Loss function

In mathematical optimization and decision theory, a loss function or cost 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 seeks to minimize a loss function. An objective function is either a loss function or its negative (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. In mathematical optimization and decision theory, a loss function or cost 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 seeks to minimize a loss function. An objective function is either a loss function or its negative (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. 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. 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. 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. In classical statistics (both frequentist and Bayesian), a loss function is typically treated as something of a background mathematical convention. Critics such as W. Edwards Deming and Nassim Nicholas Taleb have argued that loss functions require much greater attention than they have traditionally been given and that loss functions used in real world decision making need to reflect actual empirical experience. They argue that real-world loss functions are often very different from the smooth, symmetric ones used by classical convention, and are often highly asymmetric, nonlinear, and discontinuous. Leonard J. Savage argued that using non-Bayesian methods such as minimax, the loss function should be based on the idea of regret, i.e., the loss associated with a decision should be the difference between the consequences of the best decision that could have been made had the underlying circumstances been known and the decision that was in fact taken before they were known. The use of a quadratic loss function is common, for example when using least squares techniques. It is often more mathematically tractable than other loss functions because of the properties of variances, as well as being symmetric: an error above the target causes the same loss as the same magnitude of error below the target. If the target is t, then a quadratic loss function is for some constant C; the value of the constant makes no difference to a decision, and can be ignored by setting it equal to 1. Many common statistics, including t-tests, regression models, design of experiments, and much else, use least squares methods applied using linear regression theory, which is based on the quadratric loss function. The quadratic loss function is also used in linear-quadratic optimal control problems. In these problems, even in the absence of uncertainty, it may not be possible to achieve the desired values of all target variables. Often loss is expressed as a quadratic form in the deviations of the variables of interest from their desired values; this approach is tractable because it results in linear first-order conditions. In the context of stochastic control, the expected value of the quadratic form is used. In statistics and decision theory, a frequently used loss function is the 0-1 loss function