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  Right to explanation

  In the regulation of algorithms, particularly artificial intelligence and its subfield of machine learning, a right to [an] explanation is a right to be given an explanation for an output of the algorithm. Such rights primarily refer to individual rights to be given an explanation for decisions that significantly affect an individual, particularly legally or financially. For example, a person who applies for a loan and is denied may ask for an explanation, which could be "Credit bureau X reports that you declared bankruptcy last year; this is the main factor in considering you too likely to default, and thus we will not give you the loan you applied for." Some such legal rights already exist, while the scope of a general "right to explanation" is a matter of ongoing debate. There have been arguments made that a "social right to explanation" is a crucial foundation for an information society, particularly as the institutions of that society will need to use digital technologies, artificial intelligence, machine learning. In other words, that the related automated decision making systems that use explainability would be more trustworthy and transparent. Without this right, which could be constituted both legally and through professional standards, the public will be left without much recourse to challenge the decisions of automated systems. == Examples == === Credit scoring in the United States === Under the Equal Credit Opportunity Act (Regulation B of the Code of Federal Regulations), Title 12, Chapter X, Part 1002, §1002.9, creditors are required to notify applicants who are denied credit with specific reasons for the detail. As detailed in §1002.9(b)(2): (2) Statement of specific reasons. The statement of reasons for adverse action required by paragraph (a)(2)(i) of this section must be specific and indicate the principal reason(s) for the adverse action. Statements that the adverse action was based on the creditor's internal standards or policies or that the applicant, joint applicant, or similar party failed to achieve a qualifying score on the creditor's credit scoring system are insufficient. The official interpretation of this section details what types of statements are acceptable. Creditors comply with this regulation by providing a list of reasons (generally at most 4, per interpretation of regulations), consisting of a numeric reason code (as identifier) and an associated explanation, identifying the main factors affecting a credit score. An example might be: 32: Balances on bankcard or revolving accounts too high compared to credit limits === European Union === The European Union General Data Protection Regulation (GDPR, enacted 2016, taking effect 2018) extends the automated decision-making rights in the 1995 Data Protection Directive to provide a legally disputed form of a right to an explanation, stated as such in Recital 71: "[the data subject should have] the right ... to obtain an explanation of the decision reached". In full: The data subject should have the right not to be subject to a decision, which may include a measure, evaluating personal aspects relating to him or her which is based solely on automated processing and which produces legal effects concerning him or her or similarly significantly affects him or her, such as automatic refusal of an online credit application or e-recruiting practices without any human intervention. ... In any case, such processing should be subject to suitable safeguards, which should include specific information to the data subject and the right to obtain human intervention, to express his or her point of view, to obtain an explanation of the decision reached after such assessment and to challenge the decision. However, the extent to which the regulations themselves provide a "right to explanation" is heavily debated. There are two main strands of criticism. There are significant legal issues with the right as found in Article 22 — as recitals are not binding, and the right to an explanation is not mentioned in the binding articles of the text, having been removed during the legislative process. In addition, there are significant restrictions on the types of automated decisions that are covered — which must be both "solely" based on automated processing, and have legal or similarly significant effects — which significantly limits the range of automated systems and decisions to which the right would apply. In particular, the right is unlikely to apply in many of the cases of algorithmic controversy that have been picked up in the media. The UK has also recently amended its implementation of Article 22. A second potential source of such a right has been pointed to in Article 15, the "right of access by the data subject". This restates a similar provision from the 1995 Data Protection Directive, allowing the data subject access to "meaningful information about the logic involved" in the same significant, solely automated decision-making, found in Article 22. Yet this too suffers from alleged challenges that relate to the timing of when this right can be drawn upon, as well as practical challenges that mean it may not be binding in many cases of public concern. Other EU legislative instruments contain explanation rights. The European Union's Artificial Intelligence Act provides in Article 86 a "[r]ight to explanation of individual decision-making" of certain high risk systems which produce significant, adverse effects to an individual's health, safety or fundamental rights. The right provides for "clear and meaningful explanations of the role of the AI system in the decision-making procedure and the main elements of the decision taken", although only applies to the extent other law does not provide such a right. The Digital Services Act in Article 27, and the Platform to Business Regulation in Article 5, both contain rights to have the main parameters of certain recommender systems to be made clear, although these provisions have been criticised as not matching the way that such systems work. The Platform Work Directive, which provides for regulation of automation in gig economy work as an extension of data protection law, further contains explanation provisions in Article 11, using the specific language of "explanation" in a binding article rather than a recital as is the case in the GDPR. Scholars note that remains uncertainty as to whether these provisions imply sufficiently tailored explanation in practice which will need to be resolved by courts. === France === In France the 2016 Loi pour une République numérique (Digital Republic Act or loi numérique) amends the country's administrative code to introduce a new provision for the explanation of decisions made by public sector bodies about individuals. It notes that where there is "a decision taken on the basis of an algorithmic treatment", the rules that define that treatment and its "principal characteristics" must be communicated to the citizen upon request, where there is not an exclusion (e.g. for national security or defence). These should include the following: the degree and the mode of contribution of the algorithmic processing to the decision- making; the data processed and its source; the treatment parameters, and where appropriate, their weighting, applied to the situation of the person concerned; the operations carried out by the treatment. Scholars have noted that this right, while limited to administrative decisions, goes beyond the GDPR right to explicitly apply to decision support rather than decisions "solely" based on automated processing, as well as provides a framework for explaining specific decisions. Indeed, the GDPR automated decision-making rights in the European Union, one of the places a "right to an explanation" has been sought within, find their origins in French law in the late 1970s. == Criticism == Some argue that a "right to explanation" is at best unnecessary, at worst harmful, and threatens to stifle innovation. Specific criticisms include: favoring human decisions over machine decisions, being redundant with existing laws, and focusing on process over outcome. Authors of study "Slave to the Algorithm? Why a 'Right to an Explanation' Is Probably Not the Remedy You Are Looking For" Lilian Edwards and Michael Veale argue that a right to explanation is not the solution to harms caused to stakeholders by algorithmic decisions. They also state that the right of explanation in the GDPR is narrowly defined, and is not compatible with how modern machine learning technologies are being developed. With these limitations, defining transparency within the context of algorithmic accountability remains a problem. For example, providing the source code of algorithms may not be sufficient and may create other problems in terms of privacy disclosures and the gaming of technical systems. To mitigate this issue, Edwards and Veale argue that an auditing system could be more effective, to allow auditors to loo

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  XGBoost

  XGBoost (eXtreme Gradient Boosting) is an open-source software library which provides a regularizing gradient boosting framework for C++, Java, Python, R, Julia, Perl, and Scala. It works on Linux, Microsoft Windows, and macOS. From the project description, it aims to provide a "Scalable, Portable and Distributed Gradient Boosting (GBM, GBRT, GBDT) Library". It runs on a single machine, as well as the distributed processing frameworks Apache Hadoop, Apache Spark, Apache Flink, and Dask. XGBoost gained much popularity and attention in the mid-2010s as the algorithm of choice for many winning teams of machine learning competitions. == History == XGBoost initially started as a research project by Tianqi Chen as part of the Distributed (Deep) Machine Learning Community (DMLC) group at the University of Washington. Initially, it began as a terminal application which could be configured using a libsvm configuration file. It became well known in the ML competition circles after its use in the winning solution of the Higgs Machine Learning Challenge. Soon after, the Python and R packages were built, and XGBoost now has package implementations for Java, Scala, Julia, Perl, and other languages. This brought the library to more developers and contributed to its popularity among the Kaggle community, where it has been used for a large number of competitions. It was soon integrated with a number of other packages making it easier to use in their respective communities. It has now been integrated with scikit-learn for Python users and with the caret package for R users. It can also be integrated into Data Flow frameworks like Apache Spark, Apache Hadoop, and Apache Flink using the abstracted Rabit and XGBoost4J. XGBoost is also available on OpenCL for FPGAs. An efficient, scalable implementation of XGBoost has been published by Tianqi Chen and Carlos Guestrin. While the XGBoost model often achieves higher accuracy than a single decision tree, it sacrifices the intrinsic interpretability of decision trees. For example, following the path that a decision tree takes to make its decision is trivial and self-explained, but following the paths of hundreds or thousands of trees is much harder. == Features == Salient features of XGBoost which make it different from other gradient boosting algorithms include: Clever penalization of trees A proportional shrinking of leaf nodes Newton Boosting Extra randomization parameter Implementation on single, distributed systems and out-of-core computation Automatic feature selection Theoretically justified weighted quantile sketching for efficient computation Parallel tree structure boosting with sparsity Efficient cacheable block structure for decision tree training == The algorithm == XGBoost works as Newton–Raphson in function space unlike gradient boosting that works as gradient descent in function space, a second order Taylor approximation is used in the loss function to make the connection to Newton–Raphson method. A generic unregularized XGBoost algorithm is: == Parameters == XGBoost has parameters which can be specified to affect how it functions and performs. Some parameters include: Learning rate (also known as the "step size" or “"shrinkage"), it is a number between 0 and 1 (default is 0.3), which determines the rate the algorithm learns from each iteration. n_estimators, sets the number of trees to be built in the ensemble, where more trees generally increases the complexity of the model, but can lead to overfitting with too many trees. Gamma (also known as Lagrange multiplier or the minimum loss reduction parameter), controls the minimum amount of loss reduction required to make a further split on a leaf node of the tree. The default in XGBoost is 0 . max_depth, represents how deeply each tree in the boosting process can grow during training, where the default is 6. == Awards == John Chambers Award (2016) High Energy Physics meets Machine Learning award (HEP meets ML) (2016)

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  Information gain (decision tree)

  In the context of decision trees in information theory and machine learning, information gain refers to the conditional expected value of the Kullback–Leibler divergence of the univariate probability distribution of one variable from the conditional distribution of this variable given the other one. (In broader contexts, information gain can also be used as a synonym for either Kullback–Leibler divergence or mutual information, but the focus of this article is on the more narrow meaning below.) Explicitly, the information gain of a random variable X {\displaystyle X} obtained from an observation of a random variable A {\displaystyle A} taking value a {\displaystyle a} is defined as: I G ( X , a ) = D KL ( P X ∣ a ∥ P X ) {\displaystyle {\mathit {IG}}(X,a)=D_{\text{KL}}{\bigl (}P_{X\mid a}\parallel P_{X}{\bigr )}} In other words, it is the Kullback–Leibler divergence of P X ( x ) {\displaystyle P_{X}(x)} (the prior distribution for X {\displaystyle X} ) from P X ∣ a ( x ) {\displaystyle P_{X\mid a}(x)} (the posterior distribution for X {\displaystyle X} given A = a {\displaystyle A=a} ). The expected value of the information gain is the mutual information I ( X ; A ) {\displaystyle I(X;A)} : E A ⁡ [ I G ( X , A ) ] = I ( X ; A ) {\displaystyle \operatorname {E} _{A}[{\mathit {IG}}(X,A)]=I(X;A)} i.e. the reduction in the entropy of X {\displaystyle X} achieved by learning the state of the random variable A {\displaystyle A} . In machine learning, this concept can be used to define a preferred sequence of attributes to investigate to most rapidly narrow down the state of X. Such a sequence (which depends on the outcome of the investigation of previous attributes at each stage) is called a decision tree, and when applied in the area of machine learning is known as decision tree learning. Usually an attribute with high mutual information should be preferred to other attributes. == General definition == In general terms, the expected information gain is the reduction in information entropy Η from a prior state to a state that takes some information as given: I G ( T , a ) = H ( T ) − H ( T | a ) , {\displaystyle IG(T,a)=\mathrm {H} {(T)}-\mathrm {H} {(T|a)},} where H ( T | a ) {\displaystyle \mathrm {H} {(T|a)}} is the conditional entropy of T {\displaystyle T} given the value of attribute a {\displaystyle a} . This is intuitively plausible when interpreting entropy Η as a measure of uncertainty of a random variable T {\displaystyle T} : by learning (or assuming) a {\displaystyle a} about T {\displaystyle T} , our uncertainty about T {\displaystyle T} is reduced (i.e. I G ( T , a ) {\displaystyle IG(T,a)} is positive), unless of course T {\displaystyle T} is independent of a {\displaystyle a} , in which case H ( T | a ) = H ( T ) {\displaystyle \mathrm {H} (T|a)=\mathrm {H} (T)} , meaning I G ( T , a ) = 0 {\displaystyle IG(T,a)=0} . == Formal definition == Let T denote a set of training examples, each of the form ( x , y ) = ( x 1 , x 2 , x 3 , . . . , x k , y ) {\displaystyle ({\textbf {x}},y)=(x_{1},x_{2},x_{3},...,x_{k},y)} where x a ∈ v a l s ( a ) {\displaystyle x_{a}\in \mathrm {vals} (a)} is the value of the a th {\displaystyle a^{\text{th}}} attribute or feature of example x {\displaystyle {\textbf {x}}} and y is the corresponding class label. The information gain for an attribute a is defined in terms of Shannon entropy H ( − ) {\displaystyle \mathrm {H} (-)} as follows. For a value v taken by attribute a, let S a ( v ) = { x ∈ T | x a = v } {\displaystyle S_{a}{(v)}=\{{\textbf {x}}\in T|x_{a}=v\}} be defined as the set of training inputs of T for which attribute a is equal to v. Then the information gain of T for attribute a is the difference between the a priori Shannon entropy H ( T ) {\displaystyle \mathrm {H} (T)} of the training set and the conditional entropy H ( T | a ) {\displaystyle \mathrm {H} {(T|a)}} . H ( T | a ) = ∑ v ∈ v a l s ( a ) | S a ( v ) | | T | ⋅ H ( S a ( v ) ) . {\displaystyle \mathrm {H} (T|a)=\sum _{v\in \mathrm {vals} (a)}{{\frac {|S_{a}{(v)}|}{|T|}}\cdot \mathrm {H} \left(S_{a}{\left(v\right)}\right)}.} I G ( T , a ) = H ( T ) − H ( T | a ) {\displaystyle IG(T,a)=\mathrm {H} (T)-\mathrm {H} (T|a)} The mutual information is equal to the total entropy for an attribute if for each of the attribute values a unique classification can be made for the result attribute. In this case, the relative entropies subtracted from the total entropy are 0. In particular, the values v ∈ v a l s ( a ) {\displaystyle v\in vals(a)} defines a partition of the training set data T into mutually exclusive and all-inclusive subsets, inducing a categorical probability distribution P a ( v ) {\textstyle P_{a}{(v)}} on the values v ∈ v a l s ( a ) {\textstyle v\in vals(a)} of attribute a. The distribution is given P a ( v ) := | S a ( v ) | | T | {\textstyle P_{a}{(v)}:={\frac {|S_{a}{(v)}|}{|T|}}} . In this representation, the information gain of T given a can be defined as the difference between the unconditional Shannon entropy of T and the expected entropy of T conditioned on a, where the expectation value is taken with respect to the induced distribution on the values of a. I G ( T , a ) = H ( T ) − ∑ v ∈ v a l s ( a ) P a ( v ) H ( S a ( v ) ) = H ( T ) − E P a [ H ( S a ( v ) ) ] = H ( T ) − H ( T | a ) . {\displaystyle {\begin{alignedat}{2}IG(T,a)&=\mathrm {H} (T)-\sum _{v\in \mathrm {vals} (a)}{P_{a}{(v)}\mathrm {H} \left(S_{a}{(v)}\right)}\\&=\mathrm {H} (T)-\mathbb {E} _{P_{a}}{\left[\mathrm {H} {(S_{a}{(v)})}\right]}\\&=\mathrm {H} (T)-\mathrm {H} {(T|a)}.\end{alignedat}}} == Example == In engineering applications, information is analogous to signal, and entropy is analogous to noise. It determines how a decision tree chooses to split data. The leftmost figure below is very impure and has high entropy corresponding to higher disorder and lower information value. As we go to the right, the entropy decreases, and the information value increases. Now, it is clear that information gain is the measure of how much information a feature provides about a class. Let's visualize information gain in a decision tree as shown in the right: The node t is the parent node, and the sub-nodes tL and tR are child nodes. In this case, the parent node t has a collection of cancer and non-cancer samples denoted as C and NC respectively. We can use information gain to determine how good the splitting of nodes is in a decision tree. In terms of entropy, information gain is defined as: To understand this idea, let's start by an example in which we create a simple dataset and want to see if gene mutations could be related to patients with cancer. Given four different gene mutations, as well as seven samples, the training set for a decision can be created as follows: In this dataset, a 1 means the sample has the mutation (True), while a 0 means the sample does not (False). A sample with C denotes that it has been confirmed to be cancerous, while NC means it is non-cancerous. Using this data, a decision tree can be created with information gain used to determine the candidate splits for each node. For the next step, the entropy at parent node t of the above simple decision tree is computed as:H(t) = −[pC,t log2(pC,t) + pNC,t log2(pNC,t)] where, probability of selecting a class ‘C’ sample at node t, pC,t = n(t, C) / n(t), probability of selecting a class ‘NC’ sample at node t, pNC,t = n(t, NC) / n(t), n(t), n(t, C), and n(t, NC) are the number of total samples, ‘C’ samples and ‘NC’ samples at node t respectively.Using this with the example training set, the process for finding information gain beginning with H ( t ) {\displaystyle \mathrm {H} {(t)}} for Mutation 1 is as follows: pC, t = 4/7 pNC, t = 3/7 H ( t ) {\displaystyle \mathrm {H} {(t)}} = −(4/7 × log2(4/7) + 3/7 × log2(3/7)) = 0.985 Note: H ( t ) {\displaystyle \mathrm {H} {(t)}} will be the same for all mutations at the root. The relatively high value of entropy H ( t ) = 0.985 {\displaystyle \mathrm {H} {(t)}=0.985} (1 is the optimal value) suggests that the root node is highly impure and the constituents of the input at the root node would look like the leftmost figure in the above Entropy Diagram. However, such a set of data is good for learning the attributes of the mutations used to split the node. At a certain node, when the homogeneity of the constituents of the input occurs (as shown in the rightmost figure in the above Entropy Diagram), the dataset would no longer be good for learning. Moving on, the entropy at left and right child nodes of the above decision tree is computed using the formulae:H(tL) = −[pC,L log2(pC,L) + pNC,L log2(pNC,L)]H(tR) = −[pC,R log2(pC,R) + pNC,R log2(pNC,R)]where, probability of selecting a class ‘C’ sample at the left child node, pC,L = n(tL, C) / n(tL), probability of selecting a class ‘NC’ sample at the left child node, pNC,L = n(tL, NC) / n(tL), probability of selecting a class ‘C’ sample at the right child node, pC,R = n(tR, C) / n(tR), prob

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  Win–stay, lose–switch

  In psychology, game theory, statistics, and machine learning, win–stay, lose–switch (also win–stay, lose–shift or Pavlov, named after Ivan Pavlov) is a heuristic learning strategy used to model learning in decision situations. It was first invented as an improvement over randomization in bandit problems. It was later applied to the prisoner's dilemma in order to model the evolution of altruism. In most versions, it starts either with a cooperate, then proceeds as always, or starts with a "probe" of cooperate-defect-cooperate to determine the other player's strategy. A mutual cooperation is regarded as a win. The learning rule bases its decision only on the outcome of the previous play. Outcomes are divided into successes (wins) and failures (losses). If the play on the previous round resulted in a success, then the agent plays the same strategy on the next round. Alternatively, if the play resulted in a failure the agent switches to another action. A large-scale empirical study of players of the game rock, paper, scissors shows that a variation of this strategy is adopted by real-world players of the game, instead of the Nash equilibrium strategy of choosing entirely at random between the three options.

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  SlideRocket

  SlideRocket was an online presentation platform that let users create, manage, share and measure presentations. SlideRocket was provided via a SaaS model. The company was acquired by VMware in April 2011, who sold it to ClearSlide, a similar SaaS application, in March 2013. It is no longer offering independent signups, as the platform is being integrated into ClearSlide. == History == SlideRocket was founded in Jan 2006, and launched as a private beta in March 2008 at the Under The Radar Spring event. A public beta was announced in September 2008 followed shortly by public release on October 28, 2008. SlideRocket is most commonly credited with inventing the PResuMÉ or Presentation Résumé in early 2009. On April 26, 2011, SlideRocket was acquired by VMware. On March 5, 2013, VMware sold SlideRocket to ClearSlide. SlideRocket is based in San Francisco.

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  T-distributed stochastic neighbor embedding

  t-distributed stochastic neighbor embedding (t-SNE) is a statistical method for visualizing high-dimensional data by giving each datapoint a location in a two or three-dimensional map. It is based on Stochastic Neighbor Embedding originally developed by Geoffrey Hinton and Sam Roweis, where Laurens van der Maaten and Hinton proposed the t-distributed variant. It is a nonlinear dimensionality reduction technique for embedding high-dimensional data for visualization in a low-dimensional space of two or three dimensions. Specifically, it models each high-dimensional object by a two- or three-dimensional point in such a way that similar objects are modeled by nearby points and dissimilar objects are modeled by distant points with high probability. The t-SNE algorithm comprises two main stages. First, t-SNE constructs a probability distribution over pairs of high-dimensional objects in such a way that similar objects are assigned a higher probability while dissimilar points are assigned a lower probability. Second, t-SNE defines a similar probability distribution over the points in the low-dimensional map, and it minimizes the Kullback–Leibler divergence (KL divergence) between the two distributions with respect to the locations of the points in the map. While the original algorithm uses the Euclidean distance between objects as the base of its similarity metric, this can be changed as appropriate. A Riemannian variant is UMAP. t-SNE has been used for visualization in a wide range of applications, including genomics, computer security research, natural language processing, music analysis, cancer research, bioinformatics, geological domain interpretation, and biomedical signal processing. For a data set with n {\displaystyle n} elements, t-SNE runs in O ( n 2 ) {\displaystyle O(n^{2})} time and requires O ( n 2 ) {\displaystyle O(n^{2})} space. == Details == Given a set of N {\displaystyle N} high-dimensional objects x 1 , … , x N {\displaystyle \mathbf {x} _{1},\dots ,\mathbf {x} _{N}} , t-SNE first computes probabilities p i j {\displaystyle p_{ij}} that are proportional to the similarity of objects x i {\displaystyle \mathbf {x} _{i}} and x j {\displaystyle \mathbf {x} _{j}} , as follows. For i ≠ j {\displaystyle i\neq j} , define p j ∣ i = exp ⁡ ( − ‖ x i − x j ‖ 2 / 2 σ i 2 ) ∑ k ≠ i exp ⁡ ( − ‖ x i − x k ‖ 2 / 2 σ i 2 ) {\displaystyle p_{j\mid i}={\frac {\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{j}\rVert ^{2}/2\sigma _{i}^{2})}{\sum _{k\neq i}\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{k}\rVert ^{2}/2\sigma _{i}^{2})}}} and set p i ∣ i = 0 {\displaystyle p_{i\mid i}=0} . Note the above denominator ensures ∑ j p j ∣ i = 1 {\displaystyle \sum _{j}p_{j\mid i}=1} for all i {\displaystyle i} . As van der Maaten and Hinton explained: "The similarity of datapoint x j {\displaystyle x_{j}} to datapoint x i {\displaystyle x_{i}} is the conditional probability, p j | i {\displaystyle p_{j|i}} , that x i {\displaystyle x_{i}} would pick x j {\displaystyle x_{j}} as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at x i {\displaystyle x_{i}} ." Now define p i j = p j ∣ i + p i ∣ j 2 N {\displaystyle p_{ij}={\frac {p_{j\mid i}+p_{i\mid j}}{2N}}} This is motivated because p i {\displaystyle p_{i}} and p j {\displaystyle p_{j}} from the N samples are estimated as 1/N, so the conditional probability can be written as p i ∣ j = N p i j {\displaystyle p_{i\mid j}=Np_{ij}} and p j ∣ i = N p j i {\displaystyle p_{j\mid i}=Np_{ji}} . Since p i j = p j i {\displaystyle p_{ij}=p_{ji}} , you can obtain previous formula. Also note that p i i = 0 {\displaystyle p_{ii}=0} and ∑ i , j p i j = 1 {\displaystyle \sum _{i,j}p_{ij}=1} . The bandwidth of the Gaussian kernels σ i {\displaystyle \sigma _{i}} is set in such a way that the entropy of the conditional distribution equals a predefined entropy using the bisection method. As a result, the bandwidth is adapted to the density of the data: smaller values of σ i {\displaystyle \sigma _{i}} are used in denser parts of the data space. The entropy increases with the perplexity of this distribution P i {\displaystyle P_{i}} ; this relation is seen as P e r p ( P i ) = 2 H ( P i ) {\displaystyle Perp(P_{i})=2^{H(P_{i})}} where H ( P i ) {\displaystyle H(P_{i})} is the Shannon entropy H ( P i ) = − ∑ j p j | i log 2 ⁡ p j | i . {\displaystyle H(P_{i})=-\sum _{j}p_{j|i}\log _{2}p_{j|i}.} The perplexity is a hand-chosen parameter of t-SNE, and as the authors state, "perplexity can be interpreted as a smooth measure of the effective number of neighbors. The performance of SNE is fairly robust to changes in the perplexity, and typical values are between 5 and 50.". Since the Gaussian kernel uses the Euclidean distance ‖ x i − x j ‖ {\displaystyle \lVert x_{i}-x_{j}\rVert } , it is affected by the curse of dimensionality, and in high dimensional data when distances lose the ability to discriminate, the p i j {\displaystyle p_{ij}} become too similar (asymptotically, they would converge to a constant). It has been proposed to adjust the distances with a power transform, based on the intrinsic dimension of each point, to alleviate this. t-SNE aims to learn a d {\displaystyle d} -dimensional map y 1 , … , y N {\displaystyle \mathbf {y} _{1},\dots ,\mathbf {y} _{N}} (with y i ∈ R d {\displaystyle \mathbf {y} _{i}\in \mathbb {R} ^{d}} and d {\displaystyle d} typically chosen as 2 or 3) that reflects the similarities p i j {\displaystyle p_{ij}} as well as possible. To this end, it measures similarities q i j {\displaystyle q_{ij}} between two points in the map y i {\displaystyle \mathbf {y} _{i}} and y j {\displaystyle \mathbf {y} _{j}} , using a very similar approach. Specifically, for i ≠ j {\displaystyle i\neq j} , define q i j {\displaystyle q_{ij}} as q i j = ( 1 + ‖ y i − y j ‖ 2 ) − 1 ∑ k ∑ l ≠ k ( 1 + ‖ y k − y l ‖ 2 ) − 1 {\displaystyle q_{ij}={\frac {(1+\lVert \mathbf {y} _{i}-\mathbf {y} _{j}\rVert ^{2})^{-1}}{\sum _{k}\sum _{l\neq k}(1+\lVert \mathbf {y} _{k}-\mathbf {y} _{l}\rVert ^{2})^{-1}}}} and set q i i = 0 {\displaystyle q_{ii}=0} . Herein a heavy-tailed Student t-distribution (with one-degree of freedom, which is the same as a Cauchy distribution) is used to measure similarities between low-dimensional points in order to allow dissimilar objects to be modeled far apart in the map. The locations of the points y i {\displaystyle \mathbf {y} _{i}} in the map are determined by minimizing the (non-symmetric) Kullback–Leibler divergence of the distribution P {\displaystyle P} from the distribution Q {\displaystyle Q} , that is: K L ( P ∥ Q ) = ∑ i ≠ j p i j log ⁡ p i j q i j {\displaystyle \mathrm {KL} \left(P\parallel Q\right)=\sum _{i\neq j}p_{ij}\log {\frac {p_{ij}}{q_{ij}}}} The minimization of the Kullback–Leibler divergence with respect to the points y i {\displaystyle \mathbf {y} _{i}} is performed using gradient descent. The result of this optimization is a map that reflects the similarities between the high-dimensional inputs. == Output == While t-SNE plots often seem to display clusters, the visual clusters can be strongly influenced by the chosen parameterization (especially the perplexity) and so a good understanding of the parameters for t-SNE is needed. Such "clusters" can be shown to even appear in structured data with no clear clustering, and so may be false findings. Similarly, the size of clusters produced by t-SNE is not informative, and neither is the distance between clusters. Thus, interactive exploration may be needed to choose parameters and validate results. It has been shown that t-SNE can often recover well-separated clusters, and with special parameter choices, approximates a simple form of spectral clustering. == Software == A C++ implementation of Barnes-Hut is available on the github account of one of the original authors. The R package Rtsne implements t-SNE in R. ELKI contains tSNE, also with Barnes-Hut approximation scikit-learn, a popular machine learning library in Python implements t-SNE with both exact solutions and the Barnes-Hut approximation. Tensorboard, the visualization kit associated with TensorFlow, also implements t-SNE (online version) The Julia package TSne implements t-SNE

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  ARKA descriptors in QSAR

  In computational chemistry and cheminformatics, ARKA descriptors in QSAR are a class of molecular descriptors used in quantitative structure–activity relationship (QSAR) modeling (or related approaches such as QSPR and QSTR), a computational method for predicting the biological activity or toxicity of chemical compounds based on their molecular structure. Molecular descriptors are numerical values that summarize information about a molecule's structure, topology, geometry, or physicochemical properties in a form suitable for machine learning or statistical modeling. ARKA (Arithmetic Residuals in K-Groups Analysis) descriptors differ from traditional descriptors by encoding atomic-level information through recursive autoregression techniques, which aim to capture subtle structural patterns and improve predictive accuracy. They are designed to be both interpretable and well-suited to modeling nonlinear relationships in QSAR studies. == Comparisons == While QSAR is essentially a similarity-based approach, the occurrence of activity/property cliffs may greatly reduce the predictive accuracy of the developed models. The novel Arithmetic Residuals in K-groups Analysis (ARKA) approach is a supervised dimensionality reduction technique developed by the DTC Laboratory, Jadavpur University that can easily identify activity cliffs in a data set. Activity cliffs are similar in their structures but differ considerably in their activity. The basic idea of the ARKA descriptors is to group the conventional QSAR descriptors based on a predefined criterion and then assign weightage to each descriptor in each group. ARKA descriptors have also been used to develop classification-based and regression-based QSAR models with acceptable quality statistics. The ARKA descriptors have been used for the identification of activity cliffs in QSAR studies and/or model development by multiple researchers. A tutorial presentation on the ARKA descriptors is available. Recently a multi-class ARKA framework has been proposed for improved q-RASAR model generation.

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  Optimal discriminant analysis and classification tree analysis

  Optimal Discriminant Analysis (ODA) and the related classification tree analysis (CTA) are exact statistical methods that maximize predictive accuracy. For any specific sample and exploratory or confirmatory hypothesis, optimal discriminant analysis (ODA) identifies the statistical model that yields maximum predictive accuracy, assesses the exact Type I error rate, and evaluates potential cross-generalizability. Optimal discriminant analysis may be applied to > 0 dimensions, with the one-dimensional case being referred to as UniODA and the multidimensional case being referred to as MultiODA. Optimal discriminant analysis is an alternative to ANOVA (analysis of variance) and regression analysis.

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  Halite AI Programming Competition

  Halite is an open-source computer programming contest developed by the hedge fund/tech firm Two Sigma in partnership with a team at Cornell Tech. Programmers can see the game environment and learn everything they need to know about the game. Participants are asked to build bots in whichever language they choose to compete on a two-dimensional virtual battle field. == History == Benjamin Spector and Michael Truell created the first Halite competition in 2016, before partnering with Two Sigma later that year. === Halite I === Halite I asked participants to conquer territory on a grid. It launched in November 2016 and ended in February 2017. Halite I attracted about 1,500 players. === Halite II === Halite II was similar to Halite I, but with a space-war theme. It ran from October 2017 until January 2018. The second installment of the competition attracted about 6,000 individual players from more than 100 countries. Among the participants were professors, physicists and NASA engineers, as well as high school and university students. === Halite III === Halite III launched in mid-October 2018. It ran from October 2018 to January 2019, with an ocean themed playing field. Players were asked to collect and manage Halite, an energy resource. By the end of the competition, Halite III included more than 4000 players and 460 organizations. === Halite IV === Halite IV was hosted by Kaggle, and launched in mid-June 2020.

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  Autoencoder

  An autoencoder is a type of artificial neural network used to learn efficient codings of unlabeled data (unsupervised learning). An autoencoder learns two functions: an encoding function that transforms the input data, and a decoding function that recreates the input data from the encoded representation. The autoencoder learns an efficient representation (encoding) for a set of data, typically for dimensionality reduction, to generate lower-dimensional embeddings for subsequent use by other machine learning algorithms. Variants exist which aim to make the learned representations assume useful properties. Examples are regularized autoencoders (sparse, denoising and contractive autoencoders), which are effective in learning representations for subsequent classification tasks, and variational autoencoders, which can be used as generative models. Autoencoders are applied to many problems, including facial recognition, feature detection, anomaly detection, and learning the meaning of words. In terms of data synthesis, autoencoders can also be used to randomly generate new data that is similar to the input (training) data. == Mathematical principles == === Definition === An autoencoder is defined by the following components: Two sets: the space of encoded messages Z {\displaystyle {\mathcal {Z}}} ; the space of decoded messages X {\displaystyle {\mathcal {X}}} . Typically X {\displaystyle {\mathcal {X}}} and Z {\displaystyle {\mathcal {Z}}} are Euclidean spaces, that is, X = R m , Z = R n {\displaystyle {\mathcal {X}}=\mathbb {R} ^{m},{\mathcal {Z}}=\mathbb {R} ^{n}} with m > n . {\displaystyle m>n.} Two parametrized families of functions: the encoder family E ϕ : X → Z {\displaystyle E_{\phi }:{\mathcal {X}}\rightarrow {\mathcal {Z}}} , parametrized by ϕ {\displaystyle \phi } ; the decoder family D θ : Z → X {\displaystyle D_{\theta }:{\mathcal {Z}}\rightarrow {\mathcal {X}}} , parametrized by θ {\displaystyle \theta } .For any x ∈ X {\displaystyle x\in {\mathcal {X}}} , we usually write z = E ϕ ( x ) {\displaystyle z=E_{\phi }(x)} , and refer to it as the code, the latent variable, latent representation, latent vector, etc. Conversely, for any z ∈ Z {\displaystyle z\in {\mathcal {Z}}} , we usually write x ′ = D θ ( z ) {\displaystyle x'=D_{\theta }(z)} , and refer to it as the (decoded) message. Usually, both the encoder and the decoder are defined as multilayer perceptrons (MLPs). For example, a one-layer-MLP encoder E ϕ {\displaystyle E_{\phi }} is: E ϕ ( x ) = σ ( W x + b ) {\displaystyle E_{\phi }(\mathbf {x} )=\sigma (Wx+b)} where σ {\displaystyle \sigma } is an element-wise activation function, W {\displaystyle W} is a "weight" matrix, and b {\displaystyle b} is a "bias" vector. === Training an autoencoder === An autoencoder, by itself, is simply a tuple of two functions. To judge its quality, we need a task. A task is defined by a reference probability distribution μ r e f {\displaystyle \mu _{ref}} over X {\displaystyle {\mathcal {X}}} , and a "reconstruction quality" function d : X × X → [ 0 , ∞ ] {\displaystyle d:{\mathcal {X}}\times {\mathcal {X}}\to [0,\infty ]} , such that d ( x , x ′ ) {\displaystyle d(x,x')} measures how much x ′ {\displaystyle x'} differs from x {\displaystyle x} . With those, we can define the loss function for the autoencoder as L ( θ , ϕ ) := E x ∼ μ r e f [ d ( x , D θ ( E ϕ ( x ) ) ) ] {\displaystyle L(\theta ,\phi ):=\mathbb {\mathbb {E} } _{x\sim \mu _{ref}}[d(x,D_{\theta }(E_{\phi }(x)))]} The optimal autoencoder for the given task ( μ r e f , d ) {\displaystyle (\mu _{ref},d)} is then arg ⁡ min θ , ϕ L ( θ , ϕ ) {\displaystyle \arg \min _{\theta ,\phi }L(\theta ,\phi )} . The search for the optimal autoencoder can be accomplished by any mathematical optimization technique, but usually by gradient descent. This search process is referred to as "training the autoencoder". In most situations, the reference distribution is just the empirical distribution given by a dataset { x 1 , . . . , x N } ⊂ X {\displaystyle \{x_{1},...,x_{N}\}\subset {\mathcal {X}}} , so that μ r e f = 1 N ∑ i = 1 N δ x i {\displaystyle \mu _{ref}={\frac {1}{N}}\sum _{i=1}^{N}\delta _{x_{i}}} where δ x i {\displaystyle \delta _{x_{i}}} is the Dirac measure, the quality function is just L 2 {\displaystyle L^{2}} loss: d ( x , x ′ ) = ‖ x − x ′ ‖ 2 2 {\displaystyle d(x,x')=\|x-x'\|_{2}^{2}} , and ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} is the Euclidean norm. Then the problem of searching for the optimal autoencoder is just a least-squares optimization: min θ , ϕ L ( θ , ϕ ) , where L ( θ , ϕ ) = 1 N ∑ i = 1 N ‖ x i − D θ ( E ϕ ( x i ) ) ‖ 2 2 {\displaystyle \min _{\theta ,\phi }L(\theta ,\phi ),\qquad {\text{where }}L(\theta ,\phi )={\frac {1}{N}}\sum _{i=1}^{N}\|x_{i}-D_{\theta }(E_{\phi }(x_{i}))\|_{2}^{2}} === Interpretation === An autoencoder has two main parts: an encoder that maps the message to a code, and a decoder that reconstructs the message from the code. An optimal autoencoder would perform as close to perfect reconstruction as possible, with "close to perfect" defined by the reconstruction quality function d {\displaystyle d} . The simplest way to perform the copying task perfectly would be to duplicate the signal. To suppress this behavior, the code space Z {\displaystyle {\mathcal {Z}}} usually has fewer dimensions than the message space X {\displaystyle {\mathcal {X}}} . Such an autoencoder is called undercomplete. It can be interpreted as compressing the message, or reducing its dimensionality. At the limit of an ideal undercomplete autoencoder, every possible code z {\displaystyle z} in the code space is used to encode a message x {\displaystyle x} that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} , and the decoder is also perfect: D θ ( E ϕ ( x ) ) = x {\displaystyle D_{\theta }(E_{\phi }(x))=x} . This ideal autoencoder can then be used to generate messages indistinguishable from real messages, by feeding its decoder arbitrary code z {\displaystyle z} and obtaining D θ ( z ) {\displaystyle D_{\theta }(z)} , which is a message that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} . If the code space Z {\displaystyle {\mathcal {Z}}} has dimension larger than (overcomplete), or equal to, the message space X {\displaystyle {\mathcal {X}}} , or the hidden units are given enough capacity, an autoencoder can learn the identity function and become useless. However, experimental results found that overcomplete autoencoders might still learn useful features. In the ideal setting, the code dimension and the model capacity could be set on the basis of the complexity of the data distribution to be modeled. A standard way to do so is to add modifications to the basic autoencoder, to be detailed below. == Variations == === Variational autoencoder (VAE) === Variational autoencoders (VAEs) belong to the families of variational Bayesian methods. Despite the architectural similarities with basic autoencoders, VAEs are architected with different goals and have a different mathematical formulation. The latent space is, in this case, composed of a mixture of distributions instead of fixed vectors. Given an input dataset x {\displaystyle x} characterized by an unknown probability function P ( x ) {\displaystyle P(x)} and a multivariate latent encoding vector z {\displaystyle z} , the objective is to model the data as a distribution p θ ( x ) {\displaystyle p_{\theta }(x)} , with θ {\displaystyle \theta } defined as the set of the network parameters so that p θ ( x ) = ∫ z p θ ( x , z ) d z {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }(x,z)dz} . === Sparse autoencoder (SAE) === Inspired by the sparse coding hypothesis in neuroscience, sparse autoencoders (SAE) are variants of autoencoders, such that the codes E ϕ ( x ) {\displaystyle E_{\phi }(x)} for messages tend to be sparse codes, that is, E ϕ ( x ) {\displaystyle E_{\phi }(x)} is close to zero in most entries. Sparse autoencoders may include more (rather than fewer) hidden units than inputs, but only a small number of the hidden units are allowed to be active at the same time. Encouraging sparsity improves performance on classification tasks. There are two main ways to enforce sparsity. One way is to simply clamp all but the highest-k activations of the latent code to zero. This is the k-sparse autoencoder. The k-sparse autoencoder inserts the following "k-sparse function" in the latent layer of a standard autoencoder: f k ( x 1 , . . . , x n ) = ( x 1 b 1 , . . . , x n b n ) {\displaystyle f_{k}(x_{1},...,x_{n})=(x_{1}b_{1},...,x_{n}b_{n})} where b i = 1 {\displaystyle b_{i}=1} if | x i | {\displaystyle |x_{i}|} ranks in the top k, and 0 otherwise. Backpropagating through f k {\displaystyle f_{k}} is simple: set gradient to 0 for b i = 0 {\displaystyle b_{i}=0} entries, and keep gradient for b i = 1 {\displaystyle b_{i}=1} entries. This is essentially a generalized ReLU function. The other way is a relaxed version of the k-

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  Tensor product network

  A tensor product network, in artificial neural networks, is a network that exploits the properties of tensors to model associative concepts such as variable assignment. Orthonormal vectors are chosen to model the ideas (such as variable names and target assignments), and the tensor product of these vectors construct a network whose mathematical properties allow the user to easily extract the association from it.

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  Exploratory blockmodeling

  Exploratory blockmodeling is an (inductive) approach (or a group of approaches) in blockmodeling regarding the specification of an ideal blockmodel. This approach, also known as hypotheses-generating, is the simplest approach, as it "merely involves the definition of the block types permitted as well as of the number of clusters." With this approach, researcher usually defines the best possible blockmodel, which then represent the base for the analysis of the whole network. This approach is usually based on: previous analyses and theoretical considerations, using stricker blockmodel and block types, where the structural equivalence is stricker than the regular equivalence and using smaller number of classes. The opposite approach is called a confirmatory blockmodeling.

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  Medical imaging

  Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to reveal internal structures hidden by the skin and bones, as well as to diagnose and treat disease. Medical imaging also establishes a database of normal anatomy and physiology to make it possible to identify abnormalities. Although imaging of removed organs and tissues can be performed for medical reasons, such procedures are usually considered part of pathology instead of medical imaging. Measurement and recording techniques that are not primarily designed to produce images, such as electroencephalography (EEG), magnetoencephalography (MEG), electrocardiography (ECG), and others, represent other technologies that produce data susceptible to representation as a parameter graph versus time or maps that contain data about the measurement locations. In a limited comparison, these technologies can be considered forms of medical imaging in another discipline of medical instrumentation. As of 2010, 5 billion medical imaging studies had been conducted worldwide. Radiation exposure from medical imaging in 2006 made up about 50% of total ionizing radiation exposure in the United States. Medical imaging equipment is manufactured using technology from the semiconductor industry, including CMOS integrated circuit chips, power semiconductor devices, sensors such as image sensors (particularly CMOS sensors) and biosensors, and processors such as microcontrollers, microprocessors, digital signal processors, media processors and system-on-chip devices. As of 2015, annual shipments of medical imaging chips amount to 46 million units and $1.1 billion. The term "noninvasive" is used to denote a procedure where no instrument is introduced into a patient's body, which is the case for most imaging techniques used. == History == In 1972, engineer Godfrey Hounsfield from the British company EMI invented the X-ray computed tomography device for head diagnosis, which is commonly referred to as computed tomography (CT). The CT nucleus method is based on the projecting X-rays through a section of the human head, which are then processed by computer to reconstruct the cross-sectional image, known as image reconstruction. In 1975, EMI successfully developed a CT device for the entire body, enabling the clear acquisition of tomographic images of various parts of the human body. This revolutionary diagnostic technique earned Hounsfield and physicist Allan Cormack the Nobel Prize in Physiology or Medicine in 1979. Digital image processing technology for medical applications was inducted into the Space Foundation's Space Technology Hall of Fame in 1994. By 2010, over 5 billion medical imaging studies had been conducted worldwide. Radiation exposure from medical imaging in 2006 accounted for about 50% of total ionizing radiation exposure in the United States. Medical imaging equipment is manufactured using technology from the semiconductor industry, including CMOS integrated circuit chips, power semiconductor devices, sensors such as image sensors (particularly CMOS sensors) and biosensors, as well as processors like microcontrollers, microprocessors, digital signal processors, media processors and system-on-chip devices. As of 2015, annual shipments of medical imaging chips reached 46 million units, generating a market value of $1.1 billion. == Types == In the clinical context, "invisible light" medical imaging is generally equated to radiology or "clinical imaging". "Visible light" medical imaging involves digital video or still pictures that can be seen without special equipment. Dermatology and wound care are two modalities that use visible light imagery. Interpretation of medical images is generally undertaken by a physician specialising in radiology known as a radiologist; however, this may be undertaken by any healthcare professional who is trained and certified in radiological clinical evaluation. Increasingly interpretation is being undertaken by non-physicians, for example radiographers frequently train in interpretation as part of expanded practice. Diagnostic radiography designates the technical aspects of medical imaging and in particular the acquisition of medical images. The radiographer (also known as a radiologic technologist) is usually responsible for acquiring medical images of diagnostic quality; although other professionals may train in this area, notably some radiological interventions performed by radiologists are done so without a radiographer. As a field of scientific investigation, medical imaging constitutes a sub-discipline of biomedical engineering, medical physics or medicine depending on the context: Research and development in the area of instrumentation, image acquisition (e.g., radiography), modeling and quantification are usually the preserve of biomedical engineering, medical physics, and computer science; Research into the application and interpretation of medical images is usually the preserve of radiology and the medical sub-discipline relevant to medical condition or area of medical science (neuroscience, cardiology, psychiatry, psychology, etc.) under investigation. Many of the techniques developed for medical imaging also have scientific and industrial applications. === Radiography === Two forms of radiographic images are in use in medical imaging. Projection radiography and fluoroscopy, with the latter being useful for catheter guidance. These 2D techniques are still in wide use despite the advance of 3D tomography due to the low cost, high resolution, and depending on the application, lower radiation dosages with 2D technique. This imaging modality uses a wide beam of X-rays for image acquisition and is the first imaging technique available in modern medicine. Fluoroscopy produces real-time images of internal structures of the body in a similar fashion to radiography, but employs a constant input of X-rays, at a lower dose rate. Contrast media, such as barium, iodine, and air are used to visualize internal organs as they work. Fluoroscopy is also used in image-guided procedures when constant feedback during a procedure is required. An image receptor is required to convert the radiation into an image after it has passed through the area of interest. Early on, this was a fluorescing screen, which gave way to an Image Amplifier (IA) which was a large vacuum tube that had the receiving end coated with cesium iodide, and a mirror at the opposite end. Eventually the mirror was replaced with a TV camera. Projectional radiographs, more commonly known as X-rays, are often used to determine the type and extent of a fracture as well as for detecting pathological changes in the lungs. With the use of radio-opaque contrast media, such as barium, they can also be used to visualize the structure of the stomach and intestines – this can help diagnose ulcers or certain types of colon cancer. === Magnetic resonance imaging === A magnetic resonance imaging instrument (MRI scanner), or "nuclear magnetic resonance (NMR) imaging" scanner as it was originally known, uses powerful magnets to polarize and excite hydrogen nuclei (i.e., single protons) of water molecules in human tissue, producing a detectable signal that is spatially encoded, resulting in images of the body. The MRI machine emits a radio frequency (RF) pulse at the resonant frequency of the hydrogen atoms on water molecules. Radio frequency antennas ("RF coils") send the pulse to the area of the body to be examined. The RF pulse is absorbed by protons, causing their direction with respect to the primary magnetic field to change. When the RF pulse is turned off, the protons "relax" back to alignment with the primary magnet and emit radio waves in the process. This radio-frequency emission from the hydrogen atoms on water is what is detected and reconstructed into an image. The resonant frequency of a spinning magnetic dipole (of which protons are one example) is called the Larmor frequency and is determined by the strength of the main magnetic field and the chemical environment of the nuclei of interest. MRI uses three electromagnetic fields: a very strong (typically 1.5 to 3 teslas) static magnetic field to polarize the hydrogen nuclei, called the primary field; gradient fields that can be modified to vary in space and time (on the order of 1 kHz) for spatial encoding, often simply called gradients; and a spatially homogeneous radio-frequency (RF) field for manipulation of the hydrogen nuclei to produce measurable signals, collected through an RF antenna. Like CT, MRI traditionally creates a two-dimensional image of a thin "slice" of the body and is therefore considered a tomographic imaging technique. Modern MRI instruments are capable of producing images in the form of 3D blocks, which may be considered a generalization of the single-slice

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  Self-play

  Self-play is a technique for improving the performance of reinforcement learning agents. Intuitively, agents learn to improve their performance by playing "against themselves". == Definition and motivation == In multi-agent reinforcement learning experiments, researchers try to optimize the performance of a learning agent on a given task, in cooperation or competition with one or more agents. These agents learn by trial-and-error, and researchers may choose to have the learning algorithm play the role of two or more of the different agents. When successfully executed, this technique has a double advantage: It provides a straightforward way to determine the actions of the other agents, resulting in a meaningful challenge. It increases the amount of experience that can be used to improve the policy, by a factor of two or more, since the viewpoints of each of the different agents can be used for learning. Czarnecki et al argue that most of the games that people play for fun are "Games of Skill", meaning games whose space of all possible strategies looks like a spinning top. In more detail, we can partition the space of strategies into sets L 1 , L 2 , . . . , L n {\displaystyle L_{1},L_{2},...,L_{n}} , such that any i < j , π i ∈ L i , π j ∈ L j {\displaystyle i Read more →

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  Evolutionary programming

  Evolutionary programming is an evolutionary algorithm, where a share of new population is created by mutation of previous population without crossover. Evolutionary programming differs from evolution strategy ES( μ + λ {\displaystyle \mu +\lambda } ) in one detail. All individuals are selected for the new population, while in ES( μ + λ {\displaystyle \mu +\lambda } ), every individual has the same probability to be selected. It is one of the four major evolutionary algorithm paradigms. == History == It was first used by Lawrence J. Fogel in the US in 1960 in order to use simulated evolution as a learning process aiming to generate artificial intelligence. It was used to evolve finite-state machines as predictors.

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