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Inclusion of additional constraints (typically a penalty for complexity) in the model fitting process. Used to prevent overfitting / enhance predictive accuracy.

32
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3answers
12k views

(Why) do overfitted models tend to have large coefficients?

I imagine that the larger a coefficient on a variable is, the more ability the model has to "swing" in that dimension, providing an increased opportunity to fit noise. Although I think I've got a ...
7
votes
3answers
2k views

Why does regularization of coefficient magnitude improve the generalization of linear regression? [duplicate]

What is the basic argument upon which ridge and lasso regression are based on? I went through Tikhonov regularization wiki where it was mentioned that In many cases, tikhonov matrix is chosen as ...
0
votes
1answer
333 views

Is regularization required with overdetermined data

I'm doing least squares estimation on large set of data and I started to wonder whether I should regularize my OLS estimator. My professor told me that this isn't necessary, because the data is ...
3
votes
0answers
86 views

Sparsity regularization for eigenvectors

One way to think about finding the eigenvectors of a matrix $A$ is that they are the critical points of the functional $\vec x^\top A \vec x$ subject to $\|\vec x\|_2=1$. To regularize this problem, ...
0
votes
1answer
3k views

Effect of features that are highly correlated with each other on a decision tree

I have a dataset of roughly 500 features and am training a binary classifier using GBM - gradient boosted machines, an ensemble of decision trees. Of these 500 variables, I am sure some are highly ...
11
votes
1answer
8k views

Coefficients paths – comparison of ridge, lasso and elastic net regression

I would like to compare models selected with ridge, lasso and elastic net. Fig. below shows coefficients paths using all 3 methods: ridge (Fig A, alpha=0), lasso (Fig B; alpha=1) and elastic net (Fig ...
1
vote
0answers
687 views

GBDT and model building: How am I overfitting?

Here's my situation: Binary classification and I've got a training set of roughly 250k samples and 10 features, and a validation set of roughly 100k with the same number of features. I'm fitting GBDT ...
0
votes
1answer
70 views

Confusion related to regularization parameter selection by cross validation

I can see lots of paper mentioning they selected some parameters like regularization parameter $\lambda$ by cross validation. What do they mean by that?
9
votes
0answers
349 views

Asymptotic property of tuning parameter in penalized regression

I'm currently working on asymptotic properties of penalized regression. I've read a myriad of papers by now, but there is an essential issue that I cannot get my head around. To keep things simple, I'...
12
votes
4answers
5k views

What does “degree of freedom” mean in neural networks?

In Bishop's book "Pattern Classification and Machine Learning", it describes a technique for regularization in the context of neural networks. However, I don't understand a paragraph describing that ...
1
vote
0answers
213 views

Forward Stepwise selection

I am assuming the following model: $Y = \beta X + \epsilon$ Here both $X$ and $Y$ are matrices. I fit the least squares model without any regularization and get the matrix $\beta$. I would like to ...
3
votes
3answers
2k views

Alternatives to glmnet for feature selection on data with lots of NAs

I have a surgical database in which there are approximately 100,000 observations and 200 features. Each observation corresponds to a separate patient while the features correspond to either ...
3
votes
1answer
86 views

Question on the usage of regularization in applied statistics/science

I was reading the paper ``A significance test for the lasso'' by Lockhart, Tibshirani et al and was considering the issue of applying regularization in the applied sciences (for example, behavioral ...
2
votes
1answer
442 views

model selection with glmnet

I am trying to fit a multinomial logit model using glmnet. I have a few questions: How is the baseline category specified? Looking at the model coefficients using coef.glmnet, I'm thinking that many ...
4
votes
1answer
517 views

Classification with 3 groups, repeated measurements, missing values, more predictors than subjects

I am working on a classification problem with the following characteristics: Individuals belong to one of three groups. The groups are "somewhat ordinal": controls, subclinical and clinical group. ...
2
votes
1answer
177 views

Parameter estimate for linear regression with regularization

For given cost function $S(\beta) = (Y - X \beta)^T(Y - X \beta) + \lambda \beta^T \beta$, where $\lambda$ is regularization parameter, the $\beta$ that minimizes the given cost function is $\beta = [...
5
votes
1answer
705 views

Can the bias introduced by lasso change the sign of a coefficient?

L1 penalized regression introduces a bias on your regression model but decreases the variance. When this bias is introduced, is it possible that the coefficient of $B$ changes sign? This would ...
0
votes
1answer
103 views

Reducing the dimensionality of a problem

My particular application needs me to build a linear model with a strong correlation structure amongst the independent variables. The dimensions of the problem are high, for instance 1million X 200. ...
2
votes
0answers
195 views

kernelized l1 norm and the representer theorem

I'm trying to derive a kernel-ized $l_1$ penalty for logistic regression. I have been looking at the slides Learning with Sparsity Inducing Norms along with the slides on Regularization and Variable ...
5
votes
1answer
610 views

SVM optimization problem

I think I understand the main idea in support vector machines. Let us assume that we have two linear separable classes and want to apply SVMs. What SVM is doing is that it searches a hyperplane $\{\...
81
votes
6answers
32k 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 ...
1
vote
0answers
139 views

Enforcing sparsity on probability [closed]

I am trying to induce a probability distribution $Q$ by optimizing an objective function and am wondering how can one encourage sparsity for $Q$ while keeping the optimization convex. In particular, ...
0
votes
1answer
649 views

How to obtain good performance (low error rate) on massive data set?

Suppose I have massive data set (think Terabytes) is available to train a learning algorithm. Which one of the following conditions must be true to obtain good performance (low error rate) a. Using ...
3
votes
2answers
381 views

Robust regularized regression

I've been using elastic net implemented in R (via glmnet) for some modeling, but I was wondering, due to the number of outliers in my data, if there was some sort of modeling approach for regularized ...
24
votes
2answers
2k views

Advantages of doing “double lasso” or performing lasso twice?

I once heard a method of using the lasso twice (like a double-lasso) where you perform lasso on the original set of variables, say S1, obtain a sparse set called S2, and then perform lasso again on ...
19
votes
2answers
17k views

libsvm “reaching max number of iterations” warning and cross-validation

I'm using libsvm in C-SVC mode with a polynomial kernel of degree 2 and I'm required to train multiple SVMs. Each training set has 10 features and 5000 vectors. During training, I am getting this ...
2
votes
1answer
2k views

matlab gmdistribution.fit 'Regularize' - what regularization method?

I am wondering what is behind matlab 'Regularize' option for method gmdistribution.fit. If it is simply adding a 'little' value to diagonal elements of covariance matrix, so as to make covariance ...
8
votes
1answer
1k views

Manifold regularization using laplacian graph in SVM

I'm trying implement Manifold Regularization in Support Vector Machines (SVMs) in Matlab. I'm following the instructions in the paper by Belkin et al.(2006), there's the equation in it: $f^{*} = \...
10
votes
4answers
358 views

Sparsity-inducing regularization for stochastic matrices

It is well-known (e.g. in the field of compressive sensing) that the $L_1$ norm is "sparsity-inducing," in the sense that if we minimize the functional (for fixed matrix $A$ and vector $\vec{b}$) $$f_{...
20
votes
4answers
8k 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? ...
1
vote
1answer
116 views

How does the test error pattern depend on the regularizer function?

This question is regarding the role of regularizer in an objective function. Given a loss function $f(x)$, a regularizer function $r(x)$, and $\lambda$ being a trade-off function, our aim is to $\...
21
votes
3answers
6k views

Why do Lars and Glmnet give different solutions for the Lasso problem?

I want to better understand the R packages Lars and Glmnet, which are used to solve the Lasso problem: $$min_{(\beta_0 \beta) \...
11
votes
1answer
299 views

What are $\ell_p$ norms and how are they relevant to regularization?

I have been seeing a lot of papers on sparse representations lately, and most of them use the $\ell_p$ norm and do some minimization. My question is, what is the $\ell_p$ norm, and the $\ell_{p, q}$ ...
7
votes
1answer
2k views

Gradient descent and elastic-net logistic regression

I'm currently in the process of trying to understand the paper Regularization Paths for Generalized Linear Models via Coordinate Descent by Friedman et al. with regard to the regularization of ...
11
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2answers
9k views

When does LASSO select correlated predictors?

I'm using the package 'lars' in R with the following code: ...
17
votes
2answers
507 views

Frequentism and priors

Robby McKilliam says in a comment to this post: It should be pointed out that, from the frequentists point of view, there is no reason that you can't incorporate the prior knowledge into the model. ...
7
votes
0answers
993 views

Post processing random forests using regularised regression: what about bias?

I have been playing around with post processing the results of the random forest for regression machine learning algorithm in order to try and do better than the default mean of all trees prediction. ...
3
votes
1answer
661 views

Standard error of parameter estimates in regularized regression

In a regularized linear regression model (e.g., ridge regression, lasso, etc.), what is the best way to obtain standard errors for parameter estimates? If cross-validation is used, is it ...
12
votes
1answer
5k views

Regularized bayesian logistic regression in JAGS

There are several math-heavy papers that describe the Bayesian Lasso, but I want tested, correct JAGS code that I can use. Could someone post sample BUGS / JAGS code that implements regularized ...
3
votes
0answers
79 views

What's a good range of weights to evaluate for $L_2$ regularized logistic regression?

I want to find a weight that minimizes an averaged cross validated misclassification score from an $L_2$ logistic regression classifier. Obviously, the search space for the weights should be bounded ...
9
votes
2answers
365 views

Regularization $L_1$ norm and $L_2$ norm empirical study

There are many methods to perform regularization -- $L_0$, $L_1$, and $L_2$ norm based regularization for example. According to Friedman Hastie & Tibsharani, the best regularizer depends on the ...
1
vote
1answer
108 views

What does the index variable k define in the Lasso regularization function

In the Lasso L1 regularization, from where comes the value of the variable $k$ in the second part of the function? Why isn't it $n$, too? $$L(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T \cdot \beta)^2 + ...
2
votes
1answer
221 views

Model function for discovering irrelevant dimensions with L1 regularization

For homework I have been given a 20-dimensional input $x \in \mathbb{R}^{20}$, many of which are suspected to be irrelevant. I tried using L1-norm Lasso regularization to uncover which dimensions ...
4
votes
1answer
3k views

Using glmnet to solve the LASSO problem

I have recently been made aware of the Lasso algorithm and found that the package glmnet can be used to solve it. (I have the glmnet package on R). If I have a matrix $A$ and a vector $y$ how do I ...
2
votes
1answer
458 views

Non-linear regularized SVM implementation

Just a general question. Are there any good non-linear SVM (kernelized) implementations that include a regularization component (e.g. $L_1$, SCAD etc)? I've been looking around but man there are a lot ...
5
votes
4answers
1k views

Bayesian prior corresponding to penalized regression coefficients

I'm working on a Bayesian Regression problem where I would like to estimate the beta coefficients subject to this constraint (penalty): $\sum|\beta_i|<C$ or similarly $\sum \beta_i^2<C$ Which ...
3
votes
1answer
111 views

Problem specific regularization

I've been reading a lot recently about the concept of joint regularization in computer vision. Joint regularization builds on the observation that when learning multiple related concepts, for example "...
1
vote
1answer
267 views

High dimensional time series

I'm not sure what words I should look for. I have an under determined dataset of 8000 correlated variables (sales) over 12 months (ie 12 observations for each variable). And I basically want to ...
15
votes
1answer
10k views

Need for centering and standardizing data in regression

Consider linear regression with some regularization: E.g. Find $x$ that minimizes $||Ax - b||^2+\lambda||x||_1$ Usually, columns of A are standardized to have zero mean and unit norm, while $b$ is ...
5
votes
1answer
314 views

When is there a representer theorem?

The case of regularization in a hilbert space is considered---an optimization problem with an error term and a Tikhonov-regularizer. In the article "When is there a representer theorem" it is stated ...