Color gradient

In color science, a color gradient (also known as a color ramp or a color progression) specifies a range of position-dependent colors, usually used to fill a region. In assigning colors to a set of values, a gradient is a continuous colormap, a type of color scheme. In computer graphics, the term swatch has come to mean a palette of active colors. == Definitions == Color gradient is a set of colors arranged in a linear order (ordered) A continuous colormap is a curve through a colorspace === Strict definition === A colormap is a function which associate a real value r with point c in color space C {\displaystyle C} f : [ r m i n , r m a x ] ⊂ R → C {\displaystyle f:[r_{min},r_{max}]\subset \mathbf {R} \to C} which is defined by: a colorspace C an increasing sequence of sampling points r 0 < . . . < r m ∈ [ r m i n , r m a x ] {\displaystyle r_{0}<...

The Drivers Cooperative

The Drivers Cooperative or Co-Op Ride is an American ridesharing company and mobile app that is a workers cooperative, owned collectively by the drivers. The cooperative launched in May 2021 in New York City, with the first 2,500 drivers issued their ownership certificates in a media event. The cooperative was co-founded by Grenadan immigrant and for hire vehicle driver Ken Lewis, labor organizer Erik Forman, and former Uber executive Alissa Orlando. Mohammad Hossen is the first member of the drivers' advisory board, which they plan to expand democratically as more drivers are onboarded. Other staff include software and industry veterans and in addition to co-founder Lewis, there are other drivers in management roles such as ex-driver and organizer David Alexis. The Co-Op Ride app is on the iOS and Android platforms and is built on Google Maps, Stripe, and Waze. By July, the app had been downloaded by 30,000 users and the number of drivers increased to 3,400, and by August there were 40,000 users. The cooperative is owned by the drivers themselves, and takes 15% from each ride for business overhead costs, as opposed to the 25% to 40% ride hail apps like Uber or Lyft take per ride. While being ultimately owned by the driver members, not by investors, the cooperative began with seed money from the Minnesota-based Community Development Financial Institution Shared Capital Cooperative, the local Lower East Side People's Federal Credit Union, and welcomed individual donations via crowdfunding in the form of revenue sharing debt on Wefunder. Each driver is a member of the cooperative and owns one share of the company and one vote in business and leadership decisions. In addition to a larger percentage of the fees per ride driven, each driver as a part-owner will also receive a share of the company's profits after loans and other expenses are paid, in the form of weighted dividends. The drivers use their own cars. The cooperative vets its owner-members further than what is already performed by the New York City Taxi and Limousine Commission (TLC), and gives a fixed price when a car is ordered and does not engage in surge pricing. The TLC imposed a minimum payrate for mobile app ridesharing companies operating in New York city in 2018. In 2021 that is $1.26 per mile which Uber and Lyft do not pay above; the cooperative pays a minimum mileage of $1.64. The cooperative intends to be able to set aside 10% of profits to community foundations and other non-profits and community organizations. The cooperative has engaged in advocacy around a policy agenda voted on by its members. Legislation to achieve this policy goal was introduced by State Senator Julia Salazar and Assemblymember Jessica González-Rojas, with the support of a coalition led by The Drivers Cooperative, United Auto Workers Region 9 and 9A, Sunrise Movement, New York Lawyers for the Public Interest, and New York Communities for Change.

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

CrewAI

CrewAI is an open-source software framework and platform for building AI agents and multi-agent systems. Written primarily in Python, it is used to define artificial-intelligence agents, assign tasks to them, and coordinate their work through agent teams and workflows. The framework is associated with CrewAI Inc., a startup developing enterprise tools for automating business workflows with large language model-based agents. == History == CrewAI was first released on the Python Package Index in December 2023. The project was created by João Moura and later developed by CrewAI Inc. and open-source contributors. In October 2024, TechCrunch reported that CrewAI had raised $18 million across seed and Series A funding rounds from investors including Boldstart Ventures, Craft Ventures, Earl Grey Capital, and Insight Partners. The report also stated that Andrew Ng and HubSpot co-founder Dharmesh Shah had invested in the company. SiliconANGLE described the company as the developer of an open-source framework for building artificial-intelligence agents and reported that the funding consisted of a seed round led by Boldstart Ventures and a Series A led by Insight Partners. By late 2024, CrewAI had introduced commercial enterprise products built on top of its open-source components. TechCrunch reported that the company's enterprise offering added access controls, analytics, support, and templates for workflow automation. == Features == CrewAI is designed around groups of agents, sometimes called "crews", that can be assigned roles, goals, and tasks. The framework supports agent collaboration, task delegation, tool use, memory, and knowledge sources for retrieval-augmented generation workflows. The project describes two main building blocks: "Crews", which are used for autonomous agent collaboration, and "Flows", which are used for more controlled event-driven workflows. The framework is independent of LangChain and is released under the MIT License. It can be installed as a Python package and is commonly used with external large language model APIs or local models, depending on the developer's configuration. == Business model == CrewAI combines an open-source framework with commercial enterprise products. Its enterprise products are intended for organizations that need to build, monitor, and manage agent-based automations with additional security, observability, and administrative controls.

GOLOG

GOLOG is a high-level logic programming language for the specification and execution of complex actions in dynamical domains. It is based on the situation calculus. It is a first-order logical language for reasoning about action and change. GOLOG was developed at the University of Toronto. == History == The concept of situation calculus on which the GOLOG programming language is based was first proposed by John McCarthy in 1963. == Description == A GOLOG interpreter automatically maintains a direct characterization of the dynamic world being modeled, on the basis of user supplied axioms about preconditions, effects of actions and the initial state of the world. This allows the application to reason about the condition of the world and consider the impacts of different potential actions before focusing on a specific action. Golog is a logic programming language and is very different from conventional programming languages. A procedural programming language like C defines the execution of statements in advance. The programmer creates a subroutine which consists of statements, and the computer executes each statement in a linear order. In contrast, fifth-generation programming languages like Golog work with an abstract model with which the interpreter can generate the sequence of actions. The source code defines the problem and it is up to the solver to find the next action. This approach can facilitate the management of complex problems from the domain of robotics. A Golog program defines the state space in which the agent is allowed to operate. A path in the symbolic domain is found with state space search. To speed up the process, Golog programs are realized as hierarchical task networks. Apart from the original Golog language, there are some extensions available. The ConGolog language provides concurrency and interrupts. Other dialects like IndiGolog and Readylog were created for real time applications in which sensor readings are updated on the fly. == Uses == Golog has been used to model the behavior of autonomous agents. In addition to a logic-based action formalism for describing the environment and the effects of basic actions, they enable the construction of complex actions using typical programming language constructs. It is also used for applications in high level control of robots and industrial processes, virtual agents, discrete event simulation etc. It can be also used to develop Belief Desire Intention-style agent systems. == Planning and scripting == In contrast to the Planning Domain Definition Language, Golog supports planning and scripting as well. Planning means that a goal state in the world model is defined, and the solver brings a logical system into this state. Behavior scripting implements reactive procedures, which are running as a computer program. For example, suppose the idea is to authoring a story. The user defines what should be true at the end of the plot. A solver gets started and applies possible actions to the current situation until the goal state is reached. The specification of a goal state and the possible actions are realized in the logical world model. In contrast, a hardwired reactive behavior doesn't need a solver but the action sequence is provided in a scripting language. The Golog interpreter, which is written in Prolog, executes the script and this will bring the story into the goal state.

Networked Help Desk

Networked Help Desk is an open standard initiative to provide a common API for sharing customer support tickets between separate instances of issue tracking, bug tracking, customer relationship management (CRM) and project management systems to improve customer service and reduce vendor lock-in. The initiative was created by Zendesk in June 2011 in collaboration with eight other founding member organizations including Atlassian, New Relic, OTRS, Pivotal Tracker, ServiceNow and SugarCRM. The first integration, between Zendesk and Atlassian's issue tracking product, Jira, was announced at the 2011 Atlassian Summit. By August 2011, 34 member companies had joined the initiative. A year after launching, over 50 organizations had joined. Within Zendesk instances this feature is branded as ticket sharing. == Basis == Support tools are generally built around a common paradigm that begins with a customer making a request or an incident report, these create a ticket. Each ticket has a progress status and is updated with annotations and attachments. These annotations and attachments may be visible to the customer (public), or only visible to analysts (private). Customers are notified of progress made on their ticket until it is complete. If the people necessary to complete a ticket are using separate support tools, additional overhead is introduced in maintaining the relevant information in the ticket in each tool while notifying the customer of progress made by each group in completing their ticket. For example, if a customer support issue is caused by a software bug and reported to a help desk using one system, and then the fix is documented by the developers in another, and analyzed in a customer relationship management tool, keeping the records in each system up-to-date and notifying the customer manually using a swivel chair approach is unnecessarily time-consuming and error-prone. If information is not transferred correctly, a customer may have to re-explain their problem each time their ticket is transferred. For systems with the Networked Help Desk API implemented, it is possible for several different applications related to a customer's support experience to synchronize data in one uniquely identified shared ticket. While many applications in these domains have implemented APIs that allow data to be imported, exported and modified, Network Help Desk provide a common standard for customer support information to automatically synchronize between several systems. Once implemented, two systems can quickly share tickets with just a configuration change as they both understand the same interface. Communication between two instances on a specific ticket occurs in three steps, an invitation agreement, sharing of ticket data and continued synchronization of tickets. The standard allows for "full delegation" (analysts in both systems each make public and private comments and synchronize status) as well as "partial delegation" where the instance receiving the ticket can only make private comments and status changes are not synchronized. Tickets may be shared with multiple instances. == Implementation list ==

Rademacher complexity

In computational learning theory (machine learning and theory of computation), Rademacher complexity, named after Hans Rademacher, measures richness of a class of sets with respect to a probability distribution. The concept can also be extended to real valued functions. == Definitions == === Rademacher complexity of a set === Given a set A ⊆ R m {\displaystyle A\subseteq \mathbb {R} ^{m}} , the Rademacher complexity of A is defined as follows: Rad ⁡ ( A ) := 1 m E σ [ sup a ∈ A ∑ i = 1 m σ i a i ] {\displaystyle \operatorname {Rad} (A):={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{a\in A}\sum _{i=1}^{m}\sigma _{i}a_{i}\right]} where σ 1 , σ 2 , … , σ m {\displaystyle \sigma _{1},\sigma _{2},\dots ,\sigma _{m}} are independent random variables drawn from the Rademacher distribution i.e. Pr ( σ i = + 1 ) = Pr ( σ i = − 1 ) = 1 / 2 {\displaystyle \Pr(\sigma _{i}=+1)=\Pr(\sigma _{i}=-1)=1/2} for i ∈ { 1 , 2 , … , m } {\displaystyle i\in \{1,2,\dots ,m\}} , and a = ( a 1 , … , a m ) ∈ A {\displaystyle a=(a_{1},\ldots ,a_{m})\in A} . Some authors take the absolute value of the sum before taking the supremum, but if A {\displaystyle A} is symmetric this makes no difference. === Rademacher complexity of a function class === Let S = { z 1 , z 2 , … , z m } ⊆ Z {\displaystyle S=\{z_{1},z_{2},\dots ,z_{m}\}\subseteq Z} be a sample of points and consider a function class F {\displaystyle {\mathcal {F}}} of real-valued functions over Z {\displaystyle Z} . Then, the empirical Rademacher complexity of F {\displaystyle {\mathcal {F}}} given S {\displaystyle S} is defined as: Rad S ⁡ ( F ) = 1 m E σ [ sup f ∈ F | ∑ i = 1 m σ i f ( z i ) | ] {\displaystyle \operatorname {Rad} _{S}({\mathcal {F}})={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{f\in {\mathcal {F}}}\left|\sum _{i=1}^{m}\sigma _{i}f(z_{i})\right|\right]} This can also be written using the previous definition: Rad S ⁡ ( F ) = Rad ⁡ ( F ∘ S ) {\displaystyle \operatorname {Rad} _{S}({\mathcal {F}})=\operatorname {Rad} ({\mathcal {F}}\circ S)} where F ∘ S {\displaystyle {\mathcal {F}}\circ S} denotes function composition, i.e.: F ∘ S := { ( f ( z 1 ) , … , f ( z m ) ) ∣ f ∈ F } {\displaystyle {\mathcal {F}}\circ S:=\{(f(z_{1}),\ldots ,f(z_{m}))\mid f\in {\mathcal {F}}\}} The worst case empirical Rademacher complexity is Rad ¯ m ( F ) = sup S = { z 1 , … , z m } Rad S ⁡ ( F ) {\displaystyle {\overline {\operatorname {Rad} }}_{m}({\mathcal {F}})=\sup _{S=\{z_{1},\dots ,z_{m}\}}\operatorname {Rad} _{S}({\mathcal {F}})} Let P {\displaystyle P} be a probability distribution over Z {\displaystyle Z} . The Rademacher complexity of the function class F {\displaystyle {\mathcal {F}}} with respect to P {\displaystyle P} for sample size m {\displaystyle m} is: Rad P , m ⁡ ( F ) := E S ∼ P m [ Rad S ⁡ ( F ) ] {\displaystyle \operatorname {Rad} _{P,m}({\mathcal {F}}):=\mathbb {E} _{S\sim P^{m}}\left[\operatorname {Rad} _{S}({\mathcal {F}})\right]} where the above expectation is taken over an identically independently distributed (i.i.d.) sample S = ( z 1 , z 2 , … , z m ) {\displaystyle S=(z_{1},z_{2},\dots ,z_{m})} generated according to P {\displaystyle P} . == Intuition == The Rademacher complexity is typically applied on a function class of models that are used for classification, with the goal of measuring their ability to classify points drawn from a probability space under arbitrary labellings. When the function class is rich enough, it contains functions that can appropriately adapt for each arrangement of labels, simulated by the random draw of σ i {\displaystyle \sigma _{i}} under the expectation, so that this quantity in the sum is maximized. The Rademacher complexity of a set A {\displaystyle A} can be rewritten as Rad ⁡ ( A ) := 1 m E σ [ sup a ∈ A ∑ i = 1 m σ i a i ] = 1 m 2 m ∑ σ ∈ { − 1 / m , + 1 / m } m [ sup a ∈ A ⟨ σ , a ⟩ ] . {\displaystyle \operatorname {Rad} (A):={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{a\in A}\sum _{i=1}^{m}\sigma _{i}a_{i}\right]={\frac {1}{{\sqrt {m}}2^{m}}}\sum _{\sigma \in \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}}\left[\sup _{a\in A}\langle \sigma ,a\rangle \right].} Each term in the summation is the farthest distance of the set A {\displaystyle A} from the origin, along a unit-length direction σ {\displaystyle \sigma } . The directions are along the vertices of a hypercube. Thus, we can also write it as Rad ⁡ ( A ) = 1 2 m 1 2 m − 1 ∑ σ ∈ { − 1 / m , + 1 / m } m / { − 1 , + 1 } [ sup a ∈ A ⟨ σ , a ⟩ − inf a ∈ A ⟨ σ , a ⟩ ] {\displaystyle \operatorname {Rad} (A)={\frac {1}{2{\sqrt {m}}}}{\frac {1}{2^{m-1}}}\sum _{\sigma \in \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}/\{-1,+1\}}\left[\sup _{a\in A}\langle \sigma ,a\rangle -\inf _{a\in A}\langle \sigma ,a\rangle \right]} Here, the set { − 1 / m , + 1 / m } m / { − 1 , + 1 } {\displaystyle \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}/\{-1,+1\}} denotes half of the vertices of a hypercube, selected so that each diagonal has exactly one vertex selected. In words, this states that 2 m Rad ⁡ ( A ) {\displaystyle 2{\sqrt {m}}\operatorname {Rad} (A)} is precisely the average width of the set A {\displaystyle A} along all diagonal directions of a hypercube. == Examples == A singleton set has 0 width in any direction, so it has Rademacher complexity 0. The set A = { ( 1 , 1 ) , ( 1 , 2 ) } ⊆ R 2 {\displaystyle A=\{(1,1),(1,2)\}\subseteq \mathbb {R} ^{2}} has average width 1 / 2 {\displaystyle 1/{\sqrt {2}}} along the two diagonal directions of the square, so it has Rademacher complexity 1 / 4 {\displaystyle 1/4} . The unit cube [ 0 , 1 ] m {\displaystyle [0,1]^{m}} has constant width m {\displaystyle {\sqrt {m}}} along the diagonal directions, so it has Rademacher complexity 1 / 2 {\displaystyle 1/2} . Similarly, the unit cross-polytope { x ∈ R m : ‖ x ‖ 1 ≤ 1 } {\displaystyle \{x\in \mathbb {R} ^{m}:\|x\|_{1}\leq 1\}} has constant width 2 / m {\displaystyle 2/{\sqrt {m}}} along the diagonal directions, so it has Rademacher complexity 1 / m {\displaystyle 1/m} . == Using the Rademacher complexity == The Rademacher complexity can be used to derive data-dependent upper-bounds on the learnability of function classes. Intuitively, a function-class with smaller Rademacher complexity is easier to learn. === Bounding the representativeness === In machine learning, it is desired to have a training set that represents the true distribution of some sample data S {\displaystyle S} . This can be quantified using the notion of representativeness. Denote by P {\displaystyle P} the probability distribution from which the samples are drawn. Denote by H {\displaystyle H} the set of hypotheses (potential classifiers) and denote by F {\displaystyle {\mathcal {F}}} the corresponding set of error functions, i.e., for every hypothesis h ∈ H {\displaystyle h\in H} , there is a function f h ∈ F {\displaystyle f_{h}\in F} , that maps each training sample (features,label) to the error of the classifier h {\displaystyle h} (note in this case hypothesis and classifier are used interchangeably). For example, in the case that h {\displaystyle h} represents a binary classifier, the error function is a 0–1 loss function, i.e. the error function f h {\displaystyle f_{h}} returns 0 if h {\displaystyle h} correctly classifies a sample and 1 else. We omit the index and write f {\displaystyle f} instead of f h {\displaystyle f_{h}} when the underlying hypothesis is irrelevant. Define: L P ( f ) := E z ∼ P [ f ( z ) ] {\displaystyle L_{P}(f):=\mathbb {E} _{z\sim P}[f(z)]} – the expected error of some error function f ∈ F {\displaystyle f\in {\mathcal {F}}} on the real distribution P {\displaystyle P} ; L S ( f ) := 1 m ∑ i = 1 m f ( z i ) {\displaystyle L_{S}(f):={1 \over m}\sum _{i=1}^{m}f(z_{i})} – the estimated error of some error function f ∈ F {\displaystyle f\in {\mathcal {F}}} on the sample S {\displaystyle S} . The representativeness of the sample S {\displaystyle S} , with respect to P {\displaystyle P} and F {\displaystyle {\mathcal {F}}} , is defined as: Rep P ⁡ ( F , S ) := sup f ∈ F ( L P ( f ) − L S ( f ) ) {\displaystyle \operatorname {Rep} _{P}({\mathcal {F}},S):=\sup _{f\in F}(L_{P}(f)-L_{S}(f))} Smaller representativeness is better, since it provides a way to avoid overfitting: it means that the true error of a classifier is not much higher than its estimated error, and so selecting a classifier that has low estimated error will ensure that the true error is also low. Note however that the concept of representativeness is relative and hence can not be compared across distinct samples. The expected representativeness of a sample can be bounded above by the Rademacher complexity of the function class: If F {\displaystyle {\mathcal {F}}} is a set of functions with range within [ 0 , 1 ] {\displaystyle [0,1]} , then Rad P , m ⁡ ( F ) − ln ⁡ 2 2 m ≤ E S ∼ P m [ Rep P ⁡ ( F , S ) ] ≤ 2 Rad P , m ⁡ ( F ) {\displaystyle \operatorname {Rad} _{P,m}({\mathcal {F}})-{\sqrt {\frac {\ln 2}{2m}}}\leq \mathbb {E} _{S\sim P^{m}}[\operatorname {Rep} _{P}({\