Regularization

What it is

Regularization is the process of turning a problem's answer into a simpler one. The name covers two closely related jobs. The first is to make an ill-posed problem solvable: a problem where the data alone do not single out a stable answer can be made well behaved by adding a preference for simplicity. The second is to prevent overfitting, the failure mode in which a model fits its training data so closely that it no longer generalizes. The idea shows up across mathematics, statistics, finance and computer science, and is especially prominent in machine learning and in inverse problems.^[1]

How it works

In practice regularization is written as an extra term in the objective. Instead of minimizing only how badly the model fits the data, one minimizes a data-fit term plus a penalty term λR(f), where λ sets how much the penalty counts.^[1] A larger λ favors simpler models, which corresponds to a stronger prior belief about what the answer should look like; a smaller λ lets the fit to the data dominate. The trade-off is a working version of Occam's razor.^[1]

A concrete picture: fitting a curve through noisy measurements. A flexible curve can be pushed through every point, but it will swing between them and predict new points badly. Adding a penalty on the size of the curve's coefficients pulls it toward a smoother shape, accepting a slightly worse fit on the measured points in exchange for steadier behavior elsewhere.

The choice of penalty defines the family. Penalizing the squared size of the parameters is L2 regularization, or ridge regression. Ridge is explicitly a biased estimator: it trades a reduction in variance for an increase in bias.^[1] Penalizing the absolute size instead is L1 regularization, the lasso.

Google's machine learning guidance describes L2 regularization as a technique used to reduce model complexity and prevent overfitting by penalizing large weights, with a regularization rate, lambda, controlling the strength: higher values lead to simpler models, lower values raise the risk of overfitting.^[3] The same guidance notes early stopping as an alternative regularization method that ends training before the model fully converges,^[3] and regularization is listed among the standard techniques for reducing overfitting, alongside early stopping, pruning and dropout.^[2]

Where it came from

Ridge regularization carries the name of Andrey Nikolayevich Tikhonov, who applied regularization to integral equations.^[1] In statistics the ridge estimator is associated with Arthur E. Hoerl and Robert W. Kennard, whose paper "Ridge regression: Biased estimation for nonorthogonal problems" appeared in Technometrics in 1970.^[1] The lasso, or L1 regularization, was proposed by Robert Tibshirani in 1996 in the Journal of the Royal Statistical Society Series B.^[1]

Limits

The family is not free of trouble. L1 regularization can produce non-unique solutions, which motivated the elastic net, combining L1 and L2 penalties.^[1] Penalizing the number of nonzero parameters directly, L0 regularization, is NP-hard, as Natarajan showed in 1995.^[1] And every version involves the same bargain: the strength of the penalty has to be chosen, and a poorly chosen value either leaves the overfitting in place or suppresses useful signal.