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  EM algorithm and GMM model

  In statistics, EM (expectation maximization) algorithm handles latent variables, while GMM is the Gaussian mixture model. == Background == In the picture below, are shown the red blood cell hemoglobin concentration and the red blood cell volume data of two groups of people, the Anemia group and the control group (i.e. the group of people without Anemia). As expected, people with Anemia have lower red blood cell volume and lower red blood cell hemoglobin concentration than those without Anemia. x {\displaystyle x} is a random vector such as x := ( red blood cell volume , red blood cell hemoglobin concentration ) {\displaystyle x:={\big (}{\text{red blood cell volume}},{\text{red blood cell hemoglobin concentration}}{\big )}} , and from medical studies it is known that x {\displaystyle x} are normally distributed in each group, i.e. x ∼ N ( μ , Σ ) {\displaystyle x\sim {\mathcal {N}}(\mu ,\Sigma )} . z {\displaystyle z} is denoted as the group where x {\displaystyle x} belongs, with z i = 0 {\displaystyle z_{i}=0} when x i {\displaystyle x_{i}} belongs to the Anemia group and z i = 1 {\displaystyle z_{i}=1} when x i {\displaystyle x_{i}} belongs to the control group. Also z ∼ Categorical ⁡ ( k , ϕ ) {\displaystyle z\sim \operatorname {Categorical} (k,\phi )} where k = 2 {\displaystyle k=2} , ϕ j ≥ 0 , {\displaystyle \phi _{j}\geq 0,} and ∑ j = 1 k ϕ j = 1 {\displaystyle \sum _{j=1}^{k}\phi _{j}=1} . See Categorical distribution. The following procedure can be used to estimate ϕ , μ , Σ {\displaystyle \phi ,\mu ,\Sigma } . A maximum likelihood estimation can be applied: ℓ ( ϕ , μ , Σ ) = ∑ i = 1 m log ⁡ ( p ( x ( i ) ; ϕ , μ , Σ ) ) = ∑ i = 1 m log ⁡ ∑ z ( i ) = 1 k p ( x ( i ) ∣ z ( i ) ; μ , Σ ) p ( z ( i ) ; ϕ ) {\displaystyle \ell (\phi ,\mu ,\Sigma )=\sum _{i=1}^{m}\log(p(x^{(i)};\phi ,\mu ,\Sigma ))=\sum _{i=1}^{m}\log \sum _{z^{(i)}=1}^{k}p\left(x^{(i)}\mid z^{(i)};\mu ,\Sigma \right)p(z^{(i)};\phi )} As the z i {\displaystyle z_{i}} for each x i {\displaystyle x_{i}} are known, the log likelihood function can be simplified as below: ℓ ( ϕ , μ , Σ ) = ∑ i = 1 m log ⁡ p ( x ( i ) ∣ z ( i ) ; μ , Σ ) + log ⁡ p ( z ( i ) ; ϕ ) {\displaystyle \ell (\phi ,\mu ,\Sigma )=\sum _{i=1}^{m}\log p\left(x^{(i)}\mid z^{(i)};\mu ,\Sigma \right)+\log p\left(z^{(i)};\phi \right)} Now the likelihood function can be maximized by making partial derivative over μ , Σ , ϕ {\displaystyle \mu ,\Sigma ,\phi } , obtaining: ϕ j = 1 m ∑ i = 1 m 1 { z ( i ) = j } {\displaystyle \phi _{j}={\frac {1}{m}}\sum _{i=1}^{m}1\{z^{(i)}=j\}} μ j = ∑ i = 1 m 1 { z ( i ) = j } x ( i ) ∑ i = 1 m 1 { z ( i ) = j } {\displaystyle \mu _{j}={\frac {\sum _{i=1}^{m}1\{z^{(i)}=j\}x^{(i)}}{\sum _{i=1}^{m}1\left\{z^{(i)}=j\right\}}}} Σ j = ∑ i = 1 m 1 { z ( i ) = j } ( x ( i ) − μ j ) ( x ( i ) − μ j ) T ∑ i = 1 m 1 { z ( i ) = j } {\displaystyle \Sigma _{j}={\frac {\sum _{i=1}^{m}1\{z^{(i)}=j\}(x^{(i)}-\mu _{j})(x^{(i)}-\mu _{j})^{T}}{\sum _{i=1}^{m}1\{z^{(i)}=j\}}}} If z i {\displaystyle z_{i}} is known, the estimation of the parameters results to be quite simple with maximum likelihood estimation. But if z i {\displaystyle z_{i}} is unknown it is much more complicated. Being z {\displaystyle z} a latent variable (i.e. not observed), with unlabeled scenario, the expectation maximization algorithm is needed to estimate z {\displaystyle z} as well as other parameters. Generally, this problem is set as a GMM since the data in each group is normally distributed. In machine learning, the latent variable z {\displaystyle z} is considered as a latent pattern lying under the data, which the observer is not able to see very directly. x i {\displaystyle x_{i}} is the known data, while ϕ , μ , Σ {\displaystyle \phi ,\mu ,\Sigma } are the parameter of the model. With the EM algorithm, some underlying pattern z {\displaystyle z} in the data x i {\displaystyle x_{i}} can be found, along with the estimation of the parameters. The wide application of this circumstance in machine learning is what makes EM algorithm so important. == EM algorithm in GMM == The EM algorithm consists of two steps: the E-step and the M-step. Firstly, the model parameters and the z ( i ) {\displaystyle z^{(i)}} can be randomly initialized. In the E-step, the algorithm tries to guess the value of z ( i ) {\displaystyle z^{(i)}} based on the parameters, while in the M-step, the algorithm updates the value of the model parameters based on the guess of z ( i ) {\displaystyle z^{(i)}} of the E-step. These two steps are repeated until convergence is reached. The algorithm in GMM is: Repeat until convergence: 1. (E-step) For each i , j {\displaystyle i,j} , set w j ( i ) := p ( z ( i ) = j | x ( i ) ; ϕ , μ , Σ ) {\displaystyle w_{j}^{(i)}:=p\left(z^{(i)}=j|x^{(i)};\phi ,\mu ,\Sigma \right)} 2. (M-step) Update the parameters ϕ j := 1 m ∑ i = 1 m w j ( i ) {\displaystyle \phi _{j}:={\frac {1}{m}}\sum _{i=1}^{m}w_{j}^{(i)}} μ j := ∑ i = 1 m w j ( i ) x ( i ) ∑ i = 1 m w j ( i ) {\displaystyle \mu _{j}:={\frac {\sum _{i=1}^{m}w_{j}^{(i)}x^{(i)}}{\sum _{i=1}^{m}w_{j}^{(i)}}}} Σ j := ∑ i = 1 m w j ( i ) ( x ( i ) − μ j ) ( x ( i ) − μ j ) T ∑ i = 1 m w j ( i ) {\displaystyle \Sigma _{j}:={\frac {\sum _{i=1}^{m}w_{j}^{(i)}\left(x^{(i)}-\mu _{j}\right)\left(x^{(i)}-\mu _{j}\right)^{T}}{\sum _{i=1}^{m}w_{j}^{(i)}}}} With Bayes' rule, the following result is obtained by the E-step: p ( z ( i ) = j | x ( i ) ; ϕ , μ , Σ ) = p ( x ( i ) | z ( i ) = j ; μ , Σ ) p ( z ( i ) = j ; ϕ ) ∑ l = 1 k p ( x ( i ) | z ( i ) = l ; μ , Σ ) p ( z ( i ) = l ; ϕ ) {\displaystyle p\left(z^{(i)}=j|x^{(i)};\phi ,\mu ,\Sigma \right)={\frac {p\left(x^{(i)}|z^{(i)}=j;\mu ,\Sigma \right)p\left(z^{(i)}=j;\phi \right)}{\sum _{l=1}^{k}p\left(x^{(i)}|z^{(i)}=l;\mu ,\Sigma \right)p\left(z^{(i)}=l;\phi \right)}}} According to GMM setting, these following formulas are obtained: p ( x ( i ) | z ( i ) = j ; μ , Σ ) = 1 ( 2 π ) n / 2 | Σ j | 1 / 2 exp ⁡ ( − 1 2 ( x ( i ) − μ j ) T Σ j − 1 ( x ( i ) − μ j ) ) {\displaystyle p\left(x^{(i)}|z^{(i)}=j;\mu ,\Sigma \right)={\frac {1}{(2\pi )^{n/2}\left|\Sigma _{j}\right|^{1/2}}}\exp \left(-{\frac {1}{2}}\left(x^{(i)}-\mu _{j}\right)^{T}\Sigma _{j}^{-1}\left(x^{(i)}-\mu _{j}\right)\right)} p ( z ( i ) = j ; ϕ ) = ϕ j {\displaystyle p\left(z^{(i)}=j;\phi \right)=\phi _{j}} In this way, a switch between the E-step and the M-step is possible, according to the randomly initialized parameters.

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  Learning to rank

  Learning to rank (LTR) or machine-learned ranking (MLR) is the application of machine learning, often supervised, semi-supervised or reinforcement learning, in the construction of ranking models for information retrieval and recommender systems. Training data may, for example, consist of lists of items with some partial order specified between items in each list. This order is typically induced by giving a numerical or ordinal score or a binary judgment (e.g. "relevant" or "not relevant") for each item. The goal of constructing the ranking model is to rank new, unseen lists in a similar way to rankings in the training data. == Applications == === In information retrieval === Ranking is a central part of many information retrieval problems, such as document retrieval, collaborative filtering, sentiment analysis, and online advertising. A possible architecture of a machine-learned search engine is shown in the accompanying figure. Training data consists of queries and documents matching them together with the relevance degree of each match. It may be prepared manually by human assessors (or raters, as Google calls them), who check results for some queries and determine relevance of each result. It is not feasible to check the relevance of all documents, and so typically a technique called pooling is used — only the top few documents, retrieved by some existing ranking models are checked. This technique may introduce selection bias. Alternatively, training data may be derived automatically by analyzing clickthrough logs (i.e. search results which got clicks from users), query chains, or such search engines' features as Google's (since-replaced) SearchWiki. Clickthrough logs can be biased by the tendency of users to click on the top search results on the assumption that they are already well-ranked. Training data is used by a learning algorithm to produce a ranking model which computes the relevance of documents for actual queries. Typically, users expect a search query to complete in a short time (such as a few hundred milliseconds for web search), which makes it impossible to evaluate a complex ranking model on each document in the corpus, and so a two-phase scheme is used. First, a small number of potentially relevant documents are identified using simpler retrieval models which permit fast query evaluation, such as the vector space model, Boolean model, weighted AND, or BM25. This phase is called top- k {\displaystyle k} document retrieval and many heuristics were proposed in the literature to accelerate it, such as using a document's static quality score and tiered indexes. In the second phase, a more accurate but computationally expensive machine-learned model is used to re-rank these documents. === In other areas === Learning to rank algorithms have been applied in areas other than information retrieval: In machine translation for ranking a set of hypothesized translations; In computational biology for ranking candidate 3-D structures in protein structure prediction problems; In recommender systems for identifying a ranked list of related news articles to recommend to a user after he or she has read a current news article. == Feature vectors == For the convenience of MLR algorithms, query-document pairs are usually represented by numerical vectors, which are called feature vectors. Such an approach is sometimes called bag of features and is analogous to the bag of words model and vector space model used in information retrieval for representation of documents. Components of such vectors are called features, factors or ranking signals. They may be divided into three groups (features from document retrieval are shown as examples): Query-independent or static features — those features, which depend only on the document, but not on the query. For example, PageRank or document's length. Such features can be precomputed in off-line mode during indexing. They may be used to compute document's static quality score (or static rank), which is often used to speed up search query evaluation. Query-dependent or dynamic features — those features, which depend both on the contents of the document and the query, such as TF-IDF score or other non-machine-learned ranking functions. Query-level features or query features, which depend only on the query. For example, the number of words in a query. Some examples of features, which were used in the well-known LETOR dataset: TF, TF-IDF, BM25, and language modeling scores of document's zones (title, body, anchors text, URL) for a given query; Lengths and IDF sums of document's zones; Document's PageRank, HITS ranks and their variants. Selecting and designing good features is an important area in machine learning, which is called feature engineering. == Evaluation measures == There are several measures (metrics) which are commonly used to judge how well an algorithm is doing on training data and to compare the performance of different MLR algorithms. Often a learning-to-rank problem is reformulated as an optimization problem with respect to one of these metrics. Examples of ranking quality measures: Mean average precision (MAP); DCG and NDCG; Precision@n, NDCG@n, where "@n" denotes that the metrics are evaluated only on top n documents; Mean reciprocal rank; Kendall's tau; Spearman's rho. DCG and its normalized variant NDCG are usually preferred in academic research when multiple levels of relevance are used. Other metrics such as MAP, MRR and precision, are defined only for binary judgments. Recently, there have been proposed several new evaluation metrics which claim to model user's satisfaction with search results better than the DCG metric: Expected reciprocal rank (ERR); Yandex's pfound. Both of these metrics are based on the assumption that the user is more likely to stop looking at search results after examining a more relevant document, than after a less relevant document. == Approaches == Learning to Rank approaches are often categorized using one of three approaches: pointwise (where individual documents are ranked), pairwise (where pairs of documents are ranked into a relative order), and listwise (where an entire list of documents are ordered). Tie-Yan Liu of Microsoft Research Asia has analyzed existing algorithms for learning to rank problems in his book Learning to Rank for Information Retrieval. He categorized them into three groups by their input spaces, output spaces, hypothesis spaces (the core function of the model) and loss functions: the pointwise, pairwise, and listwise approach. In practice, listwise approaches often outperform pairwise approaches and pointwise approaches. This statement was further supported by a large scale experiment on the performance of different learning-to-rank methods on a large collection of benchmark data sets. In this section, without further notice, x {\displaystyle x} denotes an object to be evaluated, for example, a document or an image, f ( x ) {\displaystyle f(x)} denotes a single-value hypothesis, h ( ⋅ ) {\displaystyle h(\cdot )} denotes a bi-variate or multi-variate function and L ( ⋅ ) {\displaystyle L(\cdot )} denotes the loss function. === Pointwise approach === In this case, it is assumed that each query-document pair in the training data has a numerical or ordinal score. Then the learning-to-rank problem can be approximated by a regression problem — given a single query-document pair, predict its score. Formally speaking, the pointwise approach aims at learning a function f ( x ) {\displaystyle f(x)} predicting the real-value or ordinal score of a document x {\displaystyle x} using the loss function L ( f ; x j , y j ) {\displaystyle L(f;x_{j},y_{j})} . A number of existing supervised machine learning algorithms can be readily used for this purpose. Ordinal regression and classification algorithms can also be used in pointwise approach when they are used to predict the score of a single query-document pair, and it takes a small, finite number of values. === Pairwise approach === In this case, the learning-to-rank problem is approximated by a classification problem — learning a binary classifier h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} that can tell which document is better in a given pair of documents. The classifier shall take two documents as its input and the goal is to minimize a loss function L ( h ; x u , x v , y u , v ) {\displaystyle L(h;x_{u},x_{v},y_{u,v})} . The loss function typically reflects the number and magnitude of inversions in the induced ranking. In many cases, the binary classifier h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} is implemented with a scoring function f ( x ) {\displaystyle f(x)} . As an example, RankNet adapts a probability model and defines h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} as the estimated probability of the document x u {\displaystyle x_{u}} has higher quality than x v {\displaystyle x_{v}} : P u , v ( f ) = CDF ( f ( x u ) − f ( x v ) ) , {\displaystyle P_{u,v}(f)={\text{CDF}

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  Structured sparsity regularization

  Structured sparsity regularization is a class of methods, and an area of research in statistical learning theory, that extend and generalize sparsity regularization learning methods. Both sparsity and structured sparsity regularization methods seek to exploit the assumption that the output variable Y {\displaystyle Y} (i.e., response, or dependent variable) to be learned can be described by a reduced number of variables in the input space X {\displaystyle X} (i.e., the domain, space of features or explanatory variables). Sparsity regularization methods focus on selecting the input variables that best describe the output. Structured sparsity regularization methods generalize and extend sparsity regularization methods, by allowing for optimal selection over structures like groups or networks of input variables in X {\displaystyle X} . Common motivation for the use of structured sparsity methods are model interpretability, high-dimensional learning (where dimensionality of X {\displaystyle X} may be higher than the number of observations n {\displaystyle n} ), and reduction of computational complexity. Moreover, structured sparsity methods allow to incorporate prior assumptions on the structure of the input variables, such as overlapping groups, non-overlapping groups, and acyclic graphs. Examples of uses of structured sparsity methods include face recognition, magnetic resonance image (MRI) processing, socio-linguistic analysis in natural language processing, and analysis of genetic expression in breast cancer. == Definition and related concepts == === Sparsity regularization === Consider the linear kernel regularized empirical risk minimization problem with a loss function V ( y i , f ( x ) ) {\displaystyle V(y_{i},f(x))} and the ℓ 0 {\displaystyle \ell _{0}} "norm" as the regularization penalty: min w ∈ R d 1 n ∑ i = 1 n V ( y i , ⟨ w , x i ⟩ ) + λ ‖ w ‖ 0 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}V(y_{i},\langle w,x_{i}\rangle )+\lambda \|w\|_{0},} where x , w ∈ R d {\displaystyle x,w\in \mathbb {R^{d}} } , and ‖ w ‖ 0 {\displaystyle \|w\|_{0}} denotes the ℓ 0 {\displaystyle \ell _{0}} "norm", defined as the number of nonzero entries of the vector w {\displaystyle w} . f ( x ) = ⟨ w , x i ⟩ {\displaystyle f(x)=\langle w,x_{i}\rangle } is said to be sparse if ‖ w ‖ 0 = s < d {\displaystyle \|w\|_{0}=s 0 {\displaystyle w_{j}>0} . However, as in this case groups may overlap, we take the intersection of the complements of those groups that are not set to zero. This intersection of complements selection criteria implies the modeling choice that we allow some coefficients within a particular group g {\displaystyle g} to be set to zero, while others within the same group g {\displaystyle g} may remain positive. In other words, coefficients within a group may differ depending on the several group memberships that each variable within the group may have. ==== Union of groups: latent group Lasso ==== A different approach is to consider union of groups for variable selection. This approach captures the modeling situation where variables can be selected as long as they belong at least to one group with positive coefficients. This modeling perspective implies that we want to preserve group structure. The formulation of the union of groups approach is also referred to as latent group Lasso, and requires to modify the group ℓ 2 {\displaystyle \ell _{2}} norm considered above and introduce the following regularizer R ( w ) = i n f { ∑ g ‖ w g ‖ g : w = ∑ g = 1 G w ¯ g } {\displaystyle R(w)=inf\left\{\sum _{g}\|w_{g}\|_{g}:w=\sum _{g=1}^{G}{\bar {w}}_{g}\right\}} where w ∈ R d {\displaystyle w\in {\mathbb {R^{d}} }} , w g ∈ G g {\displaystyle w_{g}\in G_{g}} is the vector of coefficients of group g, and w ¯ g ∈ R d {\displaystyle {\bar {w}}_{g}\in {\mathbb {R^{d}} }} is a vector with coefficients w g j {\displaystyle w_{g}^{j}} for all variables j {

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  ELMo

  ELMo (embeddings from language model) is a word embedding method for representing a sequence of words as a corresponding sequence of vectors. It was created by researchers at the Allen Institute for Artificial Intelligence, and University of Washington and first released in February 2018. It is a bidirectional LSTM which takes character-level as inputs and produces word-level embeddings, trained on a corpus of about 30 million sentences and 1 billion words. The architecture of ELMo accomplishes a contextual understanding of tokens. Deep contextualized word representation is useful for many natural language processing tasks, such as coreference resolution and polysemy resolution. ELMo was historically important as a pioneer of self-supervised generative pretraining followed by fine-tuning, where a large model is trained to reproduce a large corpus, then the large model is augmented with additional task-specific weights and fine-tuned on supervised task data. It was an instrumental step in the evolution towards transformer-based language modelling. == Architecture == ELMo is a multilayered bidirectional LSTM on top of a token embedding layer. The output of all LSTMs concatenated together consists of the token embedding. The input text sequence is first mapped by an embedding layer into a sequence of vectors. Then two parts are run in parallel over it. The forward part is a 2-layered LSTM with 4096 units and 512 dimension projections, and a residual connection from the first to second layer. The backward part has the same architecture, but processes the sequence back-to-front. The outputs from all 5 components (embedding layer, two forward LSTM layers, and two backward LSTM layers) are concatenated and multiplied by a linear matrix ("projection matrix") to produce a 512-dimensional representation per input token. ELMo was pretrained on a text corpus of 1 billion words. The forward part is trained by repeatedly predicting the next token, and the backward part is trained by repeatedly predicting the previous token. After the ELMo model is pretrained, its parameters are frozen, except for the projection matrix, which can be fine-tuned to minimize loss on specific language tasks. This is an early example of the pretraining-fine-tune paradigm. The original paper demonstrated this by improving state of the art on six benchmark NLP tasks. === Contextual word representation === The architecture of ELMo accomplishes a contextual understanding of tokens. For example, the first forward LSTM of ELMo would process each input token in the context of all previous tokens, and the first backward LSTM would process each token in the context of all subsequent tokens. The second forward LSTM would then incorporate those to further contextualize each token. Deep contextualized word representation is useful for many natural language processing tasks, such as coreference resolution and polysemy resolution. For example, consider the sentenceShe went to the bank to withdraw money.In order to represent the token "bank", the model must resolve its polysemy in context. The first forward LSTM would process "bank" in the context of "She went to the", which would allow it to represent the word to be a location that the subject is going towards. The first backward LSTM would process "bank" in the context of "to withdraw money", which would allow it to disambiguate the word as referring to a financial institution. The second forward LSTM can then process "bank" using the representation vector provided by the first backward LSTM, thus allowing it to represent it to be a financial institution that the subject is going towards. == Historical context == ELMo is one link in a historical evolution of language modelling. Consider a simple problem of document classification, where we want to assign a label (e.g., "spam", "not spam", "politics", "sports") to a given piece of text. The simplest approach is the "bag of words" approach, where each word in the document is treated independently, and its frequency is used as a feature for classification. This was computationally cheap but ignored the order of words and their context within the sentence. GloVe and Word2Vec built upon this by learning fixed vector representations (embeddings) for words based on their co-occurrence patterns in large text corpora. Like BERT (but unlike "bag of words" such as Word2Vec and GloVe), ELMo word embeddings are context-sensitive, producing different representations for words that share the same spelling. It was trained on a corpus of about 30 million sentences and 1 billion words. Previously, bidirectional LSTM was used for contextualized word representation. ELMo applied the idea to a large scale, achieving state of the art performance. After the 2017 publication of Transformer architecture, the architecture of ELMo was changed from a multilayered bidirectional LSTM to a Transformer encoder, giving rise to BERT. BERT has a similar pretrain-fine-tune workflow, but uses a Transformer with implications for more parallelizable training.

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  PowerBuilder

  PowerBuilder is an integrated development environment owned by SAP since the acquisition of Sybase in 2010. On July 5, 2016, SAP and Appeon entered into an agreement whereby Appeon, an independent company, would be responsible for developing, selling, and supporting PowerBuilder. Over the years, PowerBuilder has been updated with new standards. In 2010, a major upgrade of PowerBuilder was released to provide support for the Microsoft .NET Framework. In 2014, support was added for OData, dockable windows, and 64-bit native applications. In 2019 support was added for rapidly creating RESTful Web APIs and non-visual .NET assemblies using the C# language and the .NET Core framework. And PowerScript client app development was revamped with new UI technologies and cloud architecture. In 2025 the IDE was revamped with new code editor and ultra-fast compiler. Appeon has been releasing new features every 6-12 month cycles, which per the product roadmap focus on four key focus areas: sustaining core features, modernizing application UI, improving developer productivity, and incorporating more Cloud technology. == Features == PowerBuilder has a native data-handling component called a DataWindow, which can be used to create, edit, and display data from a database. This object gives the programmer a number of tools for specifying and controlling user interface appearance and behavior, and also provides simplified access to database content and JSON or XML from Web services. To some extent, the DataWindow frees the programmer from considering the differences between Database Management Systems from different vendors. DataWindow can display data using multiple presentation styles and can connect to various data sources. == Usage == PowerBuilder is used primarily for building business-oriented CRUD applications. Although new software products are rarely built with PowerBuilder, many client-server ERP products and line-of-business applications built in the late 1980s to early 2000s with PowerBuilder still provide core database functions for large enterprises in government, higher education, manufacturing, insurance, banking, energy, and telecommunications. == History == === Early history === PowerBuilder originated from Computer Solutions Inc. (CSI), a software consulting firm founded in 1974 by Mitchell Kertzman in Massachusetts. CSI developed GrowthPower, an MRP II software package with integrated financial modules released in 1981, which ran exclusively on the HP 3000 platform and achieved over 1,000 customer installations at its peak. In the late 1980s, as demand increased for graphical user interfaces amid the rise of Microsoft Windows, Kertzman partnered with Dave Litwack, former executive vice president of product development at Cullinet Software (acquired by Computer Associates in 1989). Litwack joined the company in 1988 as head of research and development to develop a client/server GUI tool, leading to its rebranding as Powersoft Corporation in 1990. PowerBuilder 1.0 was released in July 1991 as a rapid application development tool featuring the DataWindow and PowerScript language. Powersoft went public on February 3, 1993, with shares closing at $38 from an initial $20 price. Sybase announced its acquisition of Powersoft on November 15, 1994, in a stock swap valued at approximately $940 million; the merger closed on February 14, 1995, at a revised value of about $904 million due to Sybase's stock fluctuations. === Recent history === In December 2013 SAP announced the new version going directly to number 15 and released a beta version. Key features included support for the .NET Framework v4.5, SQL Server 2012, Oracle 12, Windows 8, OData and Dockable Windows. SAP later released this as version 12.6. On May 31, 2019, PowerBuilder 2019 was released by Appeon. This release supports C# development. It provides a new C# IDE, .NET data access objects, C# migration solution, Web API client, and UI themes. On April 3, 2020, PowerBuilder 2019 R2 was launched by Appeon. This release includes a first-ever PowerScript-to-C# code converter, which can automatically migrate 80-95% of PowerBuilder business logic and DataWindows to C#. Interoperability between PowerScript and .NET programming languages is also now supported. Many existing features have also been enhanced. On January 22, 2021, PowerBuilder 2019 R3 was launched by Appeon. This release provides a groundbreaking new app deployment technology called PowerClient, which securely automates the installation and update of client apps over HTTPS. C# Web API development has been greatly enhanced with asynchronous programming and support for Amazon Aurora and Azure cloud databases. Aside from many other new features, PowerBuilder 2019 R3 is a long-term support (LTS) version that replaces previous LTS versions On August 6, 2021, PowerBuilder 2021 was launched by Appeon. The Cloud deployment capability of the PowerBuilder 2021 IDE, in conjunction with the matching PowerServer 2021 runtime, was revamped, bringing PowerBuilder up-to-date with the latest .NET technologies. The presentation layer now executes PowerScript natively on Windows devices. The middle-tier has been rebuilt around REST API standard with a pure .NET Core implementation. A new CI/CD utility that integrates with Git/SVN and Jenkins, witch compiles all PowerBuilder projects using the command-line interface, was added alongside other features. On September 4, 2022, PowerBuilder 2022 was launched by Appeon. This release brings enhancements to the productivity of developing both client/server & installable cloud apps and more security measures to safeguard your apps. It includes many new features, including Windows 11 support, introducing time-saving functionalities to the IDE, such as Tabbed Code Editor, Jump to Objects, and Quick Code Search, and supports the latest HTTP/2 and TLS 1.3 protocols and two-way TLS authentication. On August 4, 2023, PowerBuilder 2022 R2 was launched by Appeon. This release introduces a range of new features aimed at helping developers build powerful, feature-rich, and secure client/server and installable cloud apps more efficiently, including tabbed windows, fillable PDFs, and SMTP client. On January 8, 2024, PowerBuilder 2022 R3 was launched by Appeon. This release is a long-term support version. Features previously released in earlier releases have been enhanced and/or corrected. On May 7, 2025, PowerBuilder 2025 was launched by Appeon. This release delivers a revamped IDE that boosts developer productivity throughout the SLDC—from writing and extending code to debugging, automating builds, and deploying applications. It features a new-generation code editor, ultra-fast compiler, automatic REST API creation, faster GIT operations, and codeless UI modernization features. == Features == PowerBuilder is an object-oriented programming language. Nearly all of the visual and non-visual objects support inheritance, polymorphism, and encapsulation. The programmer may utilize a common code framework such as PowerBuilder Foundation Classes, also known as PFC, to inherit objects from and leverage pre-existing code. The DataWindow is the key component (and selling point) of PowerBuilder. The DataWindow offers a visual SQL painter which supports outer joins, unions and subquery operations. It can convert SQL to visual representation and back, so the developer can use native SQL if desired. DataWindow updates are automatic — it produces the proper SQL at runtime based on the DBMS to which the user is currently connected. This feature makes it easier for developers who are not experienced with SQL. The DataWindow also has the built-in ability to both retrieve data and update data via stored procedures or REST Web APIs as well as import/export JSON data. The RESTClient object introduced in PowerBuilder 2017 facilitates bridging the DataWindow with REST Web APIs and requiring minimal coding. === RDBMS interfaces === PowerBuilder offers native interfaces to all major databases, as well as ODBC and OLE-DB, in the Enterprise version. There are many connectivity options that allow performance monitoring and tuning, such as: Integrated security Tracing of all SQL Isolation level Password expiration dialog Blocking factor Number of SQL statements to cache Use connection pool Thread safety Trace ODBC API calls Due to the information about the database schema (such as primary key information) that are stored in PowerBuilder's data dictionary, the code required to implement data display and browsing is greatly simplified, because the dictionary information allows generation of the appropriate SQL behind the scenes. PowerBuilder supports the following ways of interacting with a database: DataWindow this is the simplest approach, relying on automatically generated SQL. Embedded SQL Embedded SQL supports SELECT, INSERT, UPDATE, DELETE and cursors. This option is used when the developer desires more control than is available with the

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  ASR-complete

  ASR-complete is, by analogy to "NP-completeness" in complexity theory, a term to indicate that the difficulty of a computational problem is equivalent to solving the central automatic speech recognition problem, i.e. recognize and understanding spoken language. Unlike "NP-completeness", this term is typically used informally. Such problems are hypothesised to include: Spoken natural language understanding Understanding speech from far-field microphones, i.e. handling the reverbation and background noise These problems are easy for humans to do (in fact, they are described directly in terms of imitating humans). Some systems can solve very simple restricted versions of these problems, but none can solve them in their full generality.

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  ASR-complete

  ASR-complete is, by analogy to "NP-completeness" in complexity theory, a term to indicate that the difficulty of a computational problem is equivalent to solving the central automatic speech recognition problem, i.e. recognize and understanding spoken language. Unlike "NP-completeness", this term is typically used informally. Such problems are hypothesised to include: Spoken natural language understanding Understanding speech from far-field microphones, i.e. handling the reverbation and background noise These problems are easy for humans to do (in fact, they are described directly in terms of imitating humans). Some systems can solve very simple restricted versions of these problems, but none can solve them in their full generality.

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  Symbol level

  In knowledge-based systems, agents choose actions based on the principle of rationality to move closer to a desired goal. The agent is able to make decisions based on knowledge it has about the world (see knowledge level). But for the agent to actually change its state, it must use whatever means it has available. This level of description for the agent's behavior is the symbol level. The term was coined by Allen Newell in 1982. For example, in a computer program, the knowledge level consists of the information contained in its data structures that it uses to perform certain actions. The symbol level consists of the program's algorithms, the data structures themselves, and so on.

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  Outline of machine learning

  The following outline is provided as an overview of, and topical guide to, machine learning: Machine learning (ML) is a subfield of artificial intelligence within computer science that evolved from the study of pattern recognition and computational learning theory. In 1959, Arthur Samuel defined machine learning as a "field of study that gives computers the ability to learn without being explicitly programmed". ML involves the study and construction of algorithms that can learn from and make predictions on data. These algorithms operate by building a model from a training set of example observations to make data-driven predictions or decisions expressed as outputs, rather than following strictly static program instructions. == How can machine learning be categorized? == An academic discipline A branch of science An applied science A subfield of computer science A branch of artificial intelligence A subfield of soft computing Application of statistics === Paradigms of machine learning === Supervised learning, where the model is trained on labeled data Unsupervised learning, where the model tries to identify patterns in unlabeled data Reinforcement learning, where the model learns to make decisions by receiving rewards or penalties. == Applications of machine learning == Applications of machine learning Bioinformatics Biomedical informatics Computer vision Customer relationship management Data mining Earth sciences Email filtering Inverted pendulum (balance and equilibrium system) Natural language processing Named Entity Recognition Automatic summarization Automatic taxonomy construction Dialog system Grammar checker Language recognition Handwriting recognition Optical character recognition Speech recognition Text to Speech Synthesis Speech Emotion Recognition Machine translation Question answering Speech synthesis Text mining Term frequency–inverse document frequency Text simplification Pattern recognition Facial recognition system Handwriting recognition Image recognition Optical character recognition Speech recognition Recommendation system Collaborative filtering Content-based filtering Hybrid recommender systems Search engine Search engine optimization Social engineering == Machine learning hardware == Graphics processing unit Tensor processing unit Vision processing unit == Machine learning tools == Comparison of machine learning software Comparison of deep learning software === Machine learning frameworks === ==== Proprietary machine learning frameworks ==== Amazon Machine Learning Microsoft Azure Machine Learning Studio DistBelief (replaced by TensorFlow) ==== Open source machine learning frameworks ==== Apache Singa Apache MXNet Caffe PyTorch mlpack TensorFlow Torch CNTK Accord.Net Jax MLJ.jl – A machine learning framework for Julia === Machine learning libraries === Deeplearning4j Theano scikit-learn Keras === Machine learning algorithms === == Machine learning methods == === Instance-based algorithm === K-nearest neighbors algorithm (KNN) Learning vector quantization (LVQ) Self-organizing map (SOM) === Regression analysis === Logistic regression Ordinary least squares regression (OLSR) Linear regression Stepwise regression Multivariate adaptive regression splines (MARS) Regularization algorithm Ridge regression Least Absolute Shrinkage and Selection Operator (LASSO) Elastic net Least-angle regression (LARS) Classifiers Probabilistic classifier Naive Bayes classifier Binary classifier Linear classifier Hierarchical classifier === Dimensionality reduction === Dimensionality reduction Canonical correlation analysis (CCA) Factor analysis Feature extraction Feature selection Independent component analysis (ICA) Linear discriminant analysis (LDA) Multidimensional scaling (MDS) Non-negative matrix factorization (NMF) Partial least squares regression (PLSR) Principal component analysis (PCA) Principal component regression (PCR) Projection pursuit Sammon mapping t-distributed stochastic neighbor embedding (t-SNE) === Ensemble learning === Ensemble learning AdaBoost Boosting Bootstrap aggregating (also "bagging" or "bootstrapping") Ensemble averaging Gradient boosted decision tree (GBDT) Gradient boosting Random Forest Stacked Generalization === Meta-learning === Meta-learning Inductive bias Metadata === Reinforcement learning === Reinforcement learning Q-learning State–action–reward–state–action (SARSA) Temporal difference learning (TD) Learning Automata === Supervised learning === Supervised learning Averaged one-dependence estimators (AODE) Artificial neural network Case-based reasoning Gaussian process regression Gene expression programming Group method of data handling (GMDH) Inductive logic programming Instance-based learning Lazy learning Learning Automata Learning Vector Quantization Logistic Model Tree Minimum message length (decision trees, decision graphs, etc.) Nearest Neighbor Algorithm Analogical modeling Probably approximately correct learning (PAC) learning Ripple down rules, a knowledge acquisition methodology Symbolic machine learning algorithms Support vector machines Random Forests Ensembles of classifiers Bootstrap aggregating (bagging) Boosting (meta-algorithm) Ordinal classification Conditional Random Field ANOVA Quadratic classifiers k-nearest neighbor Boosting SPRINT Bayesian networks Naive Bayes Hidden Markov models Hierarchical hidden Markov model ==== Bayesian ==== Bayesian statistics Bayesian knowledge base Naive Bayes Gaussian Naive Bayes Multinomial Naive Bayes Averaged One-Dependence Estimators (AODE) Bayesian Belief Network (BBN) Bayesian Network (BN) ==== Decision tree algorithms ==== Decision tree algorithm Decision tree Classification and regression tree (CART) Iterative Dichotomiser 3 (ID3) C4.5 algorithm C5.0 algorithm Chi-squared Automatic Interaction Detection (CHAID) Decision stump Conditional decision tree ID3 algorithm Random forest SLIQ ==== Linear classifier ==== Linear classifier Fisher's linear discriminant Linear regression Logistic regression Multinomial logistic regression Naive Bayes classifier Perceptron Support vector machine === Unsupervised learning === Unsupervised learning Expectation-maximization algorithm Vector Quantization Generative topographic map Information bottleneck method Association rule learning algorithms Apriori algorithm Eclat algorithm ==== Artificial neural networks ==== Artificial neural network Feedforward neural network Extreme learning machine Convolutional neural network Recurrent neural network Long short-term memory (LSTM) Logic learning machine Self-organizing map ==== Association rule learning ==== Association rule learning Apriori algorithm Eclat algorithm FP-growth algorithm ==== Hierarchical clustering ==== Hierarchical clustering Single-linkage clustering Conceptual clustering ==== Cluster analysis ==== Cluster analysis BIRCH DBSCAN Expectation–maximization (EM) Fuzzy clustering Hierarchical clustering k-means clustering k-medians Mean-shift OPTICS algorithm ==== Anomaly detection ==== Anomaly detection k-nearest neighbors algorithm (k-NN) Local outlier factor === Semi-supervised learning === Semi-supervised learning Active learning Generative models Low-density separation Graph-based methods Co-training Transduction === Deep learning === Deep learning Deep belief networks Deep Boltzmann machines Deep Convolutional neural networks Deep Recurrent neural networks Hierarchical temporal memory Generative Adversarial Network Style transfer Transformer Stacked Auto-Encoders === Other machine learning methods and problems === Anomaly detection Association rules Bias-variance dilemma Classification Multi-label classification Clustering Data Pre-processing Empirical risk minimization Feature engineering Feature learning Learning to rank Occam learning Online machine learning PAC learning Regression Reinforcement Learning Semi-supervised learning Statistical learning Structured prediction Graphical models Bayesian network Conditional random field (CRF) Hidden Markov model (HMM) Unsupervised learning VC theory == Machine learning research == List of artificial intelligence projects List of datasets for machine learning research == History of machine learning == History of machine learning Timeline of machine learning == Machine learning projects == Machine learning projects: DeepMind Google Brain OpenAI Meta AI Hugging Face == Machine learning organizations == === Machine learning conferences and workshops === Artificial Intelligence and Security (AISec) (co-located workshop with CCS) Conference on Neural Information Processing Systems (NIPS) ECML PKDD International Conference on Machine Learning (ICML) ML4ALL (Machine Learning For All) == Machine learning publications == === Books on machine learning === Mathematics for Machine Learning Hands-On Machine Learning Scikit-Learn, Keras, and TensorFlow The Hundred-Page Machine Learning Book === Machine learning journals === Machine Learning Journal of Machine Learning Research (JMLR) Neural Computation == Pe

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  Discovery system (artificial intelligence)

  A discovery system is an artificial intelligence system that attempts to discover new scientific concepts or laws. The aim of discovery systems is to automate scientific data analysis and the scientific discovery process. Ideally, an artificial intelligence system should be able to search systematically through the space of all possible hypotheses and yield the hypothesis - or set of equally likely hypotheses - that best describes the complex patterns in data. During the era known as the second AI summer (approximately 1978–1987), various systems akin to the era's dominant expert systems were developed to tackle the problem of extracting scientific hypotheses from data, with or without interacting with a human scientist. These systems included Autoclass, Automated Mathematician, Eurisko, which aimed at general-purpose hypothesis discovery, and more specific systems such as Dalton, which uncovers molecular properties from data. The dream of building systems that discover scientific hypotheses was pushed to the background with the second AI winter and the subsequent resurgence of subsymbolic methods such as neural networks. Subsymbolic methods emphasize prediction over explanation, and yield models which works well but are difficult or impossible to explain which has earned them the name black box AI. A black-box model cannot be considered a scientific hypothesis, and this development has even led some researchers to suggest that the traditional aim of science - to uncover hypotheses and theories about the structure of reality - is obsolete. Other researchers disagree and argue that subsymbolic methods are useful in many cases, just not for generating scientific theories. == Discovery systems from the 1970s and 1980s == Autoclass was a Bayesian Classification System written in 1986 Automated Mathematician was one of the earliest successful discovery systems. It was written in 1977 and worked by generating a modifying small Lisp programs Eurisko was a Sequel to Automated Mathematician written in 1984 Dalton is a still maintained program capable of calculating various molecular properties initially launched in 1983 and available in open source since 2017 Glauber is a scientific discovery method written in the context of computational philosophy of science launched in 1983 == Modern discovery systems (2009–present) == After a couple of decades with little interest in discovery systems, the interest in using AI to uncover natural laws and scientific explanations was renewed by the work of Michael Schmidt, then a PhD student in Computational Biology at Cornell University. Schmidt and his advisor, Hod Lipson, invented Eureqa, which they described as a symbolic regression approach to "distilling free-form natural laws from experimental data". This work effectively demonstrated that symbolic regression was a promising way forward for AI-driven scientific discovery. Since 2009, symbolic regression has matured further, and today, various commercial and open source systems are actively used in scientific research. Notable examples include Eureqa, now a part of DataRobot AI Cloud Platform, AI Feynman, and QLattice.

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  Problem solving

  Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to get from point A to B) to complex issues in business and technical fields. The former is an example of simple problem solving (SPS) addressing one issue, whereas the latter is complex problem solving (CPS) with multiple interrelated obstacles. Another classification of problem-solving tasks is into well-defined problems with specific obstacles and goals, and ill-defined problems in which the current situation is troublesome but it is not clear what kind of resolution to aim for. Similarly, one may distinguish formal or fact-based problems requiring psychometric intelligence, versus socio-emotional problems which depend on the changeable emotions of individuals or groups, such as tactful behavior, fashion, or gift choices. Solutions require sufficient resources and knowledge to attain the goal. Professionals such as lawyers, doctors, programmers, and consultants are largely problem solvers for issues that require technical skills and knowledge beyond general competence. Many businesses have found profitable markets by recognizing a problem and creating a solution: the more widespread and inconvenient the problem, the greater the opportunity to develop a scalable solution. There are many specialized problem-solving techniques and methods in fields such as science, engineering, business, medicine, mathematics, computer science, philosophy, and social organization. The mental techniques to identify, analyze, and solve problems are studied in psychology and cognitive sciences. Also widely researched are the mental obstacles that prevent people from finding solutions; problem-solving impediments include confirmation bias, mental set, and functional fixedness. == Definition == The term problem solving has a slightly different meaning depending on the discipline. For instance, it is a mental process in psychology and a computerized process in computer science. There are two different types of problems: ill-defined and well-defined; different approaches are used for each. Well-defined problems have specific end goals and clearly expected solutions, while ill-defined problems do not. Well-defined problems allow for more initial planning than ill-defined problems. Solving problems sometimes involves dealing with pragmatics (the way that context contributes to meaning) and semantics (the interpretation of the problem). The ability to understand what the end goal of the problem is, and what rules could be applied, represents the key to solving the problem. Sometimes a problem requires abstract thinking or coming up with a creative solution. Problem solving has two major domains: mathematical problem solving and personal problem solving. Each concerns some difficulty or barrier that is encountered. === Psychology === Problem solving in psychology refers to the process of finding solutions to problems encountered in life. Solutions to these problems are usually situation- or context-specific. The process starts with problem finding and problem shaping, in which the problem is discovered and simplified. The next step is to generate possible solutions and evaluate them. Finally a solution is selected to be implemented and verified. Problems have an end goal to be reached; how you get there depends upon problem orientation (problem-solving coping style and skills) and systematic analysis. Mental health professionals study the human problem-solving processes using methods such as introspection, behaviorism, simulation, computer modeling, and experiment. Social psychologists look into the person-environment relationship aspect of the problem and independent and interdependent problem-solving methods. Problem solving has been defined as a higher-order cognitive process and intellectual function that requires the modulation and control of more routine or fundamental skills. Empirical research shows many different strategies and factors influence everyday problem solving. Rehabilitation psychologists studying people with frontal lobe injuries have found that deficits in emotional control and reasoning can be re-mediated with effective rehabilitation and could improve the capacity of injured persons to resolve everyday problems. Interpersonal everyday problem solving is dependent upon personal motivational and contextual components. One such component is the emotional valence of "real-world" problems, which can either impede or aid problem-solving performance. Researchers have focused on the role of emotions in problem solving, demonstrating that poor emotional control can disrupt focus on the target task, impede problem resolution, and lead to negative outcomes such as fatigue, depression, and inertia. In conceptualization,human problem solving consists of two related processes: problem orientation, and the motivational/attitudinal/affective approach to problematic situations and problem-solving skills. People's strategies cohere with their goals and stem from the process of comparing oneself with others. === Cognitive sciences === Among the first experimental psychologists to study problem solving were the Gestaltists in Germany, such as Karl Duncker in The Psychology of Productive Thinking (1935). Perhaps best known is the work of Allen Newell and Herbert A. Simon. Experiments in the 1960s and early 1970s asked participants to solve relatively simple, well-defined, but not previously seen laboratory tasks. These simple problems, such as the Tower of Hanoi, admitted optimal solutions that could be found quickly, allowing researchers to observe the full problem-solving process. Researchers assumed that these model problems would elicit the characteristic cognitive processes by which more complex "real world" problems are solved. An outstanding problem-solving technique found by this research is the principle of decomposition. === Computer science === Much of computer science and artificial intelligence involves designing automated systems to solve a specified type of problem: to accept input data and calculate a correct or adequate response, reasonably quickly. Algorithms are recipes or instructions that direct such systems, written into computer programs. Steps for designing such systems include problem determination, heuristics, root cause analysis, de-duplication, analysis, diagnosis, and repair. Analytic techniques include linear and nonlinear programming, queuing systems, and simulation. A large, perennial obstacle is to find and fix errors in computer programs: debugging. === Logic === Formal logic concerns issues like validity, truth, inference, argumentation, and proof. In a problem-solving context, it can be used to formally represent a problem as a theorem to be proved, and to represent the knowledge needed to solve the problem as the premises to be used in a proof that the problem has a solution. The use of computers to prove mathematical theorems using formal logic emerged as the field of automated theorem proving in the 1950s. It included the use of heuristic methods designed to simulate human problem solving, as in the Logic Theory Machine, developed by Allen Newell, Herbert A. Simon and J. C. Shaw, as well as algorithmic methods such as the resolution principle developed by John Alan Robinson. In addition to its use for finding proofs of mathematical theorems, automated theorem-proving has also been used for program verification in computer science. In 1958, John McCarthy proposed the advice taker, to represent information in formal logic and to derive answers to questions using automated theorem-proving. An important step in this direction was made by Cordell Green in 1969, who used a resolution theorem prover for question-answering and for such other applications in artificial intelligence as robot planning. The resolution theorem-prover used by Cordell Green bore little resemblance to human problem solving methods. In response to criticism of that approach from researchers at MIT, Robert Kowalski developed logic programming and SLD resolution, which solves problems by problem decomposition. He has advocated logic for both computer and human problem solving and computational logic to improve human thinking. === Engineering === When products or processes fail, problem solving techniques can be used to develop corrective actions that can be taken to prevent further failures. Such techniques can also be applied to a product or process prior to an actual failure event—to predict, analyze, and mitigate a potential problem in advance. Techniques such as failure mode and effects analysis can proactively reduce the likelihood of problems. In either the reactive or the proactive case, it is necessary to build a causal explanation through a process of diagnosis. In deriving an explanation of effects in terms of causes, abduction generates new ideas or hypothes

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  Structural risk minimization

  Structural risk minimization (SRM) is an inductive principle of use in machine learning. Commonly in machine learning, a generalized model must be selected from a finite data set, with the consequent problem of overfitting – the model becoming too strongly tailored to the particularities of the training set and generalizing poorly to new data. The SRM principle addresses this problem by balancing the model's complexity against its success at fitting the training data. This principle was first set out in a 1974 book by Vladimir Vapnik and Alexey Chervonenkis and uses the VC dimension. In practical terms, Structural Risk Minimization is implemented by minimizing E t r a i n + β H ( W ) {\displaystyle E_{train}+\beta H(W)} , where E t r a i n {\displaystyle E_{train}} is the train error, the function H ( W ) {\displaystyle H(W)} is called a regularization function, and β {\displaystyle \beta } is a constant. H ( W ) {\displaystyle H(W)} is chosen such that it takes large values on parameters W {\displaystyle W} that belong to high-capacity subsets of the parameter space. Minimizing H ( W ) {\displaystyle H(W)} in effect limits the capacity of the accessible subsets of the parameter space, thereby controlling the trade-off between minimizing the training error and minimizing the expected gap between the training error and test error. The SRM problem can be formulated in terms of data. Given n data points consisting of data x and labels y, the objective J ( θ ) {\displaystyle J(\theta )} is often expressed in the following manner: J ( θ ) = 1 2 n ∑ i = 1 n ( h θ ( x i ) − y i ) 2 + λ 2 ∑ j = 1 d θ j 2 {\displaystyle J(\theta )={\frac {1}{2n}}\sum _{i=1}^{n}(h_{\theta }(x^{i})-y^{i})^{2}+{\frac {\lambda }{2}}\sum _{j=1}^{d}\theta _{j}^{2}} The first term is the mean squared error (MSE) term between the value of the learned model, h θ {\displaystyle h_{\theta }} , and the given labels y {\displaystyle y} . This term is the training error, E t r a i n {\displaystyle E_{train}} , that was discussed earlier. The second term, places a prior over the weights, to favor sparsity and penalize larger weights. The trade-off coefficient, λ {\displaystyle \lambda } , is a hyperparameter that places more or less importance on the regularization term. Larger λ {\displaystyle \lambda } encourages sparser weights at the expense of a more optimal MSE, and smaller λ {\displaystyle \lambda } relaxes regularization allowing the model to fit to data. Note that as λ → ∞ {\displaystyle \lambda \to \infty } the weights become zero, and as λ → 0 {\displaystyle \lambda \to 0} , the model typically suffers from overfitting.

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  Problem solving

  Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to get from point A to B) to complex issues in business and technical fields. The former is an example of simple problem solving (SPS) addressing one issue, whereas the latter is complex problem solving (CPS) with multiple interrelated obstacles. Another classification of problem-solving tasks is into well-defined problems with specific obstacles and goals, and ill-defined problems in which the current situation is troublesome but it is not clear what kind of resolution to aim for. Similarly, one may distinguish formal or fact-based problems requiring psychometric intelligence, versus socio-emotional problems which depend on the changeable emotions of individuals or groups, such as tactful behavior, fashion, or gift choices. Solutions require sufficient resources and knowledge to attain the goal. Professionals such as lawyers, doctors, programmers, and consultants are largely problem solvers for issues that require technical skills and knowledge beyond general competence. Many businesses have found profitable markets by recognizing a problem and creating a solution: the more widespread and inconvenient the problem, the greater the opportunity to develop a scalable solution. There are many specialized problem-solving techniques and methods in fields such as science, engineering, business, medicine, mathematics, computer science, philosophy, and social organization. The mental techniques to identify, analyze, and solve problems are studied in psychology and cognitive sciences. Also widely researched are the mental obstacles that prevent people from finding solutions; problem-solving impediments include confirmation bias, mental set, and functional fixedness. == Definition == The term problem solving has a slightly different meaning depending on the discipline. For instance, it is a mental process in psychology and a computerized process in computer science. There are two different types of problems: ill-defined and well-defined; different approaches are used for each. Well-defined problems have specific end goals and clearly expected solutions, while ill-defined problems do not. Well-defined problems allow for more initial planning than ill-defined problems. Solving problems sometimes involves dealing with pragmatics (the way that context contributes to meaning) and semantics (the interpretation of the problem). The ability to understand what the end goal of the problem is, and what rules could be applied, represents the key to solving the problem. Sometimes a problem requires abstract thinking or coming up with a creative solution. Problem solving has two major domains: mathematical problem solving and personal problem solving. Each concerns some difficulty or barrier that is encountered. === Psychology === Problem solving in psychology refers to the process of finding solutions to problems encountered in life. Solutions to these problems are usually situation- or context-specific. The process starts with problem finding and problem shaping, in which the problem is discovered and simplified. The next step is to generate possible solutions and evaluate them. Finally a solution is selected to be implemented and verified. Problems have an end goal to be reached; how you get there depends upon problem orientation (problem-solving coping style and skills) and systematic analysis. Mental health professionals study the human problem-solving processes using methods such as introspection, behaviorism, simulation, computer modeling, and experiment. Social psychologists look into the person-environment relationship aspect of the problem and independent and interdependent problem-solving methods. Problem solving has been defined as a higher-order cognitive process and intellectual function that requires the modulation and control of more routine or fundamental skills. Empirical research shows many different strategies and factors influence everyday problem solving. Rehabilitation psychologists studying people with frontal lobe injuries have found that deficits in emotional control and reasoning can be re-mediated with effective rehabilitation and could improve the capacity of injured persons to resolve everyday problems. Interpersonal everyday problem solving is dependent upon personal motivational and contextual components. One such component is the emotional valence of "real-world" problems, which can either impede or aid problem-solving performance. Researchers have focused on the role of emotions in problem solving, demonstrating that poor emotional control can disrupt focus on the target task, impede problem resolution, and lead to negative outcomes such as fatigue, depression, and inertia. In conceptualization,human problem solving consists of two related processes: problem orientation, and the motivational/attitudinal/affective approach to problematic situations and problem-solving skills. People's strategies cohere with their goals and stem from the process of comparing oneself with others. === Cognitive sciences === Among the first experimental psychologists to study problem solving were the Gestaltists in Germany, such as Karl Duncker in The Psychology of Productive Thinking (1935). Perhaps best known is the work of Allen Newell and Herbert A. Simon. Experiments in the 1960s and early 1970s asked participants to solve relatively simple, well-defined, but not previously seen laboratory tasks. These simple problems, such as the Tower of Hanoi, admitted optimal solutions that could be found quickly, allowing researchers to observe the full problem-solving process. Researchers assumed that these model problems would elicit the characteristic cognitive processes by which more complex "real world" problems are solved. An outstanding problem-solving technique found by this research is the principle of decomposition. === Computer science === Much of computer science and artificial intelligence involves designing automated systems to solve a specified type of problem: to accept input data and calculate a correct or adequate response, reasonably quickly. Algorithms are recipes or instructions that direct such systems, written into computer programs. Steps for designing such systems include problem determination, heuristics, root cause analysis, de-duplication, analysis, diagnosis, and repair. Analytic techniques include linear and nonlinear programming, queuing systems, and simulation. A large, perennial obstacle is to find and fix errors in computer programs: debugging. === Logic === Formal logic concerns issues like validity, truth, inference, argumentation, and proof. In a problem-solving context, it can be used to formally represent a problem as a theorem to be proved, and to represent the knowledge needed to solve the problem as the premises to be used in a proof that the problem has a solution. The use of computers to prove mathematical theorems using formal logic emerged as the field of automated theorem proving in the 1950s. It included the use of heuristic methods designed to simulate human problem solving, as in the Logic Theory Machine, developed by Allen Newell, Herbert A. Simon and J. C. Shaw, as well as algorithmic methods such as the resolution principle developed by John Alan Robinson. In addition to its use for finding proofs of mathematical theorems, automated theorem-proving has also been used for program verification in computer science. In 1958, John McCarthy proposed the advice taker, to represent information in formal logic and to derive answers to questions using automated theorem-proving. An important step in this direction was made by Cordell Green in 1969, who used a resolution theorem prover for question-answering and for such other applications in artificial intelligence as robot planning. The resolution theorem-prover used by Cordell Green bore little resemblance to human problem solving methods. In response to criticism of that approach from researchers at MIT, Robert Kowalski developed logic programming and SLD resolution, which solves problems by problem decomposition. He has advocated logic for both computer and human problem solving and computational logic to improve human thinking. === Engineering === When products or processes fail, problem solving techniques can be used to develop corrective actions that can be taken to prevent further failures. Such techniques can also be applied to a product or process prior to an actual failure event—to predict, analyze, and mitigate a potential problem in advance. Techniques such as failure mode and effects analysis can proactively reduce the likelihood of problems. In either the reactive or the proactive case, it is necessary to build a causal explanation through a process of diagnosis. In deriving an explanation of effects in terms of causes, abduction generates new ideas or hypothes

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  Commonsense knowledge (artificial intelligence)

  In artificial intelligence research, commonsense knowledge consists of facts about the everyday world, such as "Lemons are sour" or "Cows say moo", that all humans are expected to know. It is currently an unsolved problem in artificial general intelligence. The first AI program to address common sense knowledge was Advice Taker in 1959 by John McCarthy. Commonsense knowledge can underpin a commonsense reasoning process, to attempt inferences such as "You might bake a cake because you want people to eat the cake." A natural language processing process can be attached to the commonsense knowledge base to allow the knowledge base to attempt to answer questions about the world. Common sense knowledge also helps to solve problems in the face of incomplete information. Using widely held beliefs about everyday objects, or common sense knowledge, AI systems make common sense assumptions or default assumptions about the unknown similar to the way people do. In an AI system or in English, this is expressed as "Normally P holds", "Usually P" or "Typically P so Assume P". For example, if we know the fact "Tweety is a bird", because we know the commonly held belief about birds, "typically birds fly," without knowing anything else about Tweety, we may reasonably assume the fact that "Tweety can fly." As more knowledge of the world is discovered or learned over time, the AI system can revise its assumptions about Tweety using a truth maintenance process. If we later learn that "Tweety is a penguin" then truth maintenance revises this assumption because we also know "penguins do not fly". == Commonsense reasoning == Commonsense reasoning simulates the human ability to use commonsense knowledge to make presumptions about the type and essence of ordinary situations they encounter every day, and to change their "minds" should new information come to light. This includes time, missing or incomplete information and cause and effect. The ability to explain cause and effect is an important aspect of explainable AI. Truth maintenance algorithms automatically provide an explanation facility because they create elaborate records of presumptions. Compared with humans, all existing computer programs that attempt human-level AI perform extremely poorly on modern "commonsense reasoning" benchmark tests such as the Winograd Schema Challenge. The problem of attaining human-level competency at "commonsense knowledge" tasks is considered to probably be "AI complete" (that is, solving it would require the ability to synthesize a fully human-level intelligence), although some oppose this notion and believe compassionate intelligence is also required for human-level AI. Common sense reasoning has been applied successfully in more limited domains such as natural language processing and automated diagnosis or analysis. == Commonsense knowledge base construction == Compiling comprehensive knowledge bases of commonsense assertions (CSKBs) is a long-standing challenge in AI research. From early expert-driven efforts like CYC and WordNet, significant advances were achieved via the crowdsourced OpenMind Commonsense project, which led to the crowdsourced ConceptNet KB. Several approaches have attempted to automate CSKB construction, most notably, via text mining (WebChild, Quasimodo, TransOMCS, Ascent), as well as harvesting these directly from pre-trained language models (AutoTOMIC). These resources are significantly larger than ConceptNet, though the automated construction mostly makes them of moderately lower quality. Challenges also remain on the representation of commonsense knowledge: Most CSKB projects follow a triple data model, which is not necessarily best suited for breaking more complex natural language assertions. A notable exception here is GenericsKB, which applies no further normalization to sentences, but retains them in full. == Applications == Around 2013, MIT researchers developed BullySpace, an extension of the commonsense knowledgebase ConceptNet, to catch taunting social media comments. BullySpace included over 200 semantic assertions based around stereotypes, to help the system infer that comments like "Put on a wig and lipstick and be who you really are" are more likely to be an insult if directed at a boy than a girl. ConceptNet has also been used by chatbots and by computers that compose original fiction. At Lawrence Livermore National Laboratory, common sense knowledge was used in an intelligent software agent to detect violations of a comprehensive nuclear test ban treaty. == Data == As an example, as of 2012 ConceptNet includes these 21 language-independent relations: IsA (An "RV" is a "vehicle" | X is an instance of a Y) UsedFor (a "cake tin" is used for "making cakes" | X is used for the purpose Y) HasA (A "rabbit" has a "tail" | X possesses Y element or feature) CapableOf (a "cook" is capable of "making baked goods" | X is capable of doing Y) Desires (a "child" desires "the aroma of baking" | X has a desire for Y) CreatedBy ("cake" is created by a "baker" | X is created by Y) PartOf (a "knife" is be part of a "knife set" | X is a part of Y) Causes ("Heat" causes "cooking"| X is what causes Y) LocatedNear (the "oven" is located near the "refrigerator" | X is located near Y) AtLocation (Somewhere a "Cook" can be at a "restaurant" | X is at the location of Y) DefinedAs (a "Cupcake" is defined as a "cake" that also has the qualities of being "small", "baked within a wrapper", and "containing only one area of frosting or icing" | X is defined as Y that also has the properties A, B & C) SymbolOf (a "heart" is a symbol of "affection" | X is a symbolic representation of Y) ReceivesAction ("cake" can receive the action of being "eaten" | X is capable of receiving action Y) HasPrerequisite ("baking" has the prerequisite of obtaining the "ingredients" | X cannot do Y unless A does B) MotivatedByGoal ("baking" is motivated by the goal of "consumption"/"eating" | X has the motivation of Y goal) CausesDesire ("baking" makesYou want to "follow recipe" | X causes the desire to do Y) MadeOf ("Cake" is made of "flour"/"eggs"/"sugar"/"oil"/etc | X is made of Y) HasFirstSubevent ("baking" has first subevent "make batter" | To do X the first thing that needs to be done is Y) HasSubevent ("eat" has subevent "swallow" | Doing X will lead to Y event following) HasLastSubevent ("sleeping" has last subevent of "waking" | Doing X ends with the event Y) == Commonsense knowledge bases == Cyc Open Mind Common Sense (data source) and ConceptNet (datastore and NLP engine) Evi Graphiq

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  Hidden layer

  In artificial neural networks, a hidden layer is a layer of artificial neurons that is neither an input layer nor an output layer. The simplest examples appear in multilayer perceptrons (MLP), as illustrated in the diagram. An MLP without any hidden layer is essentially just a linear model. With hidden layers and activation functions, however, nonlinearity is introduced into the model. In typical machine learning practice, the weights and biases are initialized, then iteratively updated during training via backpropagation.

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