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Article: Manifold regularization. In machine learning, Manifold regularization is a technique for using the shape of a dataset to constrain the functions that should be learned on that dataset. In many machine learning problems, the data to be learned do not cover the entire input space. For example, a facial recogniti... | Wikipedia - Manifold regularization - Summary | 249 | 1,350 | null |
Section: Manifold regularizer > Motivation. Manifold regularization is a type of regularization, a family of techniques that reduces overfitting and ensures that a problem is well-posed by penalizing complex solutions. In particular, manifold regularization extends the technique of Tikhonov regularization as applied to... | Wikipedia - Manifold regularization - Manifold regularizer > Motivation | 222 | 974 | null |
When the algorithm considers a candidate function, it takes its norm into account in order to penalize complex functions. Formally, given a set of labeled training data ( x 1 , y 1 ) , β¦ , ( x β , y β ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{\ell },y_{\ell })} with x i β X , y i β Y {\displaystyle x_{i}\in X,y_{i}\in... | Wikipedia - Manifold regularization - Manifold regularizer > Motivation | 337 | 982 | null |
Section: Manifold regularizer > Laplacian norm. There are many possible choices for the intrinsic regularizer β f β I {\displaystyle \left\|f\right\|_{I}} . Many natural choices involve the gradient on the manifold β M {\displaystyle \nabla _{M}} , which can provide a measure of how smooth a target function is. A smoot... | Wikipedia - Manifold regularization - Manifold regularizer > Laplacian norm | 331 | 1,067 | null |
Section: Manifold regularizer > Graph-based approach of the Laplacian norm. When the distances between input points are interpreted as a graph, then the Laplacian matrix of the graph can help to estimate the marginal distribution. Suppose that the input data include β {\displaystyle \ell } labeled examples (pairs of an... | Wikipedia - Manifold regularization - Manifold regularizer > Graph-based approach of the Laplacian norm | 340 | 1,130 | null |
Then, as the number of data points β + u {\displaystyle \ell +u} increases, L {\displaystyle L} converges to the LaplaceβBeltrami operator Ξ M {\displaystyle \Delta _{M}} , which is the divergence of the gradient β M {\displaystyle \nabla _{M}} . Then, if f {\displaystyle \mathbf {f} } is a vector of the values of f {\... | Wikipedia - Manifold regularization - Manifold regularizer > Graph-based approach of the Laplacian norm | 348 | 875 | null |
Section: Manifold regularizer > Solving the regularization problem with graph-based approach. Using the weights Ξ³ A {\displaystyle \gamma _{A}} and Ξ³ I {\displaystyle \gamma _{I}} for the ambient and intrinsic regularizers, the final expression to be solved becomes: arg min f β H 1 β β i = 1 β V ( f ( x i ) , y i ) + Ξ³... | Wikipedia - Manifold regularization - Manifold regularizer > Solving the regularization problem with graph-based approach | 309 | 815 | null |
Using the weights Ξ³ A {\displaystyle \gamma _{A}} and Ξ³ I {\displaystyle \gamma _{I}} for the ambient and intrinsic regularizers, the final expression to be solved becomes: arg min f β H 1 β β i = 1 β V ( f ( x i ) , y i ) + Ξ³ A β f β K 2 + Ξ³ I ( β + u ) 2 f T L f {\displaystyle {\underset {f\in {\mathcal {H}}}{\arg \!... | Wikipedia - Manifold regularization - Manifold regularizer > Solving the regularization problem with graph-based approach | 500 | 1,358 | null |
Section: Manifold regularizer > Functional approach of the Laplacian norm. The idea beyond graph-Laplacian is to use neighbors to estimate Laplacian. This method is akin local averaging methods, that are known to scale poorly in high-dimensional problem. Indeed, graph Laplacian is known to suffer from the curse of dime... | Wikipedia - Manifold regularization - Manifold regularizer > Functional approach of the Laplacian norm | 208 | 904 | null |
The problem statement for RLS results from choosing the loss function V {\displaystyle V} in Tikhonov regularization to be the mean squared error: f β = arg min f β H 1 β β i = 1 β ( f ( x i ) β y i ) 2 + Ξ³ β f β K 2 {\displaystyle f^{*}={\underset {f\in {\mathcal {H}}}{\arg \!\min }}{\frac {1}{\ell }}\sum _{i=1}^{\ell... | Wikipedia - Manifold regularization - Applications > Laplacian Regularized Least Squares (LapRLS) | 350 | 828 | null |
, x j ) {\displaystyle K_{ij}=K(x_{i},x_{j})} , and Y {\displaystyle Y} is the vector of data labels. Adding a Laplacian term for manifold regularization gives the Laplacian RLS statement: f β = arg min f β H 1 β β i = 1 β ( f ( x i ) β y i ) 2 + Ξ³ A β f β K 2 + Ξ³ I ( β + u ) 2 f T L f {\displaystyle f^{*}={\underset {... | Wikipedia - Manifold regularization - Applications > Laplacian Regularized Least Squares (LapRLS) | 376 | 787 | null |
Letting K {\displaystyle K} be the kernel matrix as above, Y {\displaystyle Y} be the vector of data labels, and J {\displaystyle J} be the ( β + u ) Γ ( β + u ) {\displaystyle (\ell +u)\times (\ell +u)} block matrix [ I β 0 0 0 u ] {\displaystyle {\begin{bmatrix}I_{\ell }&0\\0&0_{u}\end{bmatrix}}} : Ξ± β = arg min Ξ± β ... | Wikipedia - Manifold regularization - Applications > Laplacian Regularized Least Squares (LapRLS) | 350 | 738 | null |
This can be directly expressed as a linear program, but it is also equivalent to Tikhonov regularization with the hinge loss function, V ( f ( x ) , y ) = max ( 0 , 1 β y f ( x ) ) {\displaystyle V(f(x),y)=\max(0,1-yf(x))} : f β = arg min f β H 1 β β i = 1 β max ( 0 , 1 β y i f ( x i ) ) + Ξ³ β f β K 2 {\displaystyle f^... | Wikipedia - Manifold regularization - Applications > Laplacian Support Vector Machines (LapSVM) | 343 | 757 | null |
_{i=1}^{\ell }\max(0,1-y_{i}f(x_{i}))+\gamma _{A}\left\|f\right\|_{K}^{2}+{\frac {\gamma _{I}}{(\ell +u)^{2}}}\mathbf {f} ^{\mathrm {T} }L\mathbf {f} } Again, the representer theorem allows the solution to be expressed in terms of the kernel evaluated at the data points: f β ( x ) = β i = 1 β + u Ξ± i β K ( x i , x ) {\... | Wikipedia - Manifold regularization - Applications > Laplacian Support Vector Machines (LapSVM) | 228 | 501 | null |
Again letting K {\displaystyle K} be the kernel matrix and J {\displaystyle J} be the block matrix [ I β 0 0 0 u ] {\displaystyle {\begin{bmatrix}I_{\ell }&0\\0&0_{u}\end{bmatrix}}} , the solution can be shown to be Ξ± = ( 2 Ξ³ A I + 2 Ξ³ I ( β + u ) 2 L K ) β 1 J T Y Ξ² β {\displaystyle \alpha =\left(2\gamma _{A}I+2{\frac... | Wikipedia - Manifold regularization - Applications > Laplacian Support Vector Machines (LapSVM) | 346 | 745 | null |
to}}&&\sum _{i=1}^{\ell }\beta _{i}y_{i}=0\\&&&0\leq \beta _{i}\leq {\frac {1}{\ell }}\;i=1,\ldots ,\ell \end{aligned}}} and Q {\displaystyle Q} is defined by Q = Y J K ( 2 Ξ³ A I + 2 Ξ³ I ( β + u ) 2 L K ) β 1 J T Y {\displaystyle Q=YJK\left(2\gamma _{A}I+2{\frac {\gamma _{I}}{(\ell +u)^{2}}}LK\right)^{-1}J^{\mathrm {T}... | Wikipedia - Manifold regularization - Applications > Laplacian Support Vector Machines (LapSVM) | 218 | 479 | null |
Section: Limitations. Manifold regularization assumes that data with different labels are not likely to be close together. This assumption is what allows the technique to draw information from unlabeled data, but it only applies to some problem domains. Depending on the structure of the data, it may be necessary to use... | Wikipedia - Manifold regularization - Limitations | 275 | 1,241 | null |
Section: Overview. The book outlines five approaches of machine learning: inductive reasoning, connectionism, evolutionary computation, Bayes' theorem and analogical modelling. The author explains these tribes to the reader by referring to more understandable processes of logic, connections made in the brain, natural s... | Wikipedia - The Master Algorithm - Overview | 155 | 814 | null |
Article: Matchbox Educable Noughts and Crosses Engine. The Matchbox Educable Noughts and Crosses Engine (sometimes called the Machine Educable Noughts and Crosses Engine or MENACE) was a mechanical computer made from 304 matchboxes designed and built by artificial intelligence researcher Donald Michie in 1961. It was d... | Wikipedia - Matchbox Educable Noughts and Crosses Engine - Summary | 313 | 1,561 | null |
Section: Origin. Donald Michie (1923β2007) had been on the team decrypting the German Tunny Code during World War II. Fifteen years later, he wanted to further display his mathematical and computational prowess with an early convolutional neural network. Since computer equipment was not obtainable for such uses, and Mi... | Wikipedia - Matchbox Educable Noughts and Crosses Engine - Origin | 315 | 1,572 | null |
Section: Composition. MENACE was made from 304 matchboxes glued together in an arrangement similar to a chest of drawers. Each box had a code number, which was keyed into a chart. This chart had drawings of tic-tac-toe game grids with various configurations of X, O, and empty squares, corresponding to all possible perm... | Wikipedia - Matchbox Educable Noughts and Crosses Engine - Composition | 258 | 1,226 | null |
Section: Operation. MENACE played first, as O, since all matchboxes represented permutations only relevant to the "X" player. To retrieve MENACE's choice of move, the opponent or operator located the matchbox that matched the current game state, or a rotation or mirror image of it. For example, at the start of a game, ... | Wikipedia - Matchbox Educable Noughts and Crosses Engine - Operation | 325 | 1,440 | null |
Section: Results in practice > Optimal strategy. Noughts and crosses has a well-known optimal strategy. A player must place their symbol in a way that blocks the other player from achieving any rows while simultaneously making a row themself. However, if both players use this strategy, the game always ends in a draw. I... | Wikipedia - Matchbox Educable Noughts and Crosses Engine - Results in practice > Optimal strategy | 318 | 1,417 | null |
Section: Results in practice > Correlation. Depending on the strategy employed by the human player, MENACE produces a different trend on scatter graphs of wins. Using a random turn from the human player results in an almost-perfect positive trend. Playing the optimal strategy returns a slightly slower increase. The rei... | Wikipedia - Matchbox Educable Noughts and Crosses Engine - Results in practice > Correlation | 223 | 816 | null |
Section: Legacy. Donald Michie's MENACE proved that a computer could learn from failure and success to become good at a task. It used what would become core principles within the field of machine learning before they had been properly theorised. For example, the combination of how MENACE starts with equal numbers of ty... | Wikipedia - Matchbox Educable Noughts and Crosses Engine - Legacy | 312 | 1,552 | null |
Article: Matrix regularization. In the field of statistical learning theory, matrix regularization generalizes notions of vector regularization to cases where the object to be learned is a matrix. The purpose of regularization is to enforce conditions, for example sparsity or smoothness, that can produce stable predict... | Wikipedia - Matrix regularization - Summary | 321 | 1,236 | null |
Section: Basic definition. Consider a matrix W {\displaystyle W} to be learned from a set of examples, S = ( X i t , y i t ) {\displaystyle S=(X_{i}^{t},y_{i}^{t})} , where i {\displaystyle i} goes from 1 {\displaystyle 1} to n {\displaystyle n} , and t {\displaystyle t} goes from 1 {\displaystyle 1} to T {\displaystyl... | Wikipedia - Matrix regularization - Basic definition | 263 | 717 | null |
A general model for the output y {\displaystyle y} can be posed as y i t = β¨ W , X i t β© F , {\displaystyle y_{i}^{t}=\left\langle W,X_{i}^{t}\right\rangle _{F},} where the inner product is the Frobenius inner product. For different applications the matrices X i {\displaystyle X_{i}} will have different forms, but for ... | Wikipedia - Matrix regularization - Basic definition | 347 | 1,044 | null |
Section: General applications > Matrix completion. In the problem of matrix completion, the matrix X i t {\displaystyle X_{i}^{t}} takes the form X i t = e t β e i β² , {\displaystyle X_{i}^{t}=e_{t}\otimes e_{i}',} where ( e t ) t {\displaystyle (e_{t})_{t}} and ( e i β² ) i {\displaystyle (e_{i}')_{i}} are the canonica... | Wikipedia - Matrix regularization - General applications > Matrix completion | 316 | 1,029 | null |
For example, it might be assumed that W {\displaystyle W} is low-rank, in which case the regularization penalty can take the form of a nuclear norm. R ( W ) = Ξ» β W β β = Ξ» β i | Ο i | , {\displaystyle R(W)=\lambda \left\|W\right\|_{*}=\lambda \sum _{i}\left|\sigma _{i}\right|,} where Ο i {\displaystyle \sigma _{i}} , ... | Wikipedia - Matrix regularization - General applications > Matrix completion | 162 | 458 | null |
Section: General applications > Multivariate regression. Models used in multivariate regression are parameterized by a matrix of coefficients. In the Frobenius inner product above, each matrix X {\displaystyle X} is X i t = e t β x i {\displaystyle X_{i}^{t}=e_{t}\otimes x_{i}} such that the output of the inner product... | Wikipedia - Matrix regularization - General applications > Multivariate regression | 238 | 831 | null |
One example is the squared Frobenius norm, which can be viewed as an β 2 {\displaystyle \ell ^{2}} -norm acting either entrywise, or on the singular values of the matrix: R ( W ) = Ξ» β W β F 2 = Ξ» β i β j | w i j | 2 = Ξ» Tr β‘ ( W β W ) = Ξ» β i Ο i 2 . {\displaystyle R(W)=\lambda \left\|W\right\|_{F}^{2}=\lambda \sum _{... | Wikipedia - Matrix regularization - General applications > Multivariate regression | 229 | 623 | null |
Section: General applications > Multi-task learning. The setup for multi-task learning is almost the same as the setup for multivariate regression. The primary difference is that the input variables are also indexed by task (columns of Y {\displaystyle Y} ). The representation with the Frobenius inner product is then X... | Wikipedia - Matrix regularization - General applications > Multi-task learning | 267 | 952 | null |
That is, the same solution can be found by solving the joint problem, or by solving an isolated regression problem for each column. The problems can be coupled by adding an additional regularization penalty on the covariance of solutions min W , Ξ© β X W β Y β 2 2 + Ξ» 1 β W β 2 2 + Ξ» 2 Tr β‘ ( W T Ξ© β 1 W ) {\displaystyl... | Wikipedia - Matrix regularization - General applications > Multi-task learning | 258 | 906 | null |
Section: Spectral regularization. Regularization by spectral filtering has been used to find stable solutions to problems such as those discussed above by addressing ill-posed matrix inversions (see for example Filter function for Tikhonov regularization). In many cases the regularization function acts on the input (or... | Wikipedia - Matrix regularization - Spectral regularization | 323 | 1,401 | null |
Section: Structured sparsity. Sparse optimization has become the focus of much research interest as a way to find solutions that depend on a small number of variables (see e.g. the Lasso method). In principle, entry-wise sparsity can be enforced by penalizing the entry-wise β 0 {\displaystyle \ell ^{0}} -norm of the ma... | Wikipedia - Matrix regularization - Structured sparsity | 273 | 954 | null |
The most straightforward example of structured sparsity uses the β p , q {\displaystyle \ell _{p,q}} norm with p = 2 {\displaystyle p=2} and q = 1 {\displaystyle q=1} : β W β 2 , 1 = β i β w i β 2 . {\displaystyle \left\|W\right\|_{2,1}=\sum _{i}\left\|w_{i}\right\|_{2}.} For example, the β 2 , 1 {\displaystyle \ell _{... | Wikipedia - Matrix regularization - Structured sparsity | 276 | 884 | null |
The same type of regularization can be used to enforce sparsity column-wise by taking the β 2 {\displaystyle \ell ^{2}} -norms of each column. More generally, the β 2 , 1 {\displaystyle \ell _{2,1}} norm can be applied to arbitrary groups of variables: R ( W ) = Ξ» β g G β j | G g | | w g j | 2 = Ξ» β g G β w g β g {\dis... | Wikipedia - Matrix regularization - Structured sparsity | 289 | 852 | null |
Algorithms for solving these group sparsity problems extend the more well-known Lasso and group Lasso methods by allowing overlapping groups, for example, and have been implemented via matching pursuit: and proximal gradient methods. By writing the proximal gradient with respect to a given coefficient, w g i {\displays... | Wikipedia - Matrix regularization - Structured sparsity | 309 | 833 | null |
{\displaystyle \operatorname {prox} _{\lambda ,R_{g}}\left(w_{g}\right)^{i}=\left(w_{g}^{i}-\lambda {\frac {w_{g}^{i}}{\left\|w_{g}\right\|_{g}}}\right)\mathbf {1} _{\|w_{g}\|_{g}\geq \lambda }.} where 1 β w g β g β₯ Ξ» {\displaystyle \mathbf {1} _{\|w_{g}\|_{g}\geq \lambda }} is the indicator function for group norms β₯ ... | Wikipedia - Matrix regularization - Structured sparsity | 294 | 829 | null |
Section: Multiple kernel selection. The ideas of structured sparsity and feature selection can be extended to the nonparametric case of multiple kernel learning. This can be useful when there are multiple types of input data (color and texture, for example) with different appropriate kernels for each, or when the appro... | Wikipedia - Matrix regularization - Multiple kernel selection | 273 | 884 | null |
If there are two kernels, for example, with feature maps A {\displaystyle A} and B {\displaystyle B} that lie in corresponding reproducing kernel Hilbert spaces H A , H B {\displaystyle {\mathcal {H_{A}}},{\mathcal {H_{B}}}} , then a larger space, H D {\displaystyle {\mathcal {H_{D}}}} , can be created as the sum of tw... | Wikipedia - Matrix regularization - Multiple kernel selection | 380 | 1,001 | null |
In this case the β 2 , 1 {\displaystyle \ell _{2,1}} -norm is again the sum of norms: β f β H D , 1 = β h β H A + β h β² β H B {\displaystyle \left\|f\right\|_{{\mathcal {H_{D}}},1}=\left\|h\right\|_{\mathcal {H_{A}}}+\left\|h'\right\|_{\mathcal {H_{B}}}} Thus, by choosing a matrix regularization function as this type o... | Wikipedia - Matrix regularization - Multiple kernel selection | 234 | 748 | null |
Article: Maximum inner-product search. Maximum inner-product search (MIPS) is a search problem, with a corresponding class of search algorithms which attempt to maximise the inner product between a query and the data items to be retrieved. MIPS algorithms are used in a wide variety of big data applications, including r... | Wikipedia - Maximum inner-product search - Summary | 335 | 1,383 | null |
Section: History. Chris Olah is generally credited with coining the term 'Mechanistic interpretability' and spearheading its early development. In the 2018 paper The Building Blocks of Interpretability, Olah (then at Google Brain) and his colleagues combined existing interpretability techniques, including feature visua... | Wikipedia - Mechanistic interpretability - History | 298 | 1,550 | null |
The team further elaborated this result in the March 2022 paper In-context Learning and Induction Heads. Notable results in mechanistic interpretability from 2022 include the theory of superposition wherein a model represents more features than there are directions in its representation space; a mechanistic explanation... | Wikipedia - Mechanistic interpretability - History | 350 | 1,782 | null |
Section: Cultural distinction between explainability, interpretability and mechanistic interpretability. The term mechanistic interpretability designates both a class of technical methodsβexplainability methods such as saliency maps are generally not considered mechanistic interpretability researchβand a cultural movem... | Wikipedia - Mechanistic interpretability - Cultural distinction between explainability, interpretability and mechanistic interpretability | 224 | 1,202 | null |
Article: Meta-Labeling. Meta-labeling, also known as corrective AI, is a machine learning (ML) technique utilized in quantitative finance to enhance the performance of investment and trading strategies, developed in 2017 by Marcos LΓ³pez de Prado at Guggenheim Partners and Cornell University. The core idea is to separat... | Wikipedia - Meta-Labeling - Summary | 173 | 926 | null |
Section: General architecture. Meta-labeling decouples two core components of systematic trading strategies: directional prediction and position sizing. The process involves training a primary model to generate trade signals (e.g., buy, sell, or hold) and then training a secondary model to determine whether each signal... | Wikipedia - Meta-Labeling - General architecture | 292 | 1,386 | null |
Section: General architecture > Stage 1: Forecasting side. Primary model architecture Figure 1 Figure 1 presents the architecture of a primary model. It focuses on forecasting the side of the trade. Following the example, this model (M1) takes in input data β such as open-high-low-close data and determines the side of ... | Wikipedia - Meta-Labeling - General architecture > Stage 1: Forecasting side | 273 | 1,257 | null |
Section: General architecture > Stage 2: Filtering out false positives. General meta-labeling architecture Figure 2 Next comes the phase of filtering out false positives, by applying a secondary machine learning model (M2), which is a binary classifier trained to determine if the trade will be profitable or not. The mo... | Wikipedia - Meta-Labeling - General architecture > Stage 2: Filtering out false positives | 329 | 1,690 | null |
Section: General architecture > Stage 3: Optimizing position sizes > Position sizing methods (M3). Various algorithms have been proposed for transforming predicted probabilities into trade sizes: All-or-nothing: Allocate 100% of capital if the probability exceeds a predefined threshold (e.g., 0.5); otherwise, do not tr... | Wikipedia - Meta-Labeling - General architecture > Stage 3: Optimizing position sizes > Position sizing methods (M3) | 206 | 921 | null |
Section: General architecture > Stage 3: Optimizing position sizes > Model calibration. Each machine learning algorithm used in meta-labeling tends to produce outputs with different characteristic distributions; for example, some are approximately normally distributed, whereas others exhibit a pronounced U-shape, conce... | Wikipedia - Meta-Labeling - General architecture > Stage 3: Optimizing position sizes > Model calibration | 316 | 1,686 | null |
Section: Meta-labeling architectures > Sequential meta-labeling (SMLA). The SMLA introduces multiple layers of secondary models. Each secondary model's inputs include previous secondary models' outputs and evaluation statistics. This iterative process incrementally improves the model's accuracy. Components Primary mode... | Wikipedia - Meta-Labeling - Meta-labeling architectures > Sequential meta-labeling (SMLA) | 158 | 930 | null |
Section: Meta-labeling architectures > Inverse meta-labeling. Inverse meta-labeling reverses the standard process by first identifying important features from secondary models to refine and improve the primary model. This iterative improvement cycle helps create more effective primary models before applying meta-labeli... | Wikipedia - Meta-Labeling - Meta-labeling architectures > Inverse meta-labeling | 169 | 1,009 | null |
Article: Meta-learning (computer science). Meta-learning is a subfield of machine learning where automatic learning algorithms are applied to metadata about machine learning experiments. As of 2017, the term had not found a standard interpretation, however the main goal is to use such metadata to understand how automat... | Wikipedia - Meta-learning (computer science) - Summary | 331 | 1,765 | null |
Section: Definition. A proposed definition for a meta-learning system combines three requirements: The system must include a learning subsystem. Experience is gained by exploiting meta knowledge extracted in a previous learning episode on a single dataset, or from different domains. Learning bias must be chosen dynamic... | Wikipedia - Meta-learning (computer science) - Definition | 177 | 831 | null |
Section: Examples. Some approaches which have been viewed as instances of meta-learning: Recurrent neural networks (RNNs) are universal computers. In 1993, JΓΌrgen Schmidhuber showed how "self-referential" RNNs can in principle learn by backpropagation to run their own weight change algorithm, which may be quite differe... | Wikipedia - Meta-learning (computer science) - Examples | 325 | 1,524 | null |
Model-Agnostic Meta-Learning (MAML) was introduced in 2017 by Chelsea Finn et al. Given a sequence of tasks, the parameters of a given model are trained such that few iterations of gradient descent with few training data from a new task will lead to good generalization performance on that task. MAML "trains the model t... | Wikipedia - Meta-learning (computer science) - Examples | 344 | 1,649 | null |
The metadata is formed by characteristics of the data (general, statistical, information-theoretic,... ) in the learning problem, and characteristics of the learning algorithm (type, parameter settings, performance measures,...). Another learning algorithm then learns how the data characteristics relate to the algorith... | Wikipedia - Meta-learning (computer science) - Examples | 334 | 1,825 | null |
Many different approaches exist. Inductive transfer studies how the learning process can be improved over time. Metadata consists of knowledge about previous learning episodes and is used to efficiently develop an effective hypothesis for a new task. A related approach is called learning to learn, in which the goal is ... | Wikipedia - Meta-learning (computer science) - Examples | 161 | 914 | null |
Article: MLOps. MLOps or ML Ops is a paradigm that aims to deploy and maintain machine learning models in production reliably and efficiently. It bridges the gap between machine learning development and production operations, ensuring that models are robust, scalable, and aligned with business goals. The word is a comp... | Wikipedia - MLOps - Summary | 232 | 1,240 | null |
Section: Definition. MLOps is a paradigm, including aspects like best practices, sets of concepts, as well as a development culture when it comes to the end-to-end conceptualization, implementation, monitoring, deployment, and scalability of machine learning products. Most of all, it is an engineering practice that lev... | Wikipedia - MLOps - Definition | 176 | 906 | null |
Section: History. The challenges of the ongoing use of machine learning in applications were highlighted in a 2015 paper. The predicted growth in machine learning included an estimated doubling of ML pilots and implementations from 2017 to 2018, and again from 2018 to 2020. MLOps rapidly began to gain traction among AI... | Wikipedia - MLOps - History | 157 | 826 | null |
Section: Features > V1. MobileNetV1 was published in April 2017. Its main architectural innovation was incorporation of depthwise separable convolutions. It was first developed by Laurent Sifre during an internship at Google Brain in 2013 as an architectural variation on AlexNet to improve convergence speed and model s... | Wikipedia - MobileNet - Features > V1 | 248 | 1,123 | null |
Section: Features > V2. MobileNetV2 was published in March 2019. It uses inverted residual layers and linear bottlenecks. Inverted residuals modify the traditional residual block structure. Instead of compressing the input channels before the depthwise convolution, they expand them. This expansion is followed by a 1 Γ ... | Wikipedia - MobileNet - Features > V2 | 196 | 972 | null |
Article: Model compression. Model compression is a machine learning technique for reducing the size of trained models. Large models can achieve high accuracy, but often at the cost of significant resource requirements. Compression techniques aim to compress models without significant performance reduction. Smaller mode... | Wikipedia - Model compression - Summary | 162 | 950 | null |
Section: Techniques > Quantization. Quantization reduces the numerical precision of weights and activations. For example, instead of storing weights as 32-bit floating-point numbers, they can be represented using 8-bit integers. Low-precision parameters take up less space, and takes less compute to perform arithmetic w... | Wikipedia - Model compression - Techniques > Quantization | 160 | 799 | null |
Section: Techniques > Low-rank factorization. Weight matrices can be approximated by low-rank matrices. Let W {\displaystyle W} be a weight matrix of shape m Γ n {\displaystyle m\times n} . A low-rank approximation is W β U V T {\displaystyle W\approx UV^{T}} , where U {\displaystyle U} and V {\displaystyle V} are matr... | Wikipedia - Model compression - Techniques > Low-rank factorization | 239 | 969 | null |
Section: Training. Model compression may be decoupled from training, that is, a model is first trained without regard for how it might be compressed, then it is compressed. However, it may also be combined with training. The "train big, then compress" method trains a large model for a small number of training steps (le... | Wikipedia - Model compression - Training | 231 | 1,057 | null |
Section: Technical details > Update function. For every time step: A c t i o n = [ β 1 , 0 , 1 ] {\displaystyle Action=[-1,0,1]} V e l o c i t y = V e l o c i t y + ( A c t i o n ) β 0.001 + cos β‘ ( 3 β P o s i t i o n ) β ( β 0.0025 ) {\displaystyle Velocity=Velocity+(Action)*0.001+\cos(3*Position)*(-0.0025)} P o s i ... | Wikipedia - Mountain car problem - Technical details > Update function | 165 | 406 | null |
Article: Multi-armed bandit. In probability theory and machine learning, the multi-armed bandit problem (sometimes called the K- or N-armed bandit problem) is a problem in which a decision maker iteratively selects one of multiple fixed choices (i.e., arms or actions) when the properties of each choice are only partial... | Wikipedia - Multi-armed bandit - Summary | 346 | 1,718 | null |
In the problem, each machine provides a random reward from a probability distribution specific to that machine, that is not known a priori. The objective of the gambler is to maximize the sum of rewards earned through a sequence of lever pulls. The crucial tradeoff the gambler faces at each trial is between "exploitati... | Wikipedia - Multi-armed bandit - Summary | 220 | 1,150 | null |
Section: Empirical motivation. The multi-armed bandit problem models an agent that simultaneously attempts to acquire new knowledge (called "exploration") and optimize their decisions based on existing knowledge (called "exploitation"). The agent attempts to balance these competing tasks in order to maximize their tota... | Wikipedia - Multi-armed bandit - Empirical motivation | 251 | 1,436 | null |
Section: The multi-armed bandit model. The multi-armed bandit (short: bandit or MAB) can be seen as a set of real distributions B = { R 1 , β¦ , R K } {\displaystyle B=\{R_{1},\dots ,R_{K}\}} , each distribution being associated with the rewards delivered by one of the K β N + {\displaystyle K\in \mathbb {N} ^{+}} lever... | Wikipedia - Multi-armed bandit - The multi-armed bandit model | 208 | 757 | null |
The bandit problem is formally equivalent to a one-state Markov decision process. The regret Ο {\displaystyle \rho } after T {\displaystyle T} rounds is defined as the expected difference between the reward sum associated with an optimal strategy and the sum of the collected rewards: Ο = T ΞΌ β β β t = 1 T r ^ t {\displ... | Wikipedia - Multi-armed bandit - The multi-armed bandit model | 269 | 905 | null |
Section: Variations. A common formulation is the Binary multi-armed bandit or Bernoulli multi-armed bandit, which issues a reward of one with probability p {\displaystyle p} , and otherwise a reward of zero. Another formulation of the multi-armed bandit has each arm representing an independent Markov machine. Each time... | Wikipedia - Multi-armed bandit - Variations | 204 | 995 | null |
Section: Variations > Best Arm Identification. An important variation of the classical regret minimization problem in multi-armed bandits is the one of Best Arm Identification (BAI), also known as pure exploration. This problem is crucial in various applications, including clinical trials, adaptive routing, recommendat... | Wikipedia - Multi-armed bandit - Variations > Best Arm Identification | 310 | 1,245 | null |
An algorithm in this setting is characterized by a sampling rule, a decision rule, and a stopping rule, described as follows: Sampling rule: ( a t ) t β₯ 1 {\displaystyle (a_{t})_{t\geq 1}} is a sequence of actions at each time step Stopping rule: Ο {\displaystyle \tau } is a (random) stopping time which suggests when t... | Wikipedia - Multi-armed bandit - Variations > Best Arm Identification | 371 | 1,230 | null |
Fixed confidence setting: Given a confidence level Ξ΄ β ( 0 , 1 ) {\displaystyle \delta \in (0,1)} , the objective is to identify the arm with the highest expected reward a β β arg β‘ max k ΞΌ k {\displaystyle a^{\star }\in \arg \max _{k}\mu _{k}} with the least possible amount of trials and with probability of error P ( ... | Wikipedia - Multi-armed bandit - Variations > Best Arm Identification | 322 | 994 | null |
) = M {\displaystyle \int \sum m_{1},m_{2},(...)=M} , identify M {\displaystyle M} as the sum of each attempts m 1 + m 2 {\displaystyle m_{1}+m_{2}} , (...) as needed, and from there you can get a ratio, sum or mean as quantitative probability and sample your formulation for each slots. You can also do β« β k β i N β ( ... | Wikipedia - Multi-armed bandit - Variations > Best Arm Identification | 219 | 601 | null |
. . ) {\displaystyle (m1_{x},_{y})+(m2_{x},_{y})(...)} , k {\displaystyle k} would be the total available amount in your possession, k {\displaystyle k} is relative to N {\displaystyle N} where N = n ( n a , b ) , ( n 1 a , b ) , ( n 2 a , b ) {\displaystyle N=n(n_{a},b),(n1_{a},b),(n2_{a},b)} reduced n j {\displaystyl... | Wikipedia - Multi-armed bandit - Variations > Best Arm Identification | 298 | 910 | null |
Section: Bandit strategies > Optimal solutions. In the paper "Asymptotically efficient adaptive allocation rules", Lai and Robbins (following papers of Robbins and his co-workers going back to Robbins in the year 1952) constructed convergent population selection policies that possess the fastest rate of convergence (to... | Wikipedia - Multi-armed bandit - Bandit strategies > Optimal solutions | 322 | 1,741 | null |
In this work, the authors constructed an explicit form for a class of adaptive policies with uniformly maximum convergence rate properties for the total expected finite horizon reward under sufficient assumptions of finite state-action spaces and irreducibility of the transition law. A main feature of these policies is... | Wikipedia - Multi-armed bandit - Bandit strategies > Optimal solutions | 311 | 1,596 | null |
Section: Bandit strategies > Approximate solutions > Semi-uniform strategies. Semi-uniform strategies were the earliest (and simplest) strategies discovered to approximately solve the bandit problem. All those strategies have in common a greedy behavior where the best lever (based on previous observations) is always pu... | Wikipedia - Multi-armed bandit - Bandit strategies > Approximate solutions > Semi-uniform strategies | 349 | 1,661 | null |
Adaptive epsilon-greedy strategy based on value differences (VDBE): Similar to the epsilon-decreasing strategy, except that epsilon is reduced on basis of the learning progress instead of manual tuning (Tokic, 2010). High fluctuations in the value estimates lead to a high epsilon (high exploration, low exploitation); l... | Wikipedia - Multi-armed bandit - Bandit strategies > Approximate solutions > Semi-uniform strategies | 252 | 1,269 | null |
Section: Contextual bandit > Approximate solutions for contextual bandit > Online non-linear bandits. UCBogram algorithm: The nonlinear reward functions are estimated using a piecewise constant estimator called a regressogram in nonparametric regression. Then, UCB is employed on each constant piece. Successive refineme... | Wikipedia - Multi-armed bandit - Contextual bandit > Approximate solutions for contextual bandit > Online non-linear bandits | 189 | 989 | null |
Section: Contextual bandit > Constrained contextual bandit. In practice, there is usually a cost associated with the resource consumed by each action and the total cost is limited by a budget in many applications such as crowdsourcing and clinical trials. Constrained contextual bandit (CCB) is such a model that conside... | Wikipedia - Multi-armed bandit - Contextual bandit > Constrained contextual bandit | 284 | 1,282 | null |
Section: Adversarial bandit > Example: Iterated prisoner's dilemma. An example often considered for adversarial bandits is the iterated prisoner's dilemma. In this example, each adversary has two arms to pull. They can either Deny or Confess. Standard stochastic bandit algorithms don't work very well with these iterati... | Wikipedia - Multi-armed bandit - Adversarial bandit > Example: Iterated prisoner's dilemma | 160 | 757 | null |
Section: Adversarial bandit > Approximate solutions > Exp3. Source: EXP3 is a popular algorithm for adversarial multiarmed bandits, suggested and analyzed in this setting by Auer et al. [2002b]. Recently there was an increased interest in the performance of this algorithm in the stochastic setting, due to its new appli... | Wikipedia - Multi-armed bandit - Adversarial bandit > Approximate solutions > Exp3 | 177 | 756 | null |
Section: Adversarial bandit > Approximate solutions > Exp3 > Algorithm. Parameters: Real Ξ³ β ( 0 , 1 ] {\displaystyle \gamma \in (0,1]} Initialisation: Ο i ( 1 ) = 1 {\displaystyle \omega _{i}(1)=1} for i = 1 , . . . , K {\displaystyle i=1,...,K} For each t = 1, 2, ..., T 1. Set p i ( t ) = ( 1 β Ξ³ ) Ο i ( t ) β j = 1 ... | Wikipedia - Multi-armed bandit - Adversarial bandit > Approximate solutions > Exp3 > Algorithm | 344 | 737 | null |
. . , K {\displaystyle j=1,...,K} set: x ^ j ( t ) = { x j ( t ) / p j ( t ) if j = i t 0 , otherwise {\displaystyle {\hat {x}}_{j}(t)={\begin{cases}x_{j}(t)/p_{j}(t)&{\text{if }}j=i_{t}\\0,&{\text{otherwise}}\end{cases}}} Ο j ( t + 1 ) = Ο j ( t ) exp β‘ ( Ξ³ x ^ j ( t ) / K ) {\displaystyle \omega _{j}(t+1)=\omega _{j}... | Wikipedia - Multi-armed bandit - Adversarial bandit > Approximate solutions > Exp3 > Algorithm | 198 | 356 | null |
Section: Adversarial bandit > Approximate solutions > Follow the perturbed leader (FPL) algorithm > Algorithm. Parameters: Real Ξ· {\displaystyle \eta } Initialisation: β i : R i ( 1 ) = 0 {\displaystyle \forall i:R_{i}(1)=0} For each t = 1,2,...,T 1. For each arm generate a random noise from an exponential distribution... | Wikipedia - Multi-armed bandit - Adversarial bandit > Approximate solutions > Follow the perturbed leader (FPL) algorithm > Algorithm | 309 | 761 | null |
Section: Non-stationary bandit. This framework refers to the multi-armed bandit problem in a non-stationary setting (i.e., in presence of concept drift). In the non-stationary setting, it is assumed that the expected reward for an arm k {\displaystyle k} can change at every time step t β T {\displaystyle t\in {\mathcal... | Wikipedia - Multi-armed bandit - Non-stationary bandit | 331 | 1,050 | null |
The dynamic oracle optimises the expected reward at each step t β T {\displaystyle t\in {\mathcal {T}}} by always selecting the best arm, with expected reward of ΞΌ t β {\displaystyle \mu _{t}^{*}} . Thus, the cumulative expected reward D ( T ) {\displaystyle {\mathcal {D}}(T)} for the dynamic oracle at final time step ... | Wikipedia - Multi-armed bandit - Non-stationary bandit | 294 | 812 | null |
{\displaystyle {\mathcal {D}}(T)=\sum _{t=1}^{T}{\mu _{t}^{*}}.} Hence, the regret Ο Ο ( T ) {\displaystyle \rho ^{\pi }(T)} for policy Ο {\displaystyle \pi } is computed as the difference between D ( T ) {\displaystyle {\mathcal {D}}(T)} and the cumulative expected reward at step T {\displaystyle T} for policy Ο {\dis... | Wikipedia - Multi-armed bandit - Non-stationary bandit | 339 | 772 | null |
{\displaystyle \rho ^{\pi }(T)=\sum _{t=1}^{T}{\mu _{t}^{*}}-\mathbb {E} _{\pi }^{\mu }\left[\sum _{t=1}^{T}{r_{t}}\right]={\mathcal {D}}(T)-\mathbb {E} _{\pi }^{\mu }\left[\sum _{t=1}^{T}{r_{t}}\right].} Garivier and Moulines derive some of the first results with respect to bandit problems where the underlying model c... | Wikipedia - Multi-armed bandit - Non-stationary bandit | 293 | 946 | null |
Section: Other variants > Dueling bandit. The dueling bandit variant was introduced by Yue et al. (2012) to model the exploration-versus-exploitation tradeoff for relative feedback. In this variant the gambler is allowed to pull two levers at the same time, but they only get a binary feedback telling which lever provid... | Wikipedia - Multi-armed bandit - Other variants > Dueling bandit | 204 | 980 | null |
Article: Multi-task learning. Multi-task learning (MTL) is a subfield of machine learning in which multiple learning tasks are solved at the same time, while exploiting commonalities and differences across tasks. This can result in improved learning efficiency and prediction accuracy for the task-specific models, when ... | Wikipedia - Multi-task learning - Summary | 340 | 1,828 | null |
Section: Methods > Task grouping and overlap. Within the MTL paradigm, information can be shared across some or all of the tasks. Depending on the structure of task relatedness, one may want to share information selectively across the tasks. For example, tasks may be grouped or exist in a hierarchy, or be related accor... | Wikipedia - Multi-task learning - Methods > Task grouping and overlap | 234 | 1,221 | null |
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