Questions tagged [ridge-regression]

A regularization method for regression models that shrinks coefficients towards zero.

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174
votes
3answers
127k views

When should I use lasso vs ridge?

Say I want to estimate a large number of parameters, and I want to penalize some of them because I believe they should have little effect compared to the others. How do I decide what penalization ...
104
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6answers
50k views

Why L1 norm for sparse models

I am reading the books about linear regression. There are some sentences about the L1 and L2 norm. I know them, just don't understand why L1 norm for sparse models. Can someone use give a simple ...
83
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2answers
38k views

When to use regularization methods for regression?

In what circumstances should one consider using regularization methods (ridge, lasso or least angles regression) instead of OLS? In case this helps steer the discussion, my main interest is improving ...
73
votes
2answers
19k views

Why is ridge regression called “ridge”, why is it needed, and what happens when $\lambda$ goes to infinity?

Ridge regression coefficient estimate $\hat{\beta}^R$ are the values that minimize the $$ \text{RSS} + \lambda \sum_{j=1}^p\beta_j^2. $$ My questions are: If $\lambda = 0$, then we see that the ...
65
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5answers
4k views

Unified view on shrinkage: what is the relation (if any) between Stein's paradox, ridge regression, and random effects in mixed models?

Consider the following three phenomena. Stein's paradox: given some data from multivariate normal distribution in $\mathbb R^n, \: n\ge 3$, sample mean is not a very good estimator of the true mean. ...
63
votes
5answers
14k views

What problem do shrinkage methods solve?

The holiday season has given me the opportunity to curl up next to the fire with The Elements of Statistical Learning. Coming from a (frequentist) econometrics perspective, I'm having trouble grasping ...
60
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3answers
8k views

Why does ridge estimate become better than OLS by adding a constant to the diagonal?

I understand that the ridge regression estimate is the $\beta$ that minimizes residual sum of square and a penalty on the size of $\beta$ $$\beta_\mathrm{ridge} = (\lambda I_D + X'X)^{-1}X'y = \...
56
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2answers
8k views

Why does shrinkage work?

In order to solve problems of model selection, a number of methods (LASSO, ridge regression, etc.) will shrink the coefficients of predictor variables towards zero. I am looking for an intuitive ...
52
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6answers
4k views

Is ridge regression useless in high dimensions ($n \ll p$)? How can OLS fail to overfit?

Consider a good old regression problem with $p$ predictors and sample size $n$. The usual wisdom is that OLS estimator will overfit and will generally be outperformed by the ridge regression estimator:...
42
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5answers
52k views

How to derive the ridge regression solution?

I am having some issues with the derivation of the solution for ridge regression. I know the regression solution without the regularization term: $$\beta = (X^TX)^{-1}X^Ty.$$ But after adding the ...
37
votes
3answers
8k views

Why do we only see $L_1$ and $L_2$ regularization but not other norms?

I am just curious why there are usually only $L_1$ and $L_2$ norms regularization. Are there proofs of why these are better?
37
votes
2answers
5k views

If only prediction is of interest, why use lasso over ridge?

On page 223 in An Introduction to Statistical Learning, the authors summarise the differences between ridge regression and lasso. They provide an example (Figure 6.9) of when "lasso tends to ...
36
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3answers
59k views

How to estimate shrinkage parameter in Lasso or ridge regression with >50K variables?

I want to use Lasso or ridge regression for a model with more than 50,000 variables. I want do so using software package in R. How can I estimate the shrinkage parameter ($\lambda$)? Edits: Here is ...
36
votes
1answer
6k views

When is nested cross-validation really needed and can make a practical difference?

When using cross-validation to do model selection (such as e.g. hyperparameter tuning) and to assess the performance of the best model, one should use nested cross-validation. The outer loop is to ...
35
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4answers
24k views

Ridge, lasso and elastic net

How do ridge, LASSO and elasticnet regularization methods compare? What are their respective advantages and disadvantages? Any good technical paper, or lecture notes would be appreciated as well.
35
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2answers
31k views

What is elastic net regularization, and how does it solve the drawbacks of Ridge ($L^2$) and Lasso ($L^1$)?

Is elastic net regularization always preferred to Lasso & Ridge since it seems to solve the drawbacks of these methods? What is the intuition and what is the math behind elastic net?
34
votes
1answer
19k views

Is regression with L1 regularization the same as Lasso, and with L2 regularization the same as ridge regression? And how to write “Lasso”?

I'm a software engineer learning machine learning, particularly through Andrew Ng's machine learning courses. While studying linear regression with regularization, I've found terms that are confusing: ...
33
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2answers
9k views

Is Tikhonov regularization the same as Ridge Regression?

Tikhonov regularization and ridge regression are terms often used as if they were identical. Is it possible to specify exactly what the difference is?
33
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2answers
15k views

When will L1 regularization work better than L2 and vice versa?

Note: I know that L1 has feature selection property. I am trying to understand which one to choose when feature selection is completely irrelevant. How to decide which regularization (L1 or L2) to ...
28
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2answers
4k views

Why is glmnet ridge regression giving me a different answer than manual calculation?

I'm using glmnet to calculate ridge regression estimates. I got some results that made me suspicious in that glmnet is really doing what I think it does. To check this I wrote a simple R script where ...
25
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3answers
2k views

Interpretation of ridge regularization in regression

I have several questions regarding the ridge penalty in the least squares context: $$\beta_{ridge} = (\lambda I_D + X'X)^{-1}X'y$$ 1) The expression suggests that the covariance matrix of X is ...
24
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4answers
12k views

L1 regression estimates median whereas L2 regression estimates mean?

So I was asked a question on which central measures L1 (i.e., lasso) and L2 (i.e., ridge regression) estimated. The answer is L1=median and L2=mean. Is there any type of intuitive reasoning to this? ...
22
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1answer
3k views

Bridge penalty vs. Elastic Net regularization

Some penalty functions and approximations are well studied, such as the LASSO ($L_1$) and the Ridge ($L_2$) and how these compare in regression. I've been reading about the Bridge penalty, which is ...
21
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5answers
9k views

Reason for not shrinking the bias (intercept) term in regression

For a linear model $y=\beta_0+x\beta+\varepsilon$, the shrinkage term is always $P(\beta) $. What is the reason that we do not shrink the bias (intercept) term $\beta_0$? Should we shrink the bias ...
21
votes
2answers
979 views

The limit of “unit-variance” ridge regression estimator when $\lambda\to\infty$

Consider ridge regression with an additional constraint requiring that $\hat{\mathbf y}$ has unit sum of squares (equivalently, unit variance); if needed, one can assume that $\mathbf y$ has unit sum ...
21
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2answers
5k views

What are the assumptions of ridge regression and how to test them?

Consider the standard model for multiple regression $$Y=X\beta+\varepsilon$$ where $\varepsilon \sim \mathcal N(0, \sigma^2I_n)$, so normality, homoscedasticity and uncorrelatedness of errors all hold....
21
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2answers
558 views

confidence intervals' coverage with regularized estimates

Suppose I'm trying to estimate a large number of parameters from some high-dimensional data, using some kind of regularized estimates. The regularizer introduces some bias into the estimates, but it ...
20
votes
1answer
7k views

The proof of shrinking coefficients using ridge regression through “spectral decomposition”

I have understood how ridge regression shrinks coefficients towards zero geometrically. Moreover I know how to prove that in the special "Orthonormal Case," but I am confused how that works in the ...
20
votes
2answers
9k views

Estimating R-squared and statistical significance from penalized regression model

I am using the R package penalized to obtain shrunken estimates of coefficients for a dataset where I have lots of predictors and little knowledge of which ones are important. After I've picked tuning ...
19
votes
1answer
13k views

Difference between Primal, Dual and Kernel Ridge Regression

What is the difference between Primal, Dual and Kernel Ridge Regression? People are using all three, and because of the different notation that everyone uses at different sources is difficult for me ...
19
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3answers
7k views

Relationship between ridge regression and PCA regression

I remember having read somewhere on the web a connection between ridge regression (with $\ell_2$ regularization) and PCA regression: while using $\ell_2$-regularized regression with hyperparameter $\...
19
votes
2answers
37k views

What is ridge regression? [duplicate]

I just need a simple explanation of what exactly ridge regression is so I can have a decent intuitive understanding of it. I understand it's about applying some sort of penalty to the regression ...
18
votes
3answers
11k views

How can I estimate coefficient standard errors when using ridge regression?

I am using ridge regression on highly multicollinear data. Using OLS I get large standard errors on the coefficients due to the multicollinearity. I know ridge regression is a way to deal with this ...
18
votes
2answers
5k views

Why Lasso or ElasticNet perform better than Ridge when the features are correlated

I have a set of 150 features, and many of them are highly correlated with each other. My goal is to predict the value of a discrete variable, whose range is 1-8. My sample size is 550, and I am using ...
18
votes
2answers
12k views

Why does ridge regression classifier work quite well for text classification?

During an experiment for text classification, I found ridge classifier generating results that constantly top the tests among those classifiers that are more commonly mentioned and applied for text ...
18
votes
3answers
2k views

Using regularization when doing statistical inference

I know about the benefits of regularization when building predictive models (bias vs. variance, preventing overfitting). But, I'm wondering if it is a good idea to also do regularization (lasso, ...
18
votes
1answer
611 views

Is there a clear set of conditions under which lasso, ridge, or elastic net solution paths are monotone?

The question What to conclude from this lasso plot (glmnet) demonstrates solution paths for the lasso estimator that are not monotonic. That is, some of the cofficients grow in absolute value before ...
17
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2answers
11k views

Why will ridge regression not shrink some coefficients to zero like lasso?

When explaining LASSO regression, the diagram of a diamond and circle is often used. It is said that because the shape of the constraint in LASSO is a diamond, the least squares solution obtained ...
17
votes
2answers
1k views

LASSO and ridge from the Bayesian perspective: what about the tuning parameter?

Penalized regression estimators such as LASSO and ridge are said to correspond to Bayesian estimators with certain priors. I guess (as I do not know enough about Bayesian statistics) that for a fixed ...
17
votes
3answers
5k views

Implementing ridge regression: Selecting an intelligent grid for $\lambda$?

I'm implementing Ridge Regression in a Python/C module, and I've come across this "little" problem. The idea is that I want to sample the effective degrees of freedom more or less equally spaced (like ...
16
votes
1answer
742 views

Reversing ridge regression: given response matrix and regression coefficients, find suitable predictors

Consider a standard OLS regression problem$\newcommand{\Y}{\mathbf Y}\newcommand{\X}{\mathbf X}\newcommand{\B}{\boldsymbol\beta}\DeclareMathOperator*{argmin}{argmin}$: I have matrices $\Y$ and $\X$ ...
16
votes
1answer
3k views

Under exactly what conditions is ridge regression able to provide an improvement over ordinary least squares regression?

Ridge regression estimates parameters $\boldsymbol \beta$ in a linear model $\mathbf y = \mathbf X \boldsymbol \beta$ by $$\hat{\boldsymbol \beta}_\lambda = (\mathbf X^\top \mathbf X + \lambda \mathbf ...
16
votes
2answers
420 views

Why does shrinkage really work, what's so special about 0?

There is already a post on this site talking about the same issue: Why does shrinkage work? But, even though the answers are popular, I don't believe the gist of the question is really addressed. It ...
16
votes
1answer
7k views

What's the typical range of possible values for the shrinkage parameter in penalized regression?

In lasso or ridge regression, one has to specify a shrinkage parameter, often called by $\lambda$ or $\alpha$. This value is often chosen via cross validation by checking a bunch of different values ...
15
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4answers
3k views

The proof of equivalent formulas of ridge regression

I have read the most popular books in statistical learning 1- The elements of statistical learning. 2- An introduction to statistical learning. Both mention that ridge regression has two formulas ...
15
votes
1answer
8k views

Regression in $p>n$ setting: how to choose regularization method (Lasso, PLS, PCR, ridge)?

I am trying see whether to go for ridge regression, LASSO, principal component regression (PCR), or Partial Least Squares (PLS) in a situation where there are large number of variables / features ($p$)...
15
votes
4answers
1k views

In regression, why not use regularization by default?

I remember reading somewhere in another post about the different viewpoints between people from statistics and from machine learning or neural networks, where one user was mentioning this idea as an ...
15
votes
2answers
4k views

Ridge regression – Bayesian interpretation

I have heard that ridge regression can be derived as the mean of a posterior distribution, if the prior is adequately chosen. Is the intuition that the constraints as set on the regression ...
15
votes
1answer
3k views

Regularization for ARIMA models

I am aware of LASSO, ridge and elastic-net type of regularization in linear regression models. Question: Can this (or a similar) kind of penalized estimation be applied to ARIMA modelling (with a ...
15
votes
1answer
3k views

Lagrangian relaxation in the context of ridge regression

In "The Elements of Statistical Learning" (2nd ed), p63, the authors give the following two formulations of the ridge regression problem: $$ \hat{\beta}^{ridge} = \underset{\beta}{\operatorname{...

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