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Mathematical relation assigning a probability event to a cost
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
Loss_function
Concept in machine learning
learning and mathematical optimization, loss functions for classification are computationally feasible loss functions representing the price paid for inaccuracy
Loss functions for classification
Loss_functions_for_classification
Loss function used in robust regression
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
Huber_loss
Optimization algorithm for artificial neural networks
in weight space of a feedforward neural network, with respect to a loss function. Denote: x: input (vector of features) y: target output For classification
Backpropagation
Information-theoretic measure
"Logistic loss refers to the loss function commonly used to optimize a logistic regression model. It may also be referred to as logarithmic loss (which is
Cross-entropy
Graphical depicture of loss
The Taguchi loss function is graphical depiction of loss developed by the Japanese business statistician Genichi Taguchi to describe a phenomenon affecting
Taguchi_loss_function
Loss function in machine learning
In machine learning, hinge loss is a loss function used for training classifiers. Hinge loss is used for "maximum-margin" classification, most notably
Hinge_loss
Alteration in the nucleotide sequence of a genome
mutations, are a form of loss-of-function mutations that completely prohibit the gene's function. The mutation leads to a complete loss of operation at the
Mutation
Technique to make a model more generalizable and transferable
regularization. This includes, for example, early stopping, using a robust loss function, and discarding outliers. Implicit regularization is essentially ubiquitous
Regularization_(mathematics)
Topics referred to by the same term
fiber Dielectric loss, a dielectric material's inherent dissipation of electromagnetic energy Loss function, in statistics, a function representing the
Loss
Statistical modeling technique
\tau \right\},} where 0 < τ < 1 {\displaystyle 0<\tau <1} . Define the loss function as ρ τ ( u ) = u ( τ − I ( u < 0 ) ) = { ( τ − 1 ) u , if u < 0 , τ
Quantile_regression
Framework for machine learning
The most common loss function for regression is the square loss function (also known as the L2-norm). This familiar loss function is used in Ordinary
Statistical_learning_theory
Machine learning technique
other methods by allowing optimization of an arbitrary differentiable loss function. The idea of gradient boosting originated in the observation by Leo
Gradient_boosting
Function for machine learning algorithms
Triplet loss is a machine learning loss function widely used in one-shot learning, a setting where models are trained to generalize effectively from limited
Triplet_loss
Type of neural network output and associated scoring function
Connectionist temporal classification (CTC) is a loss function and output representation used to train neural networks on sequence-labelling tasks where
Connectionist temporal classification
Connectionist_temporal_classification
Statistical property
central tendency; because a biased estimator gives a lower value of some loss function (particularly mean squared error) compared with unbiased estimators
Bias_of_an_estimator
Facial recognition system
128-dimensional Euclidean space. The system uses the triplet loss function as its cost function and introduced a new online triplet mining method. The system
FaceNet
Statistical methods to improve the quality of manufactured goods
comparisons of treatment means. However, loss functions were avoided by Ronald A. Fisher[clarification needed - loss functions weren't explicitly mentioned yet]
Taguchi_methods
Principle in statistical learning theory
function: R(h)=𝐄[L(h(x),y)]=∫L(h(x),y)dP(x,y). A loss function commonly used in theory is the 0-1 loss function: L(ŷ,y)={1 if ŷ≠y0 if ŷ=y. The ultimate goal
Empirical_risk_minimization
Line-breaking algorithm used in the TeX typesetting package
algorithm by using a discrete dynamic programming method to minimize a loss function that attempts to quantify the aesthetic qualities desired in the finished
Knuth–Plass line-breaking algorithm
Knuth–Plass_line-breaking_algorithm
Machine learning method to transfer knowledge from a large model to a smaller one
training set or consist of new, possibly unlabeled data. A cross-entropy loss function is typically used, computed between the output of the distilled model
Knowledge_distillation
Gradient boosting machine learning library
in function space unlike gradient boosting that works as gradient descent in function space, a second order Taylor approximation is used in the loss function
XGBoost
Measure for evaluating probabilistic forecasts
target variable. Scoring rules and scoring functions are often used as "cost functions" or "loss functions" of forecasting models. If a sample of forecasts
Scoring_rule
Topics referred to by the same term
function In economics, the cost curve, expressing production costs in terms of the amount produced. In mathematical optimization, the loss function,
Cost_function
Mathematical decision rule
value of a loss function (i.e., the posterior expected loss). Equivalently, it maximizes the posterior expectation of a utility function. An alternative
Bayes_estimator
Method of machine learning
this end, a square loss function V(f(xj),yj)=(f(xj)−yj)2=(⟨w,xj⟩−yj)2 is used to compute the vector w that minimizes the empirical loss In[w]=∑j=1nV(⟨w,xj⟩
Online_machine_learning
Technique to solve partial differential equations
of u(t,x) and f(t,x) can be then learned by minimizing the following loss function Ltot: Ltot=Lu+Lf where: Lu=‖u−z‖Γ is the error between the PINN u(t
Physics-informed neural networks
Physics-informed_neural_networks
Machine learning model training problem
weights are updated proportional to their partial derivative of the loss function. As the number of forward propagation steps in a network increases,
Vanishing_gradient_problem
Machine learning paradigm
(x_{i},\;y_{i})} . In order to measure how well a function fits the training data, a loss function L : Y × Y → R ≥ 0 {\displaystyle L:Y\times Y\to \mathbb
Supervised_learning
Aspect of decision and prospect theories
behavioral economics, loss aversion is a cognitive bias in which the same situation is perceived as worse if it is framed as a loss, rather than a gain
Loss_aversion
Adaptive boosting based classification algorithm
loss function based on least squares, a modified Huber loss function: ϕ(y,f(x))={−4yf(x)if yf(x)<−1,(yf(x)−1)2if −1≤yf(x)≤1.0if yf(x)>1 This function
AdaBoost
Process of finding the optimal set of variables for a machine learning algorithm
minimizes a predefined loss function on a given data set. The objective function takes a set of hyperparameters and returns the associated loss. Cross-validation
Hyperparameter_optimization
Use of machine learning to rank items
documents as its input and the goal is to minimize a loss function L(h;xu,xv,yu,v). The loss function typically reflects the number and magnitude of inversions
Learning_to_rank
Function that maps an observation to an action
evaluate the usefulness of a decision rule, it is necessary to have a loss function detailing the outcome of each action under different states. Given an
Decision_rule
Mathematical model used for classification or regression
to separate classes or to minimize prediction error under a chosen loss function. They are often contrasted with generative models, which aim to model
Discriminative_model
Machine learning technique
f(x)=∑iw(x)ifi(x). Both the experts and the weighting function are trained by minimizing some loss function, generally via gradient descent. There is much freedom
Mixture_of_experts
Variant of the SVM algorithm
{\displaystyle q} . Empirical loss function Since the expected loss function is not applicable, the following empirical loss function is selected for the training
Ranking_SVM
Middle quantile of a data set or probability distribution
risk with respect to the absolute-deviation loss function, as observed by Laplace. Other loss functions are used in statistical theory, particularly
Median
Computational model used in machine learning
long as the value of the loss function (its cost) continues to decline, the network is continuing to improve. The function typically produces a statistic
Neural network (machine learning)
Neural_network_(machine_learning)
Set of methods for supervised statistical learning
the square-loss, the target function is the conditional expectation function, fsq(x)=𝔼[yx]; For the logistic loss, it's the logit function, flog(x)=ln(px/(1−px))
Support_vector_machine
Method of estimating the parameters of a statistical model
linear-error loss respectively—which are more representative of typical loss functions—and for a continuous posterior distribution there is no loss function which
Maximum a posteriori estimation
Maximum_a_posteriori_estimation
Parameter controlling the machine learning process
a predefined loss function on given test data. The objective function takes a tuple of hyperparameters and returns the associated loss. Typically these
Hyperparameter (machine learning)
Hyperparameter_(machine_learning)
Machine learning technique
change of variables to define the "preference loss" directly as a function of the policy and uses this loss to fine-tune the model, helping it understand
Reinforcement learning from human feedback
Reinforcement_learning_from_human_feedback
Distance from a point to the boundary of a set
[non-primary source needed] A modified version of SDF was introduced as a loss function to minimise the error in interpenetration of pixels while rendering
Signed_distance_function
Process of using data analysis for predicting population data from sample data
most powerful testing) make use of loss functions, which play the role of (negative) utility functions. Loss functions need not be explicitly stated for
Statistical_inference
Notion in computational learning theory
algorithm to evaluate a learning algorithm's stability with respect to the loss function. As such, stability analysis is the application of sensitivity analysis
Stability_(learning_theory)
regularization, specifically Tikhonov regularization with the hinge loss for a loss function. This provides a theoretical framework with which to analyze SVM
Regularization perspectives on support vector machines
Regularization_perspectives_on_support_vector_machines
Deep learning generative model to encode data representation
that use gradient-based optimization, VAEs require a differentiable loss function to update the network weights through backpropagation. For variational
Variational_autoencoder
Calculation of financial risk
understanding and mitigation of systemic risk. Systemic risk Loss function Loss function § Expected loss Potential future exposure Credit valuation adjustment
Expected_loss
Measure of variation in statistics
{erf} } is the error function. The proportion that is less than or equal to a number, x, is given by the cumulative distribution function: Proportion ≤ x =
Standard_deviation
Statistical modeling method
respectively, the loss function can be rewritten as: L(D,β→)=‖Xβ→−Y‖2=(Xβ→−Y)𝖳(Xβ→−Y)=Y𝖳Y−Y𝖳Xβ→−β→𝖳X𝖳Y+β→𝖳X𝖳Xβ→ As the loss function is convex, the optimum
Linear_regression
Smoothed ramp function
softplus function is f ( x ) = ln ( 1 + e x ) . {\displaystyle f(x)=\ln(1+e^{x}).} It is a smooth approximation (in fact, an analytic function) to the
Softplus
Concept in the mathematical theory of decisions
First created in 1990 by Dr. Yiyu Yao, the extension makes use of loss functions to derive α {\displaystyle \textstyle \alpha } and β {\displaystyle
Decision-theoretic_rough_sets
Class of statistical models
response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value. Generalized
Generalized_linear_model
Optimization algorithm
we want to find the optimal control u ∗ {\displaystyle u^{*}} with loss function J ( u ) {\displaystyle J(u)} : u ∗ = arg min u ∈ U J ( u ) . {\displaystyle
Simultaneous perturbation stochastic approximation
Simultaneous_perturbation_stochastic_approximation
Statistical model for a binary dependent variable
surprising". Since the value of the logistic function is always strictly between zero and one, the log loss is always greater than zero and less than infinity
Logistic_regression
Function related to statistics and probability theory
A likelihood function (often simply called the likelihood) gives the relative merit of various statistical models for describing a data set. Often the
Likelihood_function
Mathematical function characterizing set membership
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all
Indicator_function
Tuning parameter (hyperparameter) in optimization
the step size at each iteration while moving toward a minimum of a loss function. Since it influences to what extent newly acquired information overrides
Learning_rate
Statistical classification in machine learning
the regularization and the loss function. Popular loss functions include the hinge loss (for linear SVMs) and the log loss (for linear logistic regression)
Linear_classifier
Statistical model validation technique
{\displaystyle i^{th}} candidate configuration that might be selected, then the loss function that is to be minimized can be defined as L λ i = ( 1 − γ ) Relative
Cross-validation_(statistics)
Algorithm for modelling sequential data
based on the context. The loss function for the task is typically sum of log-perplexities for the masked-out tokens: Loss=−∑t∈masked tokensln(probability
Transformer_(deep_learning)
Algorithm
Quickprop is an iterative method for determining the minimum of the loss function of an artificial neural network, following an algorithm inspired by
Quickprop
Correlation of a signal with a time-shifted copy of itself, as a function of shift
Pearson correlation between values of the process at different times, as a function of the two times or of the time lag. Let { X t } {\displaystyle \left\{X_{t}\right\}}
Autocorrelation
Technique for the generative modeling of a continuous probability distribution
integration by parts, Eq[‖fθ(x)−∇lnq(x)‖2]=Eq[‖fθ‖2+2∇⋅fθ]+Cgiving us a loss function, also known as the Hyvärinen scoring rule, that can be minimized by
Diffusion_model
Set of statistical processes for estimating the relationships among variables
independent variables. Most regression models propose that Yi is a function (regression function) of Xi and β, with ei representing an additive error term that
Regression_analysis
Probability of survival beyond any specified time
certain time. The survival function is also known as the survivor function or reliability function. The term reliability function is common in engineering
Survival_function
Reduction of the total body mass
medicine, health, or physical fitness, weight loss is a reduction of the total body mass, by a mean loss of fluid, body fat (adipose tissue), or lean mass
Weight_loss
Neural network working on two input vectors
(\cdot )} function implemented by the twin network The most common distance metric used is Euclidean distance, in case of which the loss function can be
Siamese_neural_network
Optimality condition in optimal control theory
conditions for optimality of a control with respect to a loss function. Its solution is the value function of the optimal control problem which, once known,
Hamilton–Jacobi–Bellman equation
Hamilton–Jacobi–Bellman_equation
Measure of power reflected by a discontinuity in a line or fiber
high return loss is desirable and results in a lower insertion loss. From a certain perspective "return loss" is a misnomer. The usual function of a transmission
Return_loss
Conditional probability used in Bayesian statistics
| X ) {\displaystyle p(\theta |X)} . It contrasts with the likelihood function, which is the probability of the evidence given the parameters: p ( X |
Posterior_probability
Fundamental theorem in probability theory and statistics
characteristic functions of a number of density functions becomes close to the characteristic function of the normal density as the number of density functions increases
Central_limit_theorem
Distribution function associated with the empirical measure of a sample
an empirical distribution function (a.k.a. an empirical cumulative distribution function, eCDF) is the distribution function associated with the empirical
Empirical distribution function
Empirical_distribution_function
Paralysis of all four limbs and torso
or loss of motor or sensory function in all four limbs, typically caused by damage to the cervical area of the spinal cord. A loss of motor function can
Tetraplegia
Plot of machine learning model performance over time or experience
or training set size is plotted on the x-axis, and the value of the loss function (and possibly some other metric such as the cross-validation score)
Learning curve (machine learning)
Learning_curve_(machine_learning)
Statistical theorem
{E} [L(\delta (X))]} where the "loss function" L {\displaystyle L} may be any convex function. If the loss function is twice-differentiable, as in the
Rao–Blackwell_theorem
Measure of prediction accuracy of a forecast
section below). Mean absolute percentage error is commonly used as a loss function for regression problems and in model evaluation, because of its very
Mean absolute percentage error
Mean_absolute_percentage_error
Type of feedforward neural network
evaluates the loss function L(ŷ,y), which quantifies the error between predictions and targets. The backward pass computes gradients of the loss with respect
Multilayer_perceptron
Measure of statistical dispersion
calculated by integrating the probability density function (which yields the cumulative distribution function—any other means of calculating the CDF will also
Interquartile_range
Specialized form of regression analysis, in statistics
error loss, and therefore has more leverage over the regression estimates. The Huber loss function is a robust alternative to standard square error loss that
Robust_regression
Optimization algorithm
machine learning and artificial intelligence for minimizing the cost or loss function. Gradient descent is generally attributed to Augustin-Louis Cauchy,
Gradient_descent
Approximation method in statistics
Orthogonal projection Proximal gradient methods for learning Quadratic loss function Root mean square Squared deviations from the mean Charnes, A.; Frome
Least_squares
Type of statistics
known as the Huber loss function. Tukey's biweight (also known as bisquare) function behaves in a similar way to the squared error function at first, but for
Robust_statistics
Measure of the error of an estimator
values and the true value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly
Mean_squared_error
Response to loss in humans and other animals
already begun to adapt to after the loss. If the bereaved is unable to return to their normal functioning as in before loss occurred, it is likely they will
Grief
Optimization method
by gradually improving an approximation to the Hessian matrix of the loss function, obtained only from gradient evaluations (or approximate gradient evaluations)
Broyden–Fletcher–Goldfarb–Shanno algorithm
Broyden–Fletcher–Goldfarb–Shanno_algorithm
Software agent which acts autonomously
in optimization. Loss function: Typically used in machine learning, where the goal is to minimize the loss (error). Reward function: Used in reinforcement
Intelligent_agent
Nonparametric measure of rank correlation
relationship between two variables can be described using a monotonic function. The Spearman correlation between two variables is equal to the Pearson
Spearman's rank correlation coefficient
Spearman's_rank_correlation_coefficient
Diagnostic plot of binary classifier ability
The ROC can also be thought of as a plot of the statistical power as a function of the Type I Error of the decision rule (when the performance is calculated
Receiver operating characteristic
Receiver_operating_characteristic
Measure of the asymmetry of random variables
} where Q is the quantile function (i.e., the inverse of the cumulative distribution function). The numerator is difference between the
Skewness
Machine learning paradigm
requiring it to reconstruct the same data as closely as possible. The loss function used during training typically penalizes the difference between the
Self-supervised_learning
Covariance and correlation
processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as
Cross-correlation
Method of estimating the parameters of a statistical model, given observations
{\mathbb {P} } (x)}},} and if we further assume the zero-or-one loss function, which is a same loss for all errors, the Bayes Decision rule can be reformulated
Maximum_likelihood_estimation
Statistical measure of variability
reciprocal of the quantile function Φ − 1 {\displaystyle \Phi ^{-1}} (also known as the inverse of the cumulative distribution function) for the standard normal
Median_absolute_deviation
Topics referred to by the same term
equation, a condition for optimality of a control with respect to a loss function Bellman–Ford algorithm, a method for finding shortest paths Belman (disambiguation)
Bellman
Method of improving artificial neural network
an arbitrary loss function. Suppose that ϕ is infinitely differentiable and has a bounded derivative. Assume that the objective function fLH is ζ-smooth
Batch_normalization
Mathematical function for the probability a given outcome occurs in an experiment
likely different results are. Formally, it is a probability measure: a function that assigns probabilities to events in a way that satisfies the axioms
Probability_distribution
Statistical method
median-unbiased estimators of minimum risk (with respect to an absolute loss function). Bias in the bootstrap distribution will lead to bias in the confidence
Bootstrapping_(statistics)
Statistic which divides a data set into 100 parts and analyzes it as a percentage
the percentile approximates the percentile function, the inverse of the cumulative distribution function. A related quantity is the percentile rank of
Percentile
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