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State–action–reward–state–action : Q n e w ( S t , A t ) ← ( 1 − α ) Q ( S t , A t ) + α [ R t + 1 + γ Q ( S t + 1 , A t + 1 ) ] (S_,A_)\leftarrow (1-\alpha )Q(S_,A_)+\alpha \,[R_+\gamma \,Q(S_,A_)] A SARSA agent interacts with the environment and updates the policy based on actions taken, hence this is known as an on-... |
State–action–reward–state–action : Prefrontal cortex basal ganglia working memory Sammon mapping Constructing skill trees Q-learning Temporal difference learning Reinforcement learning == References == |
Stochastic gradient descent : Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replac... |
Stochastic gradient descent : Both statistical estimation and machine learning consider the problem of minimizing an objective function that has the form of a sum: Q ( w ) = 1 n ∑ i = 1 n Q i ( w ) , \sum _^Q_(w), where the parameter w that minimizes Q ( w ) is to be estimated. Each summand function Q i is typically... |
Stochastic gradient descent : In stochastic (or "on-line") gradient descent, the true gradient of Q ( w ) is approximated by a gradient at a single sample: w := w − η ∇ Q i ( w ) . (w). As the algorithm sweeps through the training set, it performs the above update for each training sample. Several passes can be made o... |
Stochastic gradient descent : Suppose we want to fit a straight line y ^ = w 1 + w 2 x =w_+w_x to a training set with observations ( ( x 1 , y 1 ) , ( x 2 , y 2 ) … , ( x n , y n ) ) ,y_),(x_,y_)\ldots ,(x_,y_)) and corresponding estimated responses ( y ^ 1 , y ^ 2 , … , y ^ n ) _,_,\ldots ,_) using least squares. The ... |
Stochastic gradient descent : In 1951, Herbert Robbins and Sutton Monro introduced the earliest stochastic approximation methods, preceding stochastic gradient descent. Building on this work one year later, Jack Kiefer and Jacob Wolfowitz published an optimization algorithm very close to stochastic gradient descent, us... |
Stochastic gradient descent : Stochastic gradient descent is a popular algorithm for training a wide range of models in machine learning, including (linear) support vector machines, logistic regression (see, e.g., Vowpal Wabbit) and graphical models. When combined with the back propagation algorithm, it is the de facto... |
Stochastic gradient descent : Many improvements on the basic stochastic gradient descent algorithm have been proposed and used. In particular, in machine learning, the need to set a learning rate (step size) has been recognized as problematic. Setting this parameter too high can cause the algorithm to diverge; setting ... |
Stochastic gradient descent : For small learning rate η stochastic gradient descent ( w n ) n ∈ N 0 )_ _ can be viewed as a discretization of the gradient flow ODE d d t W t = − ∇ Q ( W t ) W_=-\nabla Q(W_) subject to additional stochastic noise. This approximation is only valid on a finite time-horizon in the followi... |
Stochastic gradient descent : Backtracking line search Broken Neural Scaling Law Coordinate descent – changes one coordinate at a time, rather than one example Linear classifier Online machine learning Stochastic hill climbing Stochastic variance reduction |
Stochastic gradient descent : Bottou, Léon (2004), "Stochastic Learning", Advanced Lectures on Machine Learning, LNAI, vol. 3176, Springer, pp. 146–168, ISBN 978-3-540-23122-6 Buduma, Nikhil; Locascio, Nicholas (2017), "Beyond Gradient Descent", Fundamentals of Deep Learning : Designing Next-Generation Machine Intellig... |
Stochastic gradient descent : "Gradient Descent, How Neural Networks Learn". 3Blue1Brown. October 16, 2017. Archived from the original on 2021-12-22 – via YouTube. Goh (April 4, 2017). "Why Momentum Really Works". Distill. 2 (4). doi:10.23915/distill.00006. Interactive paper explaining momentum. |
Stochastic variance reduction : (Stochastic) variance reduction is an algorithmic approach to minimizing functions that can be decomposed into finite sums. By exploiting the finite sum structure, variance reduction techniques are able to achieve convergence rates that are impossible to achieve with methods that treat t... |
Stochastic variance reduction : A function f is considered to have finite sum structure if it can be decomposed into a summation or average: f ( x ) = 1 n ∑ i = 1 n f i ( x ) , \sum _^f_(x), where the function value and derivative of each f i can be queried independently. Although variance reduction methods can be ap... |
Stochastic variance reduction : Stochastic variance reduced methods without acceleration are able to find a minima of f within accuracy ϵ > , i.e. f ( x ) − f ( x ∗ ) ≤ ϵ )\leq \epsilon in a number of steps of the order: O ( ( L μ + n ) log ( 1 ϵ ) ) . +n\right)\log \left(\right)\right). The number of steps depend... |
Stochastic variance reduction : Variance reduction approaches fall within 3 main categories: table averaging methods, full-gradient snapshot methods and dual methods. Each category contains methods designed for dealing with convex, non-smooth, and non-convex problems, each differing in hyper-parameter settings and othe... |
Stochastic variance reduction : Accelerated variance reduction methods are built upon the standard methods above. The earliest approaches make use of proximal operators to accelerate convergence, either approximately or exactly. Direct acceleration approaches have also been developed |
Stochastic variance reduction : Stochastic gradient descent Coordinate descent Online machine learning Proximal operator Stochastic optimization Stochastic approximation == References == |
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 ... |
T-distributed stochastic neighbor embedding : Given a set of N high-dimensional objects x 1 , … , x N _,\dots ,\mathbf _ , t-SNE first computes probabilities p i j that are proportional to the similarity of objects x i _ and x j _ , as follows. For i ≠ j , define p j ∣ i = exp ( − ‖ x i − x j ‖ 2 / 2 σ i 2 ) ∑... |
T-distributed stochastic neighbor embedding : 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 stru... |
T-distributed stochastic neighbor embedding : 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-SN... |
T-distributed stochastic neighbor embedding : Wattenberg, Martin; Viégas, Fernanda; Johnson, Ian (2016-10-13). "How to Use t-SNE Effectively". Distill. 1 (10): e2. doi:10.23915/distill.00002. ISSN 2476-0757.. Interactive demonstration and tutorial. Visualizing Data Using t-SNE, Google Tech Talk about t-SNE Implementati... |
Wake-sleep algorithm : The wake-sleep algorithm is an unsupervised learning algorithm for deep generative models, especially Helmholtz Machines. The algorithm is similar to the expectation-maximization algorithm, and optimizes the model likelihood for observed data. The name of the algorithm derives from its use of two... |
Wake-sleep algorithm : The goal of the wake-sleep algorithm is to find a hierarchical representation of observed data. In a graphical representation of the algorithm, data is applied to the algorithm at the bottom, while higher layers form gradually more abstract representations. Between each pair of layers are two set... |
Wake-sleep algorithm : Training consists of two phases – the “wake” phase and the “sleep” phase. It has been proven that this learning algorithm is convergent. |
Wake-sleep algorithm : Since the recognition network is limited in its flexibility, it might not be able to approximate the posterior distribution of latent variables well. To better approximate the posterior distribution, it is possible to employ importance sampling, with the recognition network as the proposal distri... |
Wake-sleep algorithm : Restricted Boltzmann machine, a type of neural net that is trained with a conceptually similar algorithm. Helmholtz machine, a neural network model trained by the wake-sleep algorithm. == References == |
Weighted majority algorithm (machine learning) : In machine learning, weighted majority algorithm (WMA) is a meta learning algorithm used to construct a compound algorithm from a pool of prediction algorithms, which could be any type of learning algorithms, classifiers, or even real human experts. The algorithm assumes... |
Weighted majority algorithm (machine learning) : Randomized weighted majority algorithm == References == |
Zero-shot learning : Zero-shot learning (ZSL) is a problem setup in deep learning where, at test time, a learner observes samples from classes which were not observed during training, and needs to predict the class that they belong to. The name is a play on words based on the earlier concept of one-shot learning, in wh... |
Zero-shot learning : The first paper on zero-shot learning in natural language processing appeared in a 2008 paper by Chang, Ratinov, Roth, and Srikumar, at the AAAI’08, but the name given to the learning paradigm there was dataless classification. The first paper on zero-shot learning in computer vision appeared at th... |
Zero-shot learning : Naturally, some form of auxiliary information has to be given about these zero-shot classes, and this type of information can be of several types. Learning with attributes: classes are accompanied by pre-defined structured description. For example, for bird descriptions, this could include "red hea... |
Zero-shot learning : The above ZSL setup assumes that at test time, only zero-shot samples are given, namely, samples from new unseen classes. In generalized zero-shot learning, samples from both new and known classes, may appear at test time. This poses new challenges for classifiers at test time, because it is very c... |
Zero-shot learning : Zero shot learning has been applied to the following fields: image classification semantic segmentation image generation object detection natural language processing computational biology |
Zero-shot learning : One-shot learning in computer vision Transfer learning Fast mapping Explanation-based learning == References == |
Loss function : In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seek... |
Loss function : In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for opt... |
Loss function : In some contexts, the value of the loss function itself is a random quantity because it depends on the outcome of a random variable X. |
Loss function : A decision rule makes a choice using an optimality criterion. Some commonly used criteria are: Minimax: Choose the decision rule with the lowest worst loss — that is, minimize the worst-case (maximum possible) loss: a r g m i n δ max θ ∈ Θ R ( θ , δ ) . \ \max _\ R(\theta ,\delta ). Invariance: Choose ... |
Loss function : Sound statistical practice requires selecting an estimator consistent with the actual acceptable variation experienced in the context of a particular applied problem. Thus, in the applied use of loss functions, selecting which statistical method to use to model an applied problem depends on knowing the ... |
Loss function : Bayesian regret Loss functions for classification Discounted maximum loss Hinge loss Scoring rule Statistical risk |
Loss function : Aretz, Kevin; Bartram, Söhnke M.; Pope, Peter F. (April–June 2011). "Asymmetric Loss Functions and the Rationality of Expected Stock Returns" (PDF). International Journal of Forecasting. 27 (2): 413–437. doi:10.1016/j.ijforecast.2009.10.008. SSRN 889323. Berger, James O. (1985). Statistical decision the... |
Cross-entropy : In information theory, the cross-entropy between two probability distributions p and q , over the same underlying set of events, measures the average number of bits needed to identify an event drawn from the set when the coding scheme used for the set is optimized for an estimated probability distribu... |
Cross-entropy : The cross-entropy of the distribution q relative to a distribution p over a given set is defined as follows: H ( p , q ) = − E p [ log q ] , _[\log q], where E p [ ⋅ ] [\cdot ] is the expected value operator with respect to the distribution p . The definition may be formulated using the Kullback... |
Cross-entropy : In information theory, the Kraft–McMillan theorem establishes that any directly decodable coding scheme for coding a message to identify one value x i out of a set of possibilities ,\ldots ,x_\ can be seen as representing an implicit probability distribution q ( x i ) = ( 1 2 ) ℓ i )=\left(\right)^ ov... |
Cross-entropy : There are many situations where cross-entropy needs to be measured but the distribution of p is unknown. An example is language modeling, where a model is created based on a training set T , and then its cross-entropy is measured on a test set to assess how accurate the model is in predicting the test... |
Cross-entropy : The cross entropy arises in classification problems when introducing a logarithm in the guise of the log-likelihood function. The section is concerned with the subject of estimation of the probability of different possible discrete outcomes. To this end, denote a parametrized family of distributions by ... |
Cross-entropy : Cross-entropy minimization is frequently used in optimization and rare-event probability estimation. When comparing a distribution q against a fixed reference distribution p , cross-entropy and KL divergence are identical up to an additive constant (since p is fixed): According to the Gibbs' inequali... |
Cross-entropy : Cross-entropy can be used to define a loss function in machine learning and optimization. Mao, Mohri, and Zhong (2023) give an extensive analysis of the properties of the family of cross-entropy loss functions in machine learning, including theoretical learning guarantees and extensions to adversarial l... |
Cross-entropy : It may be beneficial to train an ensemble of models that have diversity, such that when they are combined, their predictive accuracy is augmented. Assuming a simple ensemble of K classifiers is assembled via averaging the outputs, then the amended cross-entropy is given by e k = H ( p , q k ) − λ K ∑ j... |
Cross-entropy : Cross-entropy method Logistic regression Conditional entropy Kullback–Leibler distance Maximum-likelihood estimation Mutual information Perplexity |
Cross-entropy : de Boer, Kroese, D.P., Mannor, S. and Rubinstein, R.Y. (2005). A tutorial on the cross-entropy method. Annals of Operations Research 134 (1), 19–67. |
Huber loss : In statistics, the Huber loss is a loss function used in robust regression, that is less sensitive to outliers in data than the squared error loss. A variant for classification is also sometimes used. |
Huber loss : The Huber loss function describes the penalty incurred by an estimation procedure f. Huber (1964) defines the loss function piecewise by L δ ( a ) = (a)=&|a|\leq \delta ,\\\delta \cdot \left(|a|-\delta \right),&\end This function is quadratic for small values of a, and linear for large values, with equal v... |
Huber loss : Two very commonly used loss functions are the squared loss, L ( a ) = a 2 , and the absolute loss, L ( a ) = | a | . The squared loss function results in an arithmetic mean-unbiased estimator, and the absolute-value loss function results in a median-unbiased estimator (in the one-dimensional case, and a ... |
Huber loss : The Pseudo-Huber loss function can be used as a smooth approximation of the Huber loss function. It combines the best properties of L2 squared loss and L1 absolute loss by being strongly convex when close to the target/minimum and less steep for extreme values. The scale at which the Pseudo-Huber loss func... |
Huber loss : For classification purposes, a variant of the Huber loss called modified Huber is sometimes used. Given a prediction f ( x ) (a real-valued classifier score) and a true binary class label y ∈ , the modified Huber loss is defined as L ( y , f ( x ) ) = \max(0,1-y\,f(x))^&\,\,y\,f(x)>-1,\\-4y\,f(x)&\end T... |
Huber loss : The Huber loss function is used in robust statistics, M-estimation and additive modelling. |
Huber loss : Winsorizing Robust regression M-estimator Visual comparison of different M-estimators == References == |
Mean squared error : In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the true value. MSE is a risk... |
Mean squared error : The MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs to a sample of values of some random variable), or of an estimator (i.e., a mathematical function mapping a sample of data to an estimate of a parameter of the population from which the data is sampled). I... |
Mean squared error : In regression analysis, plotting is a more natural way to view the overall trend of the whole data. The mean of the distance from each point to the predicted regression model can be calculated, and shown as the mean squared error. The squaring is critical to reduce the complexity with negative sign... |
Mean squared error : An MSE of zero, meaning that the estimator θ ^ predicts observations of the parameter θ with perfect accuracy, is ideal (but typically not possible). Values of MSE may be used for comparative purposes. Two or more statistical models may be compared using their MSEs—as a measure of how well they e... |
Mean squared error : Minimizing MSE is a key criterion in selecting estimators: see minimum mean-square error. Among unbiased estimators, minimizing the MSE is equivalent to minimizing the variance, and the estimator that does this is the minimum variance unbiased estimator. However, a biased estimator may have lower M... |
Mean squared error : Squared error loss is one of the most widely used loss functions in statistics, though its widespread use stems more from mathematical convenience than considerations of actual loss in applications. Carl Friedrich Gauss, who introduced the use of mean squared error, was aware of its arbitrariness a... |
Mean squared error : Bias–variance tradeoff Hodges' estimator James–Stein estimator Mean percentage error Mean square quantization error Reduced chi-squared statistic Mean squared displacement Mean squared prediction error Minimum mean square error Overfitting Peak signal-to-noise ratio |
Mean squared prediction error : In statistics the mean squared prediction error (MSPE), also known as mean squared error of the predictions, of a smoothing, curve fitting, or regression procedure is the expected value of the squared prediction errors (PE), the square difference between the fitted values implied by the ... |
Mean squared prediction error : If the smoothing or fitting procedure has projection matrix (i.e., hat matrix) L, which maps the observed values vector y to predicted values vector y ^ = L y , =Ly, then PE and MSPE are formulated as: P E i = g ( x i ) − g ^ ( x i ) , =g(x_)-(x_), MSPE = E [ PE i 2 ] = ∑ i = 1 n PE ... |
Mean squared prediction error : The mean squared prediction error can be computed exactly in two contexts. First, with a data sample of length n, the data analyst may run the regression over only q of the data points (with q < n), holding back the other n – q data points with the specific purpose of using them to compu... |
Mean squared prediction error : When the model has been estimated over all available data with none held back, the MSPE of the model over the entire population of mostly unobserved data can be estimated as follows. For the model y i = g ( x i ) + σ ε i =g(x_)+\sigma \varepsilon _ where ε i ∼ N ( 0 , 1 ) \sim (0,1) , on... |
Mean squared prediction error : Akaike information criterion Bias-variance tradeoff Mean squared error Errors and residuals in statistics Law of total variance Mallows's Cp Model selection == References == |
Loss function : In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seek... |
Loss function : In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for opt... |
Loss function : In some contexts, the value of the loss function itself is a random quantity because it depends on the outcome of a random variable X. |
Loss function : A decision rule makes a choice using an optimality criterion. Some commonly used criteria are: Minimax: Choose the decision rule with the lowest worst loss — that is, minimize the worst-case (maximum possible) loss: a r g m i n δ max θ ∈ Θ R ( θ , δ ) . \ \max _\ R(\theta ,\delta ). Invariance: Choose ... |
Loss function : Sound statistical practice requires selecting an estimator consistent with the actual acceptable variation experienced in the context of a particular applied problem. Thus, in the applied use of loss functions, selecting which statistical method to use to model an applied problem depends on knowing the ... |
Loss function : Bayesian regret Loss functions for classification Discounted maximum loss Hinge loss Scoring rule Statistical risk |
Loss function : Aretz, Kevin; Bartram, Söhnke M.; Pope, Peter F. (April–June 2011). "Asymmetric Loss Functions and the Rationality of Expected Stock Returns" (PDF). International Journal of Forecasting. 27 (2): 413–437. doi:10.1016/j.ijforecast.2009.10.008. SSRN 889323. Berger, James O. (1985). Statistical decision the... |
Sum of absolute differences : In digital image processing, the sum of absolute differences (SAD) is a measure of the similarity between image blocks. It is calculated by taking the absolute difference between each pixel in the original block and the corresponding pixel in the block being used for comparison. These diff... |
Sum of absolute differences : This example uses the sum of absolute differences to identify which part of a search image is most similar to a template image. In this example, the template image is 3 by 3 pixels in size, while the search image is 3 by 5 pixels in size. Each pixel is represented by a single integer from ... |
Sum of absolute differences : Computer stereo vision Hadamard transform Motion compensation Motion estimation Object recognition (computer vision) Rate-distortion optimization |
Sum of absolute differences : E. G. Richardson, Iain (2003). H.264 and MPEG-4 Video Compression: Video Coding for Next-generation Multimedia. Chichester: John Wiley & Sons Ltd. |
Sum of absolute transformed differences : The sum of absolute transformed differences (SATD) is a block matching criterion widely used in fractional motion estimation for video compression. It works by taking a frequency transform, usually a Hadamard transform, of the differences between the pixels in the original bloc... |
Sum of absolute transformed differences : SATD is slower than the sum of absolute differences (SAD), both due to its increased complexity and the fact that SAD-specific MMX and SSE2 instructions exist, while there are no such instructions for SATD. However, SATD can still be optimized considerably with SIMD instruction... |
Sum of absolute transformed differences : Hadamard transform Motion compensation Motion estimation Rate–distortion optimization Sum of absolute differences |
Sum of absolute transformed differences : E. G. Richardson, Iain (2003). H.264 and MPEG-4 Video Compression: Video Coding for Next-generation Multimedia. Chichester: John Wiley & Sons Ltd. |
Taguchi loss function : The Taguchi loss function is graphical depiction of loss developed by the Japanese business statistician Genichi Taguchi to describe a phenomenon affecting the value of products produced by a company. Praised by Dr. W. Edwards Deming (the business guru of the 1980s American quality movement), it... |
Taguchi loss function : The Taguchi loss function is important for a number of reasons—primarily, to help engineers better understand the importance of designing for variation. |
Taguchi loss function : Taguchi methods Taguchi also focus on Robust design of model. == References == |
Log-linear model : A log-linear model is a mathematical model that takes the form of a function whose logarithm equals a linear combination of the parameters of the model, which makes it possible to apply (possibly multivariate) linear regression. That is, it has the general form exp ( c + ∑ i w i f i ( X ) ) w_f_(X)... |
Log-linear model : Log-linear analysis General linear model Generalized linear model Boltzmann distribution Elasticity |
Log-linear model : Gujarati, Damodar N.; Porter, Dawn C. (2009). "How to Measure Elasticity: The Log-Linear Model". Basic Econometrics. New York: McGraw-Hill/Irwin. pp. 159–162. ISBN 978-0-07-337577-9. |
Boosting (machine learning) : In machine learning (ML), boosting is an ensemble metaheuristic for primarily reducing bias (as opposed to variance). It can also improve the stability and accuracy of ML classification and regression algorithms. Hence, it is prevalent in supervised learning for converting weak learners to... |
Boosting (machine learning) : While boosting is not algorithmically constrained, most boosting algorithms consist of iteratively learning weak classifiers with respect to a distribution and adding them to a final strong classifier. When they are added, they are weighted in a way that is related to the weak learners' ac... |
Boosting (machine learning) : Given images containing various known objects in the world, a classifier can be learned from them to automatically classify the objects in future images. Simple classifiers built based on some image feature of the object tend to be weak in categorization performance. Using boosting methods... |
Boosting (machine learning) : Boosting algorithms can be based on convex or non-convex optimization algorithms. Convex algorithms, such as AdaBoost and LogitBoost, can be "defeated" by random noise such that they can't learn basic and learnable combinations of weak hypotheses. This limitation was pointed out by Long & ... |
Boosting (machine learning) : scikit-learn, an open source machine learning library for Python Orange, a free data mining software suite, module Orange.ensemble Weka is a machine learning set of tools that offers variate implementations of boosting algorithms like AdaBoost and LogitBoost R package GBM (Generalized Boos... |
Boosting (machine learning) : Freund, Yoav; Schapire, Robert E. (1997). "A Decision-Theoretic Generalization of On-line Learning and an Application to Boosting" (PDF). Journal of Computer and System Sciences. 55 (1): 119–139. doi:10.1006/jcss.1997.1504. Schapire, Robert E. (1990). "The strength of weak learnability". M... |
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