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Loss function information


In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) [1] 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 seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy.

In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century.[2] In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in the 1920s.[3] In optimal control, the loss is the penalty for failing to achieve a desired value. In financial risk management, the function is mapped to a monetary loss.

Comparison of common loss functions used for regression
  1. ^ Hastie, Trevor; Tibshirani, Robert; Friedman, Jerome H. (2001). The Elements of Statistical Learning. Springer. p. 18. ISBN 0-387-95284-5.
  2. ^ Wald, A. (1950). Statistical Decision Functions. Wiley.
  3. ^ Cramér, H. (1930). On the mathematical theory of risk. Centraltryckeriet.

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Loss function

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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...

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Loss functions for classification

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learning and mathematical optimization, loss functions for classification are computationally feasible loss functions representing the price paid for inaccuracy...

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Huber loss

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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...

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Taguchi loss function

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The Taguchi loss function is graphical depiction of loss developed by the Japanese business statistician Genichi Taguchi to describe a phenomenon affecting...

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Backpropagation

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context of control theory. Backpropagation computes the gradient of a loss function with respect to the weights of the network for a single input–output...

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Mutation

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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...

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Triplet loss

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Triplet loss is a loss function for machine learning algorithms where a reference input (called anchor) is compared to a matching input (called positive)...

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Hinge loss

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In machine learning, the hinge loss is a loss function used for training classifiers. The hinge loss is used for "maximum-margin" classification, most...

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Loss

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fiber Dielectric loss, a dielectric material's inherent dissipation of electromagnetic energy Loss function, in statistics, a function representing the...

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Quantile regression

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\right\}} where τ ∈ ( 0 , 1 ) . {\displaystyle \tau \in (0,1).} Define the loss function as ρ τ ( m ) = m ( τ − I ( m < 0 ) ) {\displaystyle \rho _{\tau }(m)=m(\tau...

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Taguchi methods

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comparisons of treatment means. However, loss functions were avoided by Ronald A. Fisher[clarification needed - loss functions weren't explicitly mentioned yet]...

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Cost function

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function In economics, the cost curve, expressing production costs in terms of the amount produced. In mathematical optimization, the loss function,...

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Gradient boosting

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other methods by allowing optimization of an arbitrary differentiable loss function. The idea of gradient boosting originated in the observation by Leo...

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FaceNet

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128-dimensional Euclidean space. The system used the triplet loss function as the cost function and introduced a new online triplet mining method. The system...

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Statistical learning theory

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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...

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Support vector machine

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between the hinge loss and these other loss functions is best stated in terms of target functions - the function that minimizes expected risk for a given...

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Bias of an estimator

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central tendency; because a biased estimator gives a lower value of some loss function (particularly mean squared error) compared with unbiased estimators...

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Median

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risk with respect to the absolute-deviation loss function, as observed by Laplace. Other loss functions are used in statistical theory, particularly...

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Expected loss

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understanding and mitigation of systemic risk. Systemic risk Loss function Loss function § Expected loss Potential future exposure Sandra Thompson/Voon Hoe Chen...

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Tetraplegia

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as the dysfunction or loss of motor and/or sensory function in the cervical area of the spinal cord. A loss of motor function can present as either weakness...

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Supervised learning

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(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...

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Hyperparameter optimization

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predefined loss function on given independent data. The objective function takes a tuple of hyperparameters and returns the associated loss. Cross-validation...

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Mean squared error

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values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly...

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Bayes estimator

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value of a loss function (i.e., the posterior expected loss). Equivalently, it maximizes the posterior expectation of a utility function. An alternative...

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Likelihood function

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likelihood function (often simply called the likelihood) is the joint probability mass (or probability density) of observed data viewed as a function of the...

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Maximum a posteriori estimation

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linear-error loss respectively—which are more representative of typical loss functions—and for a continuous posterior distribution there is no loss function which...

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Survival function

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certain time. The survival function is also known as the survivor function or reliability function. The term reliability function is common in engineering...

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Scoring rule

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or the median. Scoring rules and scoring functions can be thought of as "cost functions" or "loss functions". They are evaluated as empirical mean of...

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