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A loss function assigns a real-valued cost to outcomes; optimization usually minimizes it, while objective functions may be defined as the negative of loss and therefore maximized.
In mathematical optimization and decision theory, a loss (or cost/error) function maps an event or variable values to a real number that represents the “cost” of that outcome. Optimization problems typically aim to minimize the loss. An objective function is closely related: in many settings it is either the loss itself or its opposite (e.g., reward/profit/utility), in which case the objective is maximized. Loss functions appear across fields such as statistics (parameter estimation), classification (penalty for incorrect labels), economics (economic cost or regret), actuarial science (insurance modeling), optimal control (penalty for failing to reach desired targets), and financial risk management (monetary loss).
A loss function assigns a real-valued cost to outcomes; optimization usually minimizes it, while objective functions may be defined as the negative of loss and therefore maximized.
In statistics, loss depends on the discrepancy between estimated and true quantities; expected loss leads to risk functions (frequentist) and Bayes risk (Bayesian).
Common loss choices include quadratic (squared error) and 0-1 loss (Hamming distortion in information theory), and the choice of loss strongly affects sensitivity to outliers and the resulting optimal decisions.
A function that maps an event or variable values to a real number representing the cost associated with that outcome.
A scalar function to optimize, which may be the loss function or its opposite (e.g., reward/utility), leading to minimization or maximization respectively.
The expectation of the loss with respect to the sampling distribution of the observed data under a fixed parameter, often denoted R(θ,δ).
The expected loss averaged over both the parameter (using a prior) and the data, minimized by the Bayes decision rule.
A classification loss that is 0 when the prediction is correct and 1 when it is incorrect.
A loss of the form λ(x)=C(t−x)^2 that penalizes squared deviations from a target and is widely used in least-squares methods.
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