Loss Functions in Decision and Statistics

Definition of loss and objective functions

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).

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Optimization as loss minimization

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