L0 regularization

See also: Machine learning terms

Introduction

In machine learning, L0 regularization is a regularization method that penalizes the non-zero weights in a model. It is an example of sparsity-inducing regularization, where the goal is to minimize features used in the model. As such, L0 regularization can be employed as a feature selection technique, decreasing features used and potentially improving interpretability and reducing overfitting.

L0 regularization is rarely used when compared to L1 regularization and L2 regularization.

Mathematical Definition

Let the model parameters be represented as a vector x, and the loss function as L(x). The goal of L0 regularization is to minimize the sum of this loss function plus a penalty term that encourages sparsity among model parameters. This can be formalized as follows:

minimize L(x) + λ * ||x||₀

where λ is a regularization parameter that controls the strength of the sparsity constraint and ||x||₀ is the L0 "norm" of x, which is simply the number of non-zero elements in the vector.

Implementation

L0 regularization is a non-convex optimization problem, making it computationally intensive to solve. A common approach for implementing L0 regularization involves using a relaxation of the L0 norm called the L1 norm - which is a convex optimization problem and easily solved. In this way, the L1 regularization term serves as a surrogate for the more challenging L0 regularization term while still promoting sparsity in model parameters while still enabling efficient optimization.

Another method for implementing L0 regularization is to combine L1 and L2 regularization, where the L1 serves as a surrogate for L0, while L2 helps smooth out the optimization landscape and boosts convergence.

Challenges

The primary disadvantage of L0 regularization is its computational cost. The optimization problem is NP-hard, meaning finding an optimal solution cannot be done within a reasonable amount of time for large datasets. Furthermore, since it may contain multiple local minima, finding the global minimum can prove challenging.

Another potential drawback of L0 regularization is that it may lead to overfitting if not set correctly. If the term is set too high, the model may use too few features and underfit; conversely, if set too low, too many features would be included, leading to overfitting.

Explain Like I'm 5 (ELI5)

L0 regularization is like a game where you need to build a tower with blocks, but only use the largest blocks, which are limited. The objective is to build the tallest tower possible while using only those large pieces. In similar fashion, L0 regularization tells the computer only to use certain features when making predictions; this helps it make better predictions but also makes it harder for it to build tall towers since there are fewer pieces available.