gradientregularization

Gradient regularization refers to a family of regularization techniques in machine learning that penalize large gradients of a model's predictions or loss with respect to its inputs. By discouraging rapid changes in response to small input perturbations, gradient regularization aims to produce smoother functions, improve generalization, and enhance robustness.

Two common formulations are used in supervised learning. The first penalizes the gradient of the loss with respect to the input: add a term lambda times the squared norm of grad_x L(θ, x, y). The second penalizes the Frobenius norm of the Jacobian of the model outputs with respect to the inputs: add a term lambda times ||J_f(x)||_F^2, where J_f(x) = ∂f(x)/∂x. A related idea is the contractive autoencoder, which adds a penalty on the Jacobian of the hidden representation with respect to the input to encourage invariance.

Compared with conventional weight-based regularization (such as L2 on weights), gradient regularization targets the smoothness of the input–output map itself. It can improve robustness to adversarial perturbations and calibration but may increase computational cost because additional gradient computations are required during training. The strength of the regularization is controlled by a hyperparameter lambda and may interact with data augmentation, model architecture, and optimization dynamics.

Applications span supervised classification, regression, and representation learning, with particular interest in settings where input perturbations are likely or where smoother decision boundaries are desired. Related approaches include Jacobian regularization, contractive autoencoders, and gradient penalties used in training some generative models.