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Training

Objective Function

ActivePublished: 24 October 2026Updated: 24 October 2026Published
Key innovation
Reduces model learning to the optimization of a single scalar quality measure, providing one unambiguous signal against which all parameters are adjusted.
Category
Training
Abstraction level
Primitive
Operation level
TrainingPost-training
Use cases
Gradient-based neural network trainingClassification (cross-entropy)Regression (mean squared error)Reinforcement learning (reward maximization)Preference tuning of language models (RLHF, DPO)Regularization and overfitting control

How it works

The objective function J(θ) takes model parameters θ and returns a scalar. In supervised learning it is typically the average loss over the training data (empirical risk), often with an added regularization term that limits model complexity (structural risk minimization). An optimization algorithm (e.g. gradient descent) computes the gradient of the objective with respect to the parameters and updates them in the opposite direction (for minimization), repeating until convergence. The chosen form of the objective - cross-entropy for classification, mean squared error for regression, a reward function for reinforcement learning - determines the behaviour the model develops.

Problem solved

Training a model requires an unambiguous, measurable criterion of success. The objective function reduces an informal goal (e.g. classify well) to a single, usually differentiable number that can be optimized systematically. Without it there would be no signal against which an algorithm could improve parameters.

Components

Loss / data-fidelity termPrimary learning signal

Measures the discrepancy between model predictions and target values (e.g. cross-entropy, mean squared error). Averaged over the training data it forms the empirical risk.

Regularization termComplexity and overfitting control

An optional component that penalizes model complexity (e.g. L2 norm of weights) to limit overfitting. The sum of the loss term and the regularizer forms the full objective (structural risk minimization).

Evolution

1950
Formalization in statistical decision theory
Inflection point

Abraham Wald, in Statistical Decision Functions (1950), reintroduced the loss function into statistics as the basis for decisions; the concept itself dates back to Laplace.

1986
Gradient-based minimization in neural networks
Inflection point

The popularization of backpropagation (Rumelhart, Hinton, Williams, 1986) made the differentiable objective function the central element of neural network training.

Hyperparameters (configurable axes)

Loss function formCritical

Choice of the specific loss (cross-entropy, MSE, hinge, contrastive, ...) depending on the task and data distribution.

Regularization strength (lambda)High

Weight of the regularization term relative to the loss term; controls the bias-variance trade-off.