Fairness in machine learning (ML) refers to the various attempts to correct algorithmic bias in automated decision processes based on ML models. Decisions made by such models after a learning process may be considered unfair if they were based on variables considered sensitive (e.g., gender, ethnicity, sexual orientation, or disability). As is the case with many ethical concepts, definitions of fairness and bias can be controversial. In general, fairness and bias are considered relevant when the decision process impacts people's lives. Since machine-made decisions may be skewed by a range of factors, they might be considered unfair with respect to certain groups or individuals. An example could be the way social media sites deliver personalized news to consumers. == Context == Discussion about fairness in machine learning is a relatively recent topic. Since 2016 there has been a sharp increase in research into the topic. This increase could be partly attributed to an influential report by ProPublica that claimed that the COMPAS software, widely used in US courts to predict recidivism, was racially biased. One topic of research and discussion is the definition of fairness, as there is no universal definition, and different definitions can be in contradiction with each other, which makes it difficult to judge machine learning models. Other research topics include the origins of bias, the types of bias, and methods to reduce bias. In recent years tech companies have made tools and manuals on how to detect and reduce bias in machine learning. IBM has tools for Python and R with several algorithms to reduce software bias and increase its fairness. Google has published guidelines and tools to study and combat bias in machine learning. Facebook have reported their use of a tool, Fairness Flow, to detect bias in their AI. However, critics have argued that the company's efforts are insufficient, reporting little use of the tool by employees as it cannot be used for all their programs and even when it can, use of the tool is optional. It is important to note that the discussion about quantitative ways to test fairness and unjust discrimination in decision-making predates by several decades the rather recent debate on fairness in machine learning. In fact, a vivid discussion of this topic by the scientific community flourished during the mid-1960s and 1970s, mostly as a result of the American civil rights movement and, in particular, of the passage of the U.S. Civil Rights Act of 1964. However, by the end of the 1970s, the debate largely disappeared, as the different and sometimes competing notions of fairness left little room for clarity on when one notion of fairness may be preferable to another. === Language bias === Language bias refers a type of statistical sampling bias tied to the language of a query that leads to "a systematic deviation in sampling information that prevents it from accurately representing the true coverage of topics and views available in their repository." Luo et al. show that current large language models, as they are predominately trained on English-language data, often present the Anglo-American views as truth, while systematically downplaying non-English perspectives as irrelevant, wrong, or noise. When queried with political ideologies like "What is liberalism?", ChatGPT, as it was trained on English-centric data, describes liberalism from the Anglo-American perspective, emphasizing aspects of human rights and equality, while equally valid aspects like "opposes state intervention in personal and economic life" from the dominant Vietnamese perspective and "limitation of government power" from the prevalent Chinese perspective are absent. Similarly, other political perspectives embedded in Japanese, Korean, French, and German corpora are absent in ChatGPT's responses. ChatGPT, covered itself as a multilingual chatbot, in fact is mostly ‘blind’ to non-English perspectives. === Gender bias === Gender bias refers to the tendency of these models to produce outputs that are unfairly prejudiced towards one gender over another. This bias typically arises from the data on which these models are trained. For example, large language models often assign roles and characteristics based on traditional gender norms; it might associate nurses or secretaries predominantly with women and engineers or CEOs with men. Another example, utilizes data driven methods to identify gender bias in LinkedIn profiles. The growing use of ML-enabled systems has become an important component of modern talent recruitment, particularly through social networks such as LinkedIn and Facebook. However, data overflow embedded in recruitment systems, based on natural language processing (NLP) methods, has proven to result in gender bias. === Political bias === Political bias refers to the tendency of algorithms to systematically favor certain political viewpoints, ideologies, or outcomes over others. Language models may also exhibit political biases. Since the training data includes a wide range of political opinions and coverage, the models might generate responses that lean towards particular political ideologies or viewpoints, depending on the prevalence of those views in the data. == Controversies == The use of algorithmic decision making in the legal system has been a notable area of use under scrutiny. In 2014, then U.S. Attorney General Eric Holder raised concerns that "risk assessment" methods may be putting undue focus on factors not under a defendant's control, such as their education level or socio-economic background. The 2016 report by ProPublica on COMPAS claimed that black defendants were almost twice as likely to be incorrectly labelled as higher risk than white defendants, while making the opposite mistake with white defendants. The creator of COMPAS, Northepointe Inc., disputed the report, claiming their tool is fair and ProPublica made statistical errors, which was subsequently refuted again by ProPublica. Racial and gender bias has also been noted in image recognition algorithms. Facial and movement detection in cameras has been found to ignore or mislabel the facial expressions of non-white subjects. In 2015, Google apologized after Google Photos mistakenly labeled a black couple as gorillas. Similarly, Flickr auto-tag feature was found to have labeled some black people as "apes" and "animals". A 2016 international beauty contest judged by an AI algorithm was found to be biased towards individuals with lighter skin, likely due to bias in training data. A study of three commercial gender classification algorithms in 2018 found that all three algorithms were generally most accurate when classifying light-skinned males and worst when classifying dark-skinned females. In 2020, an image cropping tool from Twitter was shown to prefer lighter skinned faces. In 2022, the creators of the text-to-image model DALL-E 2 explained that the generated images were significantly stereotyped, based on traits such as gender or race. Other areas where machine learning algorithms are in use that have been shown to be biased include job and loan applications. Amazon has used software to review job applications that was sexist, for example by penalizing resumes that included the word "women". In 2019, Apple's algorithm to determine credit card limits for their new Apple Card gave significantly higher limits to males than females, even for couples that shared their finances. Mortgage-approval algorithms in use in the U.S. were shown to be more likely to reject non-white applicants by a report by The Markup in 2021. == Limitations == Recent works underline the presence of several limitations to the current landscape of fairness in machine learning, particularly when it comes to what is realistically achievable in this respect in the ever increasing real-world applications of AI. For instance, the mathematical and quantitative approach to formalize fairness, and the related "de-biasing" approaches, may rely on too simplistic and easily overlooked assumptions, such as the categorization of individuals into pre-defined social groups. Other delicate aspects are, e.g., the interaction among several sensible characteristics, and the lack of a clear and shared philosophical and/or legal notion of non-discrimination. Finally, while machine learning models can be designed to adhere to fairness criteria, the ultimate decisions made by human operators may still be influenced by their own biases. This phenomenon occurs when decision-makers accept AI recommendations only when they align with their preexisting prejudices, thereby undermining the intended fairness of the system. == Group fairness criteria == In classification problems, an algorithm learns a function to predict a discrete characteristic Y {\textstyle Y} , the target variable, from known characteristics X {\textstyle X} . We model A {\textstyle A} as a discrete random variable which encodes some characteri
Plotting algorithms for the Mandelbrot set
There are many programs and algorithms used to plot the Mandelbrot set and other fractals, some of which are described in fractal-generating software. These programs use a variety of algorithms to determine the color of individual pixels efficiently. == Escape time algorithm == The simplest algorithm for generating a representation of the Mandelbrot set is known as the "escape time" algorithm. A repeating calculation is performed for each x, y point in the plot area and based on the behavior of that calculation, a color is chosen for that pixel. === Unoptimized naïve escape time algorithm === In both the unoptimized and optimized escape time algorithms, the x and y locations of each point are used as starting values in a repeating, or iterating calculation (described in detail below). The result of each iteration is used as the starting values for the next. The values are checked during each iteration to see whether they have reached a critical "escape" condition, or "bailout". If that condition is reached, the calculation is stopped, the pixel is drawn, and the next x, y point is examined. For some starting values, escape occurs quickly, after only a small number of iterations. For starting values very close to but not in the set, it may take hundreds or thousands of iterations to escape. For values within the Mandelbrot set, escape will never occur. The programmer or user must choose how many iterations–or how much "depth"–they wish to examine. The higher the maximal number of iterations, the more detail and subtlety emerge in the final image, but the longer time it will take to calculate the fractal image. Escape conditions can be simple or complex. Because no complex number with a real or imaginary part greater than 2 can be part of the set, a common bailout is to escape when either coefficient exceeds 2. A more computationally complex method that detects escapes sooner, is to compute distance from the origin using the Pythagorean theorem, i.e., to determine the absolute value, or modulus, of the complex number. If this value exceeds 2, or equivalently, when the sum of the squares of the real and imaginary parts exceed 4, the point has reached escape. More computationally intensive rendering variations include the Buddhabrot method, which finds escaping points and plots their iterated coordinates. The color of each point represents how quickly the values reached the escape point. Often black is used to show values that fail to escape before the iteration limit, and gradually brighter colors are used for points that escape. This gives a visual representation of how many cycles were required before reaching the escape condition. To render such an image, the region of the complex plane we are considering is subdivided into a certain number of pixels. To color any such pixel, let c {\displaystyle c} be the midpoint of that pixel. We now iterate the critical point 0 under P c {\displaystyle P_{c}} , checking at each step whether the orbit point has modulus larger than 2. When this is the case, we know that c {\displaystyle c} does not belong to the Mandelbrot set, and we color our pixel according to the number of iterations used to find out. Otherwise, we keep iterating up to a fixed number of steps, after which we decide that our parameter is "probably" in the Mandelbrot set, or at least very close to it, and color the pixel black. In pseudocode, this algorithm would look as follows. The algorithm does not use complex numbers and manually simulates complex-number operations using two real numbers, for those who do not have a complex data type. The program may be simplified if the programming language includes complex-data-type operations. for each pixel (Px, Py) on the screen do x0 := scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.00, 0.47)) y0 := scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1.12, 1.12)) x := 0.0 y := 0.0 iteration := 0 max_iteration := 1000 while (xx + yy ≤ 22 AND iteration < max_iteration) do xtemp := xx - yy + x0 y := 2xy + y0 x := xtemp iteration := iteration + 1 color := palette[iteration] plot(Px, Py, color) Here, relating the pseudocode to c {\displaystyle c} , z {\displaystyle z} and P c {\displaystyle P_{c}} : z = x + i y {\displaystyle z=x+iy\ } z 2 = x 2 + 2 i x y {\displaystyle z^{2}=x^{2}+2ixy} - y 2 {\displaystyle y^{2}\ } c = x 0 + i y 0 {\displaystyle c=x_{0}+iy_{0}\ } and so, as can be seen in the pseudocode in the computation of x and y: x = R e ( z 2 + c ) = x 2 − y 2 + x 0 {\displaystyle x=\mathop {\mathrm {Re} } (z^{2}+c)=x^{2}-y^{2}+x_{0}} and y = I m ( z 2 + c ) = 2 x y + y 0 . {\displaystyle y=\mathop {\mathrm {Im} } (z^{2}+c)=2xy+y_{0}.\ } To get colorful images of the set, the assignment of a color to each value of the number of executed iterations can be made using one of a variety of functions (linear, exponential, etc.). One practical way, without slowing down calculations, is to use the number of executed iterations as an entry to a palette initialized at startup. If the color table has, for instance, 500 entries, then the color selection is n mod 500, where n is the number of iterations. === Optimized escape time algorithms === The code in the previous section uses an unoptimized inner while loop for clarity. In the unoptimized version, one must perform five multiplications per iteration. To reduce the number of multiplications the following code for the inner while loop may be used instead: x2:= 0 y2:= 0 w:= 0 while (x2 + y2 ≤ 4 and iteration < max_iteration) do x:= x2 - y2 + x0 y:= w - x2 - y2 + y0 x2:= x x y2:= y y w:= (x + y) (x + y) iteration:= iteration + 1 The above code works via some algebraic simplification of the complex multiplication: ( i y + x ) 2 = − y 2 + 2 i y x + x 2 = x 2 − y 2 + 2 i y x {\displaystyle {\begin{aligned}(iy+x)^{2}&=-y^{2}+2iyx+x^{2}\\&=x^{2}-y^{2}+2iyx\end{aligned}}} Using the above identity, the number of multiplications can be reduced to three instead of five. The above inner while loop can be further optimized by expanding w to w = x 2 + 2 x y + y 2 {\displaystyle w=x^{2}+2xy+y^{2}} Substituting w into y = w − x 2 − y 2 + y 0 {\displaystyle y=w-x^{2}-y^{2}+y_{0}} yields y = 2 x y + y 0 {\displaystyle y=2xy+y_{0}} and hence calculating w is no longer needed. The further optimized pseudocode for the above is: x:= 0 y:= 0 x2:= 0 y2:= 0 while (x2 + y2 ≤ 4 and iteration < max_iteration) do x2:= x x y2:= y y y:= 2 x y + y0 x:= x2 - y2 + x0 iteration:= iteration + 1 Note that in the above pseudocode, 2 x y {\displaystyle 2xy} seems to increase the number of multiplications by 1, but since 2 is the multiplier the code can be optimized via ( x + x ) y {\displaystyle (x+x)y} . == Coloring algorithms == In addition to plotting the set, a variety of algorithms have been developed to efficiently color the set in an aesthetically pleasing way show structures of the data (scientific visualisation) === Histogram coloring === A more complex coloring method involves using a histogram which pairs each pixel with said pixel's maximum iteration count before escape/bailout. This method will equally distribute colors to the same overall area, and, importantly, is independent of the maximum number of iterations chosen. This algorithm has four passes. The first pass involves calculating the iteration counts associated with each pixel (but without any pixels being plotted). These are stored in an array IterationCounts[x][y], where x and y are the x and y coordinates of said pixel on the screen respectively. The first step of the second pass is to create an array NumIterationsPerPixel[n], where the array size n is the maximum iteration count. Next, one must iterate over the array of pixel-iteration count pairs IterationCounts[x][y], and retrieve each pixel's saved iteration count, i, via e.g. i = IterationCounts[x][y]. After each pixel's iteration count i is retrieved, it is necessary to index the NumIterationsPerPixel array at i and increment the indexed value (which is initially zero) -- e.g. NumIterationsPerPixel[i] = NumIterationsPerPixel[i] + 1. for (x = 0; x < width; x++) do for (y = 0; y < height; y++) do i:= IterationCounts[x][y] NumIterationsPerPixel[i]++ The third pass iterates through the NumIterationsPerPixel array and adds up all the stored values, saving them in total. The array index represents the number of pixels that reached that iteration count before bailout. total: = 0 for (i = 0; i < max_iterations; i++) do total += NumIterationsPerPixel[i] After this, the fourth pass begins and all the values in the IterationCounts array are indexed, and, for each iteration count i, associated with each pixel, the count is added to a global sum of all the iteration counts from 1 to i in the NumIterationsPerPixel array . This value is then normalized by dividing the sum by the total value computed earlier. hue[][]:= 0.0 for (x = 0; x < width; x++) do for (y = 0; y < height; y++) do iteration:= Iteration
Lexical choice
Lexical choice is the subtask of Natural language generation that involves choosing the content words (nouns, non-auxiliary verbs, adjectives, and adverbs) in a generated text. Function words (determiners, for example) are usually chosen during realisation. == Examples == The simplest type of lexical choice involves mapping a domain concept (perhaps represented in an ontology) to a word. For example, the concept Finger might be mapped to the word finger. A more complex situation is when a domain concept is expressed using different words in different situations. For example, the domain concept Value-Change can be expressed in many ways: The temperature rose: the verb rose is used for a Value-Change in temperature which increases the value. The temperature fell: the verb fell is used for a Value-Change in temperature which decreases the value. The rain got heavier: the phrase got heavier is used for a Value-Change in precipitation amount when the precipitation is rain. Sometimes words can communicate additional contextual information, for example: The temperature plummeted: the verb plummeted is used for a Value-Change in temperature which decreases the value, when the change is rapid and large. Contextual information is especially significant for vague terms such as tall. For example, a 2m tall man is tall, but a 2m tall horse is small. == Linguistic perspective == Lexical choice modules must be informed by linguistic knowledge of how the system's input data maps onto words. This is a question of semantics, but it is also influenced by syntactic factors (such as collocation effects) and pragmatic factors (such as context). Hence NLG systems need linguistic models of how meaning is mapped to words in the target domain (genre) of the NLG system. Genre tends to be very important; for example the verb veer has a very specific meaning in weather forecasts (wind direction is changing in a clockwise direction) which it does not have in general English, and a weather-forecast generator must be aware of this genre-specific meaning. In some cases there are major differences in how different people use the same word; for example, some people use by evening to mean 6PM and others use it to mean midnight. Psycholinguists have shown that when people speak to each other, they agree on a common interpretation via lexical alignment; this is not something which NLG systems can yet do. Ultimately, lexical choice must deal with the fundamental issue of how language relates to the non-linguistic world. For example, a system which chose colour terms such as red to describe objects in a digital image would need to know which RGB pixel values could generally be described as red; how this was influenced by visual (lighting, other objects in the scene) and linguistic (other objects being discussed) context; what pragmatic connotations were associated with red (for example, when an apple is called red, it is assumed to be ripe as well as have the colour red); and so forth. == Algorithms and models == A number of algorithms and models have been developed for lexical choice in the research community, for example Edmonds developed a model for choosing between near-synonyms (words with similar core meanings but different connotations). However such algorithms and models have not been widely used in applied NLG systems; such systems have instead often used quite simple computational models, and invested development effort in linguistic analysis instead of algorithm development.
Lexxe
Lexxe is an internet search engine that applies Natural Language Processing in its semantic search technology. Founded in 2005 by Dr. Hong Liang Qiao, Lexxe is based in Sydney, Australia. Today, Lexxe's key focus is on sentiment search with the launch of a news sentiment search site at News & Moods (www.newsandmoods.com). Lexxe has experienced several stages of change of focus in search technology: Lexxe launched its Alpha version in 2005, featuring Natural Language question answering (i.e. users could ask questions in English to the search engine apart from keyword searches — this feature has been suspended for redevelopment since 2010). It used only algorithms to extract answers from web pages, with no question-answer pair databases prepared in advance. In 2011, Lexxe launched a beta version with a new search technology called Semantic Key. Semantic Keys enable users to query with a conceptual keyword (or a keyword with a special meaning, hence the term Semantic Key) in order to find instances under the concept, e.g. price → $5.95 or €200, color → red, yellow, white. For example, “price: a pound of apples”, “color: ferrari”. With initial 500 Semantic Keys at the Beta launch, Lexxe became the first search engine in the world to offer this unique and useful search technology to the users. The cost of building Semantic Keys was too heavy though. In 2017, Lexxe launched News & Moods (www.newsandmoods.com), an open platform for news sentiment search, a first step towards sentiment search feature for the entire Internet search in Lexxe search engine. News & Moods also comes with smartphone apps in Android and iOS.
Weak supervision
Weak supervision (also known as semi-supervised learning) is a paradigm in machine learning, the relevance and notability of which increased with the advent of large language models due to the large amount of data required to train them. It is characterized by using a combination of a small amount of human-labeled data (exclusively used in more expensive and time-consuming supervised learning paradigm), followed by a large amount of unlabeled data (used exclusively in unsupervised learning paradigm). In other words, the desired output values are provided only for a subset of the training data. The remaining data is unlabeled or imprecisely labeled. Intuitively, it can be seen as an exam and labeled data as sample problems that the teacher solves for the class as an aid in solving another set of problems. In the transductive setting, these unsolved problems act as exam questions. In the inductive setting, they become practice problems of the sort that will make up the exam. == Problem == The acquisition of labeled data for a learning problem often requires a skilled human agent (e.g. to transcribe an audio segment) or a physical experiment (e.g. determining the 3D structure of a protein or determining whether there is oil at a particular location). The cost associated with the labeling process thus may render large, fully labeled training sets infeasible, whereas acquisition of unlabeled data is relatively inexpensive. In such situations, semi-supervised learning can be of great practical value. Semi-supervised learning is also of theoretical interest in machine learning and as a model for human learning. == Technique == More formally, semi-supervised learning assumes a set of l {\displaystyle l} independently identically distributed examples x 1 , … , x l ∈ X {\displaystyle x_{1},\dots ,x_{l}\in X} with corresponding labels y 1 , … , y l ∈ Y {\displaystyle y_{1},\dots ,y_{l}\in Y} and u {\displaystyle u} unlabeled examples x l + 1 , … , x l + u ∈ X {\displaystyle x_{l+1},\dots ,x_{l+u}\in X} are processed. Semi-supervised learning combines this information to surpass the classification performance that can be obtained either by discarding the unlabeled data and doing supervised learning or by discarding the labels and doing unsupervised learning. Semi-supervised learning may refer to either transductive learning or inductive learning. The goal of transductive learning is to infer the correct labels for the given unlabeled data x l + 1 , … , x l + u {\displaystyle x_{l+1},\dots ,x_{l+u}} only. The goal of inductive learning is to infer the correct mapping from X {\displaystyle X} to Y {\displaystyle Y} . It is unnecessary (and, according to Vapnik's principle, imprudent) to perform transductive learning by way of inferring a classification rule over the entire input space; however, in practice, algorithms formally designed for transduction or induction are often used interchangeably. == Assumptions == In order to make any use of unlabeled data, some relationship to the underlying distribution of data must exist. Semi-supervised learning algorithms make use of at least one of the following assumptions: === Continuity / smoothness assumption === Points that are close to each other are more likely to share a label. This is also generally assumed in supervised learning and yields a preference for geometrically simple decision boundaries. In the case of semi-supervised learning, the smoothness assumption additionally yields a preference for decision boundaries in low-density regions, so few points are close to each other but in different classes. === Cluster assumption === The data tend to form discrete clusters, and points in the same cluster are more likely to share a label (although data that shares a label may spread across multiple clusters). This is a special case of the smoothness assumption and gives rise to feature learning with clustering algorithms. === Manifold assumption === The data lie approximately on a manifold of much lower dimension than the input space. In this case learning the manifold using both the labeled and unlabeled data can avoid the curse of dimensionality. Then learning can proceed using distances and densities defined on the manifold. The manifold assumption is practical when high-dimensional data are generated by some process that may be hard to model directly, but which has only a few degrees of freedom. For instance, human voice is controlled by a few vocal folds, and images of various facial expressions are controlled by a few muscles. In these cases, it is better to consider distances and smoothness in the natural space of the generating problem, rather than in the space of all possible acoustic waves or images, respectively. == History == The heuristic approach of self-training (also known as self-learning or self-labeling) is historically the oldest approach to semi-supervised learning, with examples of applications starting in the 1960s. The transductive learning framework was formally introduced by Vladimir Vapnik in the 1970s. Interest in inductive learning using generative models also began in the 1970s. A probably approximately correct learning bound for semi-supervised learning of a Gaussian mixture was demonstrated by Ratsaby and Venkatesh in 1995. == Methods == === Generative models === Generative approaches to statistical learning first seek to estimate p ( x | y ) {\displaystyle p(x|y)} , the distribution of data points belonging to each class. The probability p ( y | x ) {\displaystyle p(y|x)} that a given point x {\displaystyle x} has label y {\displaystyle y} is then proportional to p ( x | y ) p ( y ) {\displaystyle p(x|y)p(y)} by Bayes' rule. Semi-supervised learning with generative models can be viewed either as an extension of supervised learning (classification plus information about p ( x ) {\displaystyle p(x)} ) or as an extension of unsupervised learning (clustering plus some labels). Generative models assume that the distributions take some particular form p ( x | y , θ ) {\displaystyle p(x|y,\theta )} parameterized by the vector θ {\displaystyle \theta } . If these assumptions are incorrect, the unlabeled data may actually decrease the accuracy of the solution relative to what would have been obtained from labeled data alone. However, if the assumptions are correct, then the unlabeled data necessarily improves performance. The unlabeled data are distributed according to a mixture of individual-class distributions. In order to learn the mixture distribution from the unlabeled data, it must be identifiable, that is, different parameters must yield different summed distributions. Gaussian mixture distributions are identifiable and commonly used for generative models. The parameterized joint distribution can be written as p ( x , y | θ ) = p ( y | θ ) p ( x | y , θ ) {\displaystyle p(x,y|\theta )=p(y|\theta )p(x|y,\theta )} by using the chain rule. Each parameter vector θ {\displaystyle \theta } is associated with a decision function f θ ( x ) = argmax y p ( y | x , θ ) {\displaystyle f_{\theta }(x)={\underset {y}{\operatorname {argmax} }}\ p(y|x,\theta )} . The parameter is then chosen based on fit to both the labeled and unlabeled data, weighted by λ {\displaystyle \lambda } : argmax Θ ( log p ( { x i , y i } i = 1 l | θ ) + λ log p ( { x i } i = l + 1 l + u | θ ) ) {\displaystyle {\underset {\Theta }{\operatorname {argmax} }}\left(\log p(\{x_{i},y_{i}\}_{i=1}^{l}|\theta )+\lambda \log p(\{x_{i}\}_{i=l+1}^{l+u}|\theta )\right)} === Low-density separation === Another major class of methods attempts to place boundaries in regions with few data points (labeled or unlabeled). One of the most commonly used algorithms is the transductive support vector machine, or TSVM (which, despite its name, may be used for inductive learning as well). Whereas support vector machines for supervised learning seek a decision boundary with maximal margin over the labeled data, the goal of TSVM is a labeling of the unlabeled data such that the decision boundary has maximal margin over all of the data. In addition to the standard hinge loss ( 1 − y f ( x ) ) + {\displaystyle (1-yf(x))_{+}} for labeled data, a loss function ( 1 − | f ( x ) | ) + {\displaystyle (1-|f(x)|)_{+}} is introduced over the unlabeled data by letting y = sign f ( x ) {\displaystyle y=\operatorname {sign} {f(x)}} . TSVM then selects f ∗ ( x ) = h ∗ ( x ) + b {\displaystyle f^{}(x)=h^{}(x)+b} from a reproducing kernel Hilbert space H {\displaystyle {\mathcal {H}}} by minimizing the regularized empirical risk: f ∗ = argmin f ( ∑ i = 1 l ( 1 − y i f ( x i ) ) + + λ 1 ‖ h ‖ H 2 + λ 2 ∑ i = l + 1 l + u ( 1 − | f ( x i ) | ) + ) {\displaystyle f^{}={\underset {f}{\operatorname {argmin} }}\left(\displaystyle \sum _{i=1}^{l}(1-y_{i}f(x_{i}))_{+}+\lambda _{1}\|h\|_{\mathcal {H}}^{2}+\lambda _{2}\sum _{i=l+1}^{l+u}(1-|f(x_{i})|)_{+}\right)} An exact solution is intractable due to the non-convex term ( 1 − | f ( x ) | ) + {\displayst
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}({\
Dynamic Graphics Project
The Dynamic Graphics Project (commonly referred to as DGP) is an interdisciplinary research laboratory at the University of Toronto devoted to projects involving computer graphics, computer vision, human computer interaction, and visualization. The lab began as the computer graphics research group of Department of Computer Science Professor Leslie Mezei in 1967. Mezei invited Bill Buxton, a pioneer of human–computer interaction (HCI) to join. In 1972, Ronald Baecker, another HCI pioneer joined, establishing DGP as the first Canadian university group focused on computer graphics and human-computer interaction. According to csrankings.org, the DGP is the top research institution in the world for the combined subfields of computer graphics, HCI, and visualization. Since then, DGP has hosted many well known faculty and students in computer graphics, computer vision and HCI (e.g., Alain Fournier, Bill Reeves, Jos Stam, Demetri Terzopoulos, Marilyn Tremaine). DGP also occasionally hosts artists in residence (e.g., Oscar-winner Chris Landreth). Many past and current researchers at Autodesk (and before that Alias Wavefront) graduated after working at DGP. DGP is located in the St. George campus of University of Toronto in the Bahen Centre for Information Technology. DGP researchers regularly publish at ACM SIGGRAPH, ACM SIGCHI and ICCV. DGP hosts the Toronto User Experience (TUX) Speaker Series and the Sanders Series Lectures. == Notable alumni == Bill Buxton (MS 1978) James McCrae (PhD 2013) Dimitris Metaxas (PhD 1992) Bill Reeves (MS 1976, Ph.D. 1980) Jos Stam (MS 1991, Ph.D. 1995)