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# lasso

Regularized least-squares regression using lasso or elastic net algorithms

## Syntax

B = lasso(X,Y)
B = lasso(X,Y,Name,Value)
[B,FitInfo] = lasso(___)

## Description

B = lasso(X,Y) returns fitted least-squares regression coefficients for a set of regularization coefficients Lambda.

B = lasso(X,Y,Name,Value) fits regularized regressions with additional options specified by one or more Name,Value pair arguments.

[B,FitInfo] = lasso(___), for any previous input syntax, also returns a structure containing information about the fits.

## Input Arguments

 X Numeric matrix. Each row represents one observation, and each column represents one predictor (variable). Y Numeric vector of length n, where n is the number of rows of X. Y(i) is the response to row i of X.

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

'AbsTol'

Absolute error tolerance used to determine convergence of ADMM Algorithm. The algorithm converges when successive estimates of the coefficient vector differ by an amount less than AbsTol.

### Note

This option only applies when using lasso on tall arrays. See Extended Capabilities for more information.

Default: 1e-4

'Alpha'

Scalar value in the interval (0,1] representing the weight of lasso (L1) versus ridge (L2) optimization. Alpha = 1 represents lasso regression, Alpha close to 0 approaches ridge regression, and other values represent elastic net optimization. See Definitions.

Default: 1

'B0'

Initial values for x-coefficients in ADMM Algorithm.

### Note

This option only applies when using lasso on tall arrays. See Extended Capabilities for more information.

Default: Vector of zeros

'CV'

Method lasso uses to estimate mean squared error:

• K, a positive integer — lasso uses K-fold cross-validation.

• cvp, a cvpartition object — lasso uses the cross-validation method expressed in cvp. You cannot use a 'leaveout' partition with lasso.

• 'resubstitution'lasso uses X and Y to fit the model and to estimate the mean squared error without cross-validation.

Default: 'resubstitution'

'DFmax'

Maximum number of nonzero coefficients in the model. lasso returns results only for Lambda values that satisfy this criterion.

Default: Inf

'Lambda'

Vector of nonnegative Lambda values. See Definitions.

• If you do not supply Lambda, lasso calculates the largest value of Lambda that gives a nonnull model. In this case, LambdaRatio gives the ratio of the smallest to the largest value of the sequence, and NumLambda gives the length of the vector.

• If you supply Lambda, lasso ignores LambdaRatio and NumLambda.

Default: Geometric sequence of NumLambda values, the largest just sufficient to produce B = 0

'LambdaRatio'

Positive scalar, the ratio of the smallest to the largest Lambda value when you do not set Lambda.

If you set LambdaRatio = 0, lasso generates a default sequence of Lambda values, and replaces the smallest one with 0.

Default: 1e-4

'MaxIter'

Maximum number of iterations allowed, specified as a positive integer. If the algorithm executes MaxIter iterations before reaching the convergence tolerance, then the function stops iterating and returns a warning message. The function can return more than one warning when NumLambda is greater than 1.

Default: 1e5 (standard), 1e4 (for tall arrays)

'MCReps'

Positive integer, the number of Monte Carlo repetitions for cross-validation.

• If CV is 'resubstitution' or a cvpartition of type 'resubstitution', MCReps must be 1.

• If CV is a cvpartition of type 'holdout', MCReps must be greater than 1.

Default: 1

'NumLambda'

Positive integer, the number of Lambda values lasso uses when you do not set Lambda. lasso can return fewer than NumLambda fits if the residual error of the fits drops below a threshold fraction of the variance of Y.

Default: 100

'Options'

Structure that specifies whether to cross-validate in parallel, and specifies the random streams. Create the Options structure with statset. Option fields:

• UseParallel — Set to true to compute in parallel. Default is false.

• UseSubstreams — Set to true to compute in parallel in a reproducible fashion. To compute reproducibly, set Streams to a type allowing substreams: 'mlfg6331_64' or 'mrg32k3a'. Default is false.

• Streams — A RandStream object or cell array consisting of one such object. If you do not specify Streams, lasso uses the default stream.

'PredictorNames'

Cell array of character vectors representing names of the predictor variables, in the order in which they appear in X. For an example, see Remove Redundant Predictors by Using Cross-Validated Fits.

Default: {}

'RelTol'

Convergence threshold for the coordinate descent algorithm [3]. The algorithm terminates when successive estimates of the coefficient vector differ in the L2 norm by a relative amount less than RelTol.

Default: 1e-4

'Rho'

Augmented Lagrangian parameter ρ for ADMM Algorithm.

### Note

This option only applies when using lasso on tall arrays. See Extended Capabilities for more information.

Default: Automatic selection

'Standardize'

Boolean value specifying whether lasso scales X before fitting the models. This affects whether the regularization is applied to the coefficients on the standardized scale or original scale. The results are always presented on the original data scale.

X and Y are always centered.

Default: true

'U0'

Initial value of scaled dual variable u in ADMM Algorithm.

### Note

This option only applies when using lasso on tall arrays. See Extended Capabilities for more information.

Default: Vector of zeros

'Weights'

Observation weights, a nonnegative vector of length n, where n is the number of rows of X. lasso scales Weights to sum to 1.

Default: 1/n * ones(n,1)

## Output Arguments

B

Fitted coefficients, a p-by-L matrix, where p is the number of predictors (columns) in X, and L is the number of Lambda values.

FitInfo

Structure containing information about the model fits.

Field in FitInfoDescription
InterceptIntercept term β0 for each linear model, a 1-by-L vector
LambdaLambda parameters in ascending order, a 1-by-L vector
AlphaValue of Alpha parameter, a scalar
DFNumber of nonzero coefficients in B for each value of Lambda, a 1-by-L vector
MSEMean squared error (MSE), a 1-by-L vector

If you set the CV name-value pair to cross-validate, the FitInfo structure contains additional fields.

Field in FitInfoDescription
SEThe standard error of MSE for each Lambda, as calculated during cross-validation, a 1-by-L vector
LambdaMinMSEThe Lambda value with minimum MSE, a scalar
Lambda1SEThe largest Lambda such that MSE is within one standard error of the minimum MSE, a scalar
IndexMinMSEThe index of Lambda with value LambdaMinMSE, a scalar
Index1SEThe index of Lambda with value Lambda1SE, a scalar

## Examples

collapse all

Construct a data set with redundant predictors and identify those predictors by using lasso.

Create a matrix X of 100 five-dimensional normal variables. Create a response vector Y from just two components of X and add a small amount of noise.

rng default % For reproducibility
X = randn(100,5);
r = [0;2;0;-3;0]; % Only two nonzero coefficients
Y = X*r + randn(100,1)*.1; % Small added noise

Construct the default lasso fit.

B = lasso(X,Y);

Find the coefficient vector for the 25th value in B.

B(:,25)
ans =

0
1.6093
0
-2.5865
0

lasso identifies and removes the redundant predictors.

Construct a data set with redundant predictors and identify those predictors by using cross-validated lasso.

Create a matrix X of 100 five-dimensional normal variables. Create a response vector Y from two components of X and add a small amount of noise.

rng default % For reproducibility
X = randn(100,5);
r = [0;2;0;-3;0]; % Only two nonzero coefficients
Y = X*r + randn(100,1)*.1; % Small added noise

Construct the lasso fit by using tenfold cross-validation with labeled predictor variables.

[B,FitInfo] = lasso(X,Y,'CV',10,'PredictorNames',{'x1','x2','x3','x4','x5'});

Display the variables in the model that corresponds to the minimum cross-validated mean squared error (MSE).

minMSEModel = FitInfo.PredictorNames(B(:,FitInfo.IndexMinMSE)~=0)
minMSEModel =

1x2 cell array

{'x2'}    {'x4'}

Display the variables in the sparsest model within one standard error of the minimum MSE.

sparseModel = FitInfo.PredictorNames(B(:,FitInfo.Index1SE)~=0)
sparseModel =

1x2 cell array

{'x2'}    {'x4'}

In this example, lasso identifies the same predictors for the two models and removes the redundant predictors. However, in general, lasso can choose a different set of predictors.

Visually examine the cross-validated error of various levels of regularization.

Load the sample data.

Prepare the design matrix for a lasso fit with interactions.

X = [x1 x2 x3];
D = x2fx(X,'interaction');
D(:,1) = []; % No constant term

Construct the lasso fit using ten-fold cross-validation. Include the FitInfo output so you can plot the result.

rng default % For reproducibility
[B,FitInfo] = lasso(D,y,'CV',10);

Plot the cross-validated fits.

lassoPlot(B,FitInfo,'PlotType','CV');

The green circle and dashed line locate the Lambda with minimum cross-validation error. The blue circle and dashed line locate the point with minimum cross-validation error plus one standard deviation.

collapse all

### Lasso

For a given value of λ, a nonnegative parameter, lasso solves the problem

$\underset{{\beta }_{0},\beta }{\mathrm{min}}\left(\frac{1}{2N}\sum _{i=1}^{N}{\left({y}_{i}-{\beta }_{0}-{x}_{i}^{T}\beta \right)}^{2}+\lambda \sum _{j=1}^{p}|{\beta }_{j}|\right).$

• N is the number of observations.

• yi is the response at observation i.

• xi is data, a vector of p values at observation i.

• λ is a nonnegative regularization parameter corresponding to one value of Lambda.

• The parameters β0 and β are a scalar and a vector of length p, respectively.

As λ increases, the number of nonzero components of β decreases.

The lasso problem involves the L1 norm of β, as contrasted with the elastic net algorithm.

### Elastic Net

For an α strictly between 0 and 1, and a nonnegative λ, elastic net solves the problem

$\underset{{\beta }_{0},\beta }{\mathrm{min}}\left(\frac{1}{2N}\sum _{i=1}^{N}{\left({y}_{i}-{\beta }_{0}-{x}_{i}^{T}\beta \right)}^{2}+\lambda {P}_{\alpha }\left(\beta \right)\right),$

where

${P}_{\alpha }\left(\beta \right)=\frac{\left(1-\alpha \right)}{2}{‖\beta ‖}_{2}^{2}+\alpha {‖\beta ‖}_{1}=\sum _{j=1}^{p}\left(\frac{\left(1-\alpha \right)}{2}{\beta }_{j}^{2}+\alpha |{\beta }_{j}|\right).$

Elastic net is the same as lasso when α = 1. As α shrinks toward 0, elastic net approaches ridge regression. For other values of α, the penalty term Pα(β) interpolates between the L1 norm of β and the squared L2 norm of β.

## Algorithms

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When operating on tall arrays, lasso uses an algorithm based on the Alternating Direction Method of Multipliers (ADMM) [5]. The notation used here is the same as in the reference paper. This method solves problems of the form

Minimize $l\left(x\right)+g\left(z\right)$

Subject to $Ax+Bz=c$

Using this notation the lasso regression problem is

Minimize $l\left(x\right)+g\left(z\right)=\frac{1}{2}{‖Ax-b‖}_{2}^{2}+\lambda {‖z‖}_{1}$

Subject to $x-z=0$

Since the loss function $l\left(x\right)=\frac{1}{2}{‖Ax-b‖}_{2}^{2}$ is quadratic, the iterative updates performed by the algorithm amount to solving a linear system of equations with a single coefficient matrix but several right-hand sides. The updates performed by the algorithm during each iteration are

$\begin{array}{l}{x}^{k+1}={\left({A}^{T}A+\rho I\right)}^{-1}\left({A}^{T}b+\rho \left({z}^{k}-{u}^{k}\right)\right)\\ {z}^{k+1}={S}_{\lambda /\rho }\left({x}^{k+1}+{u}^{k}\right)\\ {u}^{k+1}={u}^{k}+{x}^{k+1}-{z}^{k+1}.\end{array}$

A is the dataset (a tall array), x contains the coefficients, ρ is the penalty parameter (augmented Lagrangian parameter), b is the response (a tall array), and S is the soft thresholding operator.

${S}_{\kappa }\left(a\right)=\left\{\begin{array}{c}\begin{array}{cc}a-\kappa ,\text{\hspace{0.17em}}& a>\kappa \end{array}\\ \begin{array}{cc}0,\text{\hspace{0.17em}}& |a|\text{\hspace{0.17em}}\le \kappa \text{\hspace{0.17em}}\end{array}\\ \begin{array}{cc}a+\kappa ,\text{\hspace{0.17em}}& a<\kappa \text{\hspace{0.17em}}\end{array}\end{array}.$

lasso solves the linear system using Cholesky factorization since the coefficient matrix ${A}^{T}A+\rho I$ is symmetric and positive definite. Since $\rho$ does not change between iterations, the Cholesky factorization is cached between iterations instead of solving from scratch.

Even though A and b are tall arrays, they appear only in the terms ${A}^{T}A$ and ${A}^{T}b$. The results of these two matrix multiplications are small enough to fit in memory, so they are precomputed and the iterative updates between iterations are performed entirely within memory.

## References

[1] Tibshirani, R. “Regression Shrinkage and Selection via the Lasso.” Journal of the Royal Statistical Society. Series B, Vol. 58, No. 1, 1996, pp. 267–288.

[2] Zou, H. and T. Hastie. “Regularization and Variable Selection via the Elastic Net.” Journal of the Royal Statistical Society. Series B, Vol. 67, No. 2, 2005, pp. 301–320.

[3] Friedman, J., R. Tibshirani, and T. Hastie. “Regularization Paths for Generalized Linear Models via Coordinate Descent.” Journal of Statistical Software. Vol. 33, No. 1, 2010. http://www.jstatsoft.org/v33/i01

[4] Hastie, T., R. Tibshirani, and J. Friedman. The Elements of Statistical Learning. 2nd edition. New York: Springer, 2008.

[5] Boyd, S. “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers”. Foundations and Trends in Machine Learning. Vol 3, No. 1, 2010, pp. 1–122.