c ∈ A S i c β ( p ) {\displaystyle \beta _{i}(p_{S_{i}})\,=\displaystyle \sum \limits _{p_{S_{i}^{c}}\in A_{S_{i}^{c}}}\beta (p)} Here S i c {\displaystyle S_{i}^{c}} is the complement of S i {\displaystyle S_{i}} with respect to { 1 , . . . , n } {\displaystyle \mathbf {\{} 1,...\,,n\}} and the β i ( p S i ) {\displaystyle \beta _{i}(p_{S_{i}})} is called the i t h {\displaystyle i^{th}} objective function, or the objective function at S i {\displaystyle S_{i}} . It can observed that the computation of the i t h {\displaystyle i^{th}} objective function in the obvious way needs M q 1 q 2 q 3 ⋯ q n {\displaystyle Mq_{1}q_{2}q_{3}\cdots q_{n}} operations. This is because there are q 1 q 2 ⋯ q n {\displaystyle q_{1}q_{2}\cdots q_{n}} additions and ( M − 1 ) q 1 q 2 . . . q n {\displaystyle (M-1)q_{1}q_{2}...q_{n}} multiplications needed in the computation of the i th {\displaystyle i^{\text{th}}} objective function. The GDL algorithm which is explained in the next section can reduce this computational complexity. The following is an example of the MPF problem. Let p 1 , p 2 , p 3 , p 4 , {\displaystyle p_{1},\,p_{2},\,p_{3},\,p_{4},} and p 5 {\displaystyle p_{5}} be variables such that p 1 ∈ A 1 , p 2 ∈ A 2 , p 3 ∈ A 3 , p 4 ∈ A 4 , {\displaystyle p_{1}\in
Information
Information is an abstract concept that refers to something which has the power to inform. At the most fundamental level, it pertains to the interpretation (perhaps formally) of that which may be sensed, or their abstractions. Any natural process that is not completely random and any observable pattern in any medium can be said to convey some amount of information. Whereas digital signals and other data use discrete signs to convey information, other phenomena and artifacts such as analogue signals, poems, pictures, music or other sounds, and currents convey information in a more continuous form. Information is not knowledge itself, but the meaning that may be derived from a representation through interpretation. The concept of information is relevant to and connected with various concepts, including constraint, communication, control, data, form, education, knowledge, meaning, understanding, mental stimuli, pattern, perception, proposition, representation, and entropy. Information is often processed iteratively: Data available at one step are processed into information to be interpreted and processed at the next step. For example, in written text each symbol or letter conveys information relevant to the word it is part of, each word conveys information relevant to the phrase it is part of, each phrase conveys information relevant to the sentence it is part of, and so on until at the final step information is interpreted and becomes knowledge in a given domain. In a digital signal, bits may be interpreted into the symbols, letters, numbers, or structures that convey the information available at the next level up. The key characteristic of information is that it is subject to interpretation and processing. The derivation of information from a signal or message may be thought of as the resolution of ambiguity or uncertainty that arises during the interpretation of patterns within the signal or message. Information may be structured as data. Redundant data can be compressed up to an optimal size, which is the theoretical limit of compression. The information available through a collection of data may be derived by analysis. For example, a restaurant collects data from every customer order. That information may be analyzed to produce knowledge that is put to use when the business subsequently wants to identify the most popular or least popular dish. Information can be transmitted in time, via data storage, and space, via communication and telecommunication. Information is expressed either as the content of a message or through direct or indirect observation. That which is perceived can be construed as a message in its own right, and in that sense, all information is always conveyed as the content of a message. Information can be encoded into various forms for transmission and interpretation (for example, information may be encoded into a sequence of signs, or transmitted via a signal). It can also be encrypted for safe storage and communication. The uncertainty of an event is measured by its probability of occurrence. Uncertainty is proportional to the negative logarithm of the probability of occurrence. Information theory takes advantage of this by concluding that more uncertain events require more information to resolve their uncertainty. The bit is the standard unit of information. It is 'that which reduces uncertainty by half'. Other units such as the nat may be used. For example, the information encoded in one "fair" coin flip is log2(2/1) = 1 bit, and in two fair coin flips is log2(4/1) = 2 bits. A 2011 Science article estimates that 97% of technologically stored information was already in digital bits in 2007 and that the year 2002 was the beginning of the digital age for information storage (with digital storage capacity bypassing analogue for the first time). == Etymology and history of the concept == The English word "information" comes from Middle French enformacion/informacion/information 'a criminal investigation' and its etymon, Latin informatiō(n) 'conception, teaching, creation'. In English, "information" is an uncountable mass noun. References on "formation or molding of the mind or character, training, instruction, teaching" date from the 14th century in both English (according to Oxford English Dictionary) and other European languages. In the transition from Middle Ages to Modernity the use of the concept of information reflected a fundamental turn in epistemological basis – from "giving a (substantial) form to matter" to "communicating something to someone". Peters (1988, pp. 12–13) concludes: Information was readily deployed in empiricist psychology (though it played a less important role than other words such as impression or idea) because it seemed to describe the mechanics of sensation: objects in the world inform the senses. But sensation is entirely different from "form" – the one is sensual, the other intellectual; the one is subjective, the other objective. My sensation of things is fleeting, elusive, and idiosyncratic. For Hume, especially, sensory experience is a swirl of impressions cut off from any sure link to the real world... In any case, the empiricist problematic was how the mind is informed by sensations of the world. At first informed meant shaped by; later it came to mean received reports from. As its site of action drifted from cosmos to consciousness, the term's sense shifted from unities (Aristotle's forms) to units (of sensation). Information came less and less to refer to internal ordering or formation, since empiricism allowed for no preexisting intellectual forms outside of sensation itself. Instead, information came to refer to the fragmentary, fluctuating, haphazard stuff of sense. Information, like the early modern worldview in general, shifted from a divinely ordered cosmos to a system governed by the motion of corpuscles. Under the tutelage of empiricism, information gradually moved from structure to stuff, from form to substance, from intellectual order to sensory impulses. In the modern era, the most important influence on the concept of information is derived from the Information theory developed by Claude Shannon and others. This theory, however, reflects a fundamental contradiction. Northrup (1993) wrote: Thus, actually two conflicting metaphors are being used: The well-known metaphor of information as a quantity, like water in the water-pipe, is at work, but so is a second metaphor, that of information as a choice, a choice made by :an information provider, and a forced choice made by an :information receiver. Actually, the second metaphor implies that the information sent isn't necessarily equal to the information received, because any choice implies a comparison with a list of possibilities, i.e., a list of possible meanings. Here, meaning is involved, thus spoiling the idea of information as a pure "Ding an sich." Thus, much of the confusion regarding the concept of information seems to be related to the basic confusion of metaphors in Shannon's theory: is information an autonomous quantity, or is information always per SE information to an observer? Actually, I don't think that Shannon himself chose one of the two definitions. Logically speaking, his theory implied information as a subjective phenomenon. But this had so wide-ranging epistemological impacts that Shannon didn't seem to fully realize this logical fact. Consequently, he continued to use metaphors about information as if it were an objective substance. This is the basic, inherent contradiction in Shannon's information theory." (Northrup, 1993, p. 5). In their seminal book The Study of Information: Interdisciplinary Messages, Almach and Mansfield (1983) collected key views on the interdisciplinary controversy in computer science, artificial intelligence, library and information science, linguistics, psychology, and physics, as well as in the social sciences. Almach (1983, p. 660) himself disagrees with the use of the concept of information in the context of signal transmission, the basic senses of information in his view all referring "to telling something or to the something that is being told. Information is addressed to human minds and is received by human minds." All other senses, including its use with regard to nonhuman organisms as well to society as a whole, are, according to Machlup, metaphoric and, as in the case of cybernetics, anthropomorphic. Hjørland (2007) describes the fundamental difference between objective and subjective views of information and argues that the subjective view has been supported by, among others, Bateson, Yovits, Span-Hansen, Brier, Buckland, Goguen, and Hjørland. Hjørland provided the following example: A stone on a field could contain different information for different people (or from one situation to another). It is not possible for information systems to map all the stone's possible information for every individual. Nor is any one mapping the one "true" mapping. But peop
Prompt engineering
Prompt engineering is the process of structuring natural language inputs (known as prompts) to produce specified outputs from a generative artificial intelligence (GenAI) model. Context engineering is the related area of software engineering that focuses on the management of non-prompt contexts supplied to the GenAI model, such as metadata, API tools, and tokens. It can also be defined as the practice of designing and refining input instructions given to a generative AI model to produce more accurate, relevant, or useful outputs. Effective prompt engineering involves understanding how a model interprets language, and may include techniques such as few-shot prompting, chain-of-thought prompting, and role assignment. It is increasingly considered a skill for working with large language models (LLMs) in both research and professional contexts. During the 2020s AI boom, prompt engineering became regarded as a business capability across corporations and industries. Employees with the title prompt engineer were hired to create prompts that would increase productivity and efficacy, although the individual title has since lost traction amid AI models that produce better prompts than humans and corporate training in prompting for general employees. Common prompting techniques include multi-shot, chain-of-thought, and tree-of-thought prompting, as well as the use of assigning roles to the model. Automated prompt generation methods, such as retrieval-augmented generation (RAG), provide for greater accuracy and a wider scope of functions for prompt engineers. Prompt injection is a type of cybersecurity attack that targets machine learning models through malicious prompts. == Terminology == The Oxford English Dictionary defines prompt engineering as "The action or process of formulating and refining prompts for an artificial intelligence program, algorithm, etc., in order to optimize its output or to achieve a desired outcome; the discipline or profession concerned with this." In 2023, prompt ("an instruction given to an artificial intelligence program, algorithm, etc., which determines or influences the content it generates") was the runner-up to Oxford's word of the year. === Prompt === A prompt is some natural language text that describes and prescribes the task that an artificial intelligence (AI) should perform. A prompt for a text-to-text language model can be a query, a command, or a longer statement referencing context, instructions, and conversation history. The process of prompt engineering may involve designing clear queries, refining wording, providing relevant context, specifying the style of output, and assigning a character for the AI to mimic in order to guide the model toward more accurate, useful, and consistent responses. When communicating with a text-to-image or a text-to-audio model, a typical prompt contains a description of a desired output such as "a high-quality photo of an astronaut riding a horse" or "Lo-fi slow BPM electro chill with organic samples". Prompt engineering may be applied to text-to-image models to achieve a desired subject, style, layout, lighting, and aesthetic. === Techniques === Common terms used to describe various specific prompt engineering techniques include chain-of-thought, tree-of-thought, and retrieval-augmented generation (RAG). A 2024 survey of the field identified over 50 distinct text-based prompting techniques, 40 multimodal variants, and a vocabulary of 33 terms used across prompting research, highlighting a present lack of standardised terminology for prompt engineering. Vibe coding is an AI-assisted software development method where a user prompts an LLM with a description of what they want and lets it generate or edit the code. In 2025, "vibe coding" was the Collins Dictionary word of the year. === Context engineering === Context engineering is a related process that focuses on the context elements that accompany user prompts, which include system instructions, retrieved knowledge, tool definitions, conversation summaries, and task metadata. Context engineering is performed to improve reliability, provenance and token efficiency in production LLM systems. The concept emphasises operational practices such as token budgeting, provenance tags, versioning of context artifacts, observability (logging which context was supplied), and context regression tests to ensure that changes to supplied context do not silently alter system behaviour. == Rationale == Research has found that the performance of large language models (LLMs) is highly sensitive to choices such as the ordering of examples, the quality of demonstration labels, and even small variations in phrasing. In some cases, reordering examples in a prompt produced accuracy shifts of more than 40 percent. === In-context learning === A model's ability to temporarily learn from prompts is known as in-context learning. In-context learning is an emergent ability of large language models. It is an emergent property of model scale, meaning that breaks in scaling laws occur, leading to its efficacy increasing at a different rate in larger models than in smaller models. Unlike training and fine-tuning, which produce lasting changes, in-context learning is temporary. Training models to perform in-context learning can be viewed as a form of meta-learning, or "learning to learn". === Prompting to estimate model sensitivity === Research consistently demonstrates that LLMs are highly sensitive to subtle variations in prompt formatting, structure, and linguistic properties. Some studies have shown up to 76 accuracy points across formatting changes in few-shot settings. Linguistic features significantly influence prompt effectiveness—such as morphology, syntax, and lexico-semantic changes—which meaningfully enhance task performance across a variety of tasks. Clausal syntax, for example, improves consistency and reduces uncertainty in knowledge retrieval. This sensitivity persists even with larger model sizes, additional few-shot examples, or instruction tuning. To address sensitivity of models and make them more robust, several evaluative methods have been proposed. FormatSpread facilitates systematic analysis by evaluating a range of plausible prompt formats, offering a more comprehensive performance interval. Similarly, PromptEval estimates performance distributions across diverse prompts, enabling robust metrics such as performance quantiles and accurate evaluations under constrained budgets. == Prompting techniques == === Multi-shot === A prompt may include a few examples for a model to learn from in context, an approach called few-shot learning. For example, the prompt may ask the model to complete "maison → house, chat → cat, chien →", with the expected response being dog. === Chain-of-thought === Chain-of-thought (CoT) prompting is a technique that allows large language models (LLMs) to solve a problem as a series of intermediate steps before giving a final answer. In 2022, Google Brain reported that chain-of-thought prompting improves reasoning ability by inducing the model to answer a multi-step problem with steps of reasoning that mimic a train of thought. Chain-of-thought techniques were developed to help LLMs handle multi-step reasoning tasks, such as arithmetic or commonsense reasoning questions. When applied to PaLM, a 540 billion parameter language model, according to Google, CoT prompting significantly aided the model, allowing it to perform comparably with task-specific fine-tuned models on several tasks, achieving state-of-the-art results at the time on the GSM8K mathematical reasoning benchmark. It is possible to fine-tune models on CoT reasoning datasets to enhance this capability further and stimulate better interpretability. As originally proposed by Google, each CoT prompt is accompanied by a set of input/output examples—called exemplars—to demonstrate the desired model output, making it a few-shot prompting technique. However, according to a later paper from researchers at Google and the University of Tokyo, simply appending the words "Let's think step-by-step" was also effective, which allowed for CoT to be employed as a zero-shot technique. ==== Self-consistency ==== Self-consistency performs several chain-of-thought rollouts, then selects the most commonly reached conclusion out of all the rollouts. === Tree-of-thought === Tree-of-thought prompting generalizes chain-of-thought by generating multiple lines of reasoning in parallel, with the ability to backtrack or explore other paths. It can use tree search algorithms like breadth-first, depth-first, or beam. === Text-to-image prompting === In 2022, text-to-image models like DALL-E 2, Stable Diffusion, and Midjourney were released to the public. These models take text prompts as input and use them to generate images. Early text-to-image models typically do not understand negation, grammar and sentence structure in the same way as large language models, and may thus requi
Algorithm IMED
In multi-armed bandit problems, IMED (for Indexed Minimum Empirical Divergence) is an algorithm developed in 2015 by Junya Honda and Akimichi Takemura. It is the first algorithm proved to be asymptotically optimal respect to the problem-dependant Lai–Robbins lower bound for distributions in ( − ∞ , 1 ] {\displaystyle (-\infty ,1]} . == Multi-armed bandit problem == The Multi-armed bandit problem is a sequential game where one player has to choose at each turn between K {\displaystyle K} actions (arms). Behind every arm a {\displaystyle a} there is an unknown distribution ν a {\displaystyle \nu _{a}} that lies in a set D {\displaystyle {\mathcal {D}}} known by the player (for example, D {\displaystyle {\mathcal {D}}} can be the set of Gaussian distributions or Bernoulli distributions). At each turn t {\displaystyle t} the player chooses (pulls) an arm a t {\displaystyle a_{t}} , he then gets an observation X t {\displaystyle X_{t}} of the distribution ν a t {\displaystyle \nu _{a_{t}}} . === Regret minimization === The goal is to minimize the regret at time T {\displaystyle T} that is defined as R T := ∑ a = 1 K Δ a E [ N a ( T ) ] {\displaystyle R_{T}:=\sum _{a=1}^{K}\Delta _{a}\mathbb {E} [N_{a}(T)]} where μ a := E [ ν a ] {\displaystyle \mu _{a}:=\mathbb {E} [\nu _{a}]} is the mean of arm a {\displaystyle a} μ ∗ := max a μ a {\displaystyle \mu ^{}:=\max _{a}\mu _{a}} is the highest mean Δ a := μ ∗ − μ a {\displaystyle \Delta _{a}:=\mu ^{}-\mu _{a}} N a ( t ) {\displaystyle N_{a}(t)} is the number of pulls of arm a {\displaystyle a} up to turn t {\displaystyle t} The player has to find an algorithm that chooses at each turn t {\displaystyle t} which arm to pull based on the previous actions and observations ( a s , X s ) s < t {\displaystyle (a_{s},X_{s})_{s