**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Regularized least squares

Summary

Regularized least squares (RLS) is a family of methods for solving the least-squares problem while using regularization to further constrain the resulting solution.
RLS is used for two main reasons. The first comes up when the number of variables in the linear system exceeds the number of observations. In such settings, the ordinary least-squares problem is ill-posed and is therefore impossible to fit because the associated optimization problem has infinitely many solutions. RLS allows the introduction of further constraints that uniquely determine the solution.
The second reason for using RLS arises when the learned model suffers from poor generalization. RLS can be used in such cases to improve the generalizability of the model by constraining it at training time. This constraint can either force the solution to be "sparse" in some way or to reflect other prior knowledge about the problem such as information about correlations between features. A Bayesian understanding of this can be reached by showing that RLS methods are often equivalent to priors on the solution to the least-squares problem.
Consider a learning setting given by a probabilistic space , . Let denote a training set of pairs i.i.d. with respect to . Let be a loss function. Define as the space of the functions such that expected risk:
is well defined.
The main goal is to minimize the expected risk:
Since the problem cannot be solved exactly there is a need to specify how to measure the quality of a solution. A good learning algorithm should provide an estimator with a small risk.
As the joint distribution is typically unknown, the empirical risk is taken. For regularized least squares the square loss function is introduced:
However, if the functions are from a relatively unconstrained space, such as the set of square-integrable functions on , this approach may overfit the training data, and lead to poor generalization. Thus, it should somehow constrain or penalize the complexity of the function .

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (5)

Related courses (38)

Related lectures (449)

Regularized least squares

Regularized least squares (RLS) is a family of methods for solving the least-squares problem while using regularization to further constrain the resulting solution. RLS is used for two main reasons. The first comes up when the number of variables in the linear system exceeds the number of observations. In such settings, the ordinary least-squares problem is ill-posed and is therefore impossible to fit because the associated optimization problem has infinitely many solutions.

Elastic net regularization

In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the lasso and ridge methods. The elastic net method overcomes the limitations of the LASSO (least absolute shrinkage and selection operator) method which uses a penalty function based on Use of this penalty function has several limitations. For example, in the "large p, small n" case (high-dimensional data with few examples), the LASSO selects at most n variables before it saturates.

Ridge regression

Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. Also known as Tikhonov regularization, named for Andrey Tikhonov, it is a method of regularization of ill-posed problems. It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters.

CH-242(b): Statistical thermodynamics

The course covers two topics: an introduction to interfacial chemistry, and statistical thermodynamics. The second part includes concepts like the Boltzmann distribution law, partition functions, ense

DH-406: Machine learning for DH

This course aims to introduce the basic principles of machine learning in the context of the digital humanities. We will cover both supervised and unsupervised learning techniques, and study and imple

PHYS-467: Machine learning for physicists

Machine learning and data analysis are becoming increasingly central in sciences including physics. In this course, fundamental principles and methods of machine learning will be introduced and practi

Statistical Thermodynamics: Density of StatesCH-242(b): Statistical thermodynamics

Explores density of states in statistical thermodynamics and the use of Heaviside functions for energy level probabilities.

Statistical Thermodynamics: Particles and LevelsCH-242(b): Statistical thermodynamics

Covers statistical thermodynamics of particles and levels, permutations, and entropy in rubber band elasticity.

Statistical Thermodynamics: Partition Function and Stirling ApproximationCH-242(b): Statistical thermodynamics

Explores the partition function and Stirling approximation in statistical thermodynamics, emphasizing the importance of recognizing higher order terms.