Lasso Regression (home made)

Again, this post is related to my MAT7381 course, where we will see that it is actually possible to write our own code to compute Lasso regression, \min\left\lbrace\frac{1}{2}\|\mathbf{y}-\mathbf{X}\mathbf{\beta}\|_{\ell_2}^2+\lambda\|\mathbf{\beta}\|_{\ell_1}\right\rbraceWe have to define the soft-thresholding functionS(z,\gamma)=\text{sign}(z)\cdot(|z|-\gamma)_+=\begin{cases}z-\gamma&\text{ if }\gamma>|z|\text{ and }z<0\\z+\gamma&\text{ if }\gamma<|z|\text{ and }z<0 \\0&\text{ if }\gamma\geq|z|\end{cases}The R function would be

soft_thresholding = function(x,a){
sign(x) * pmax(abs(x)-a,0)
}

To solve our optimization problem, set\mathbf{r}_j=\mathbf{y} - \left(\beta_0\mathbf{1}+\sum_{k\neq j}\beta_k\mathbf{x}_k\right)=\mathbf{y}-\widehat{\mathbf{y}}^{(j)}
so that the optimization problem can be written, equivalently
\min\left\lbrace\frac{1}{2n}\sum_{j=1}^p [\mathbf{r}_j-\beta_j\mathbf{x}_j]^2+\lambda |\beta_j|\right\rbrace
hence\min\left\lbrace\frac{1}{2n}\sum_{j=1}^p \beta_j^2\|\mathbf{x}_j\|-2\beta_j\mathbf{r}_j^T\mathbf{x}_j+\lambda |\beta_j|\right\rbrace
and one gets
\beta_{j,\lambda} = \frac{1}{\|\mathbf{x}_j\|^2}S(\mathbf{r}_j^T\mathbf{x}_j,n\lambda)
or, if we develop
\beta_{j,\lambda} = \frac{1}{\sum_i x_{ij}^2}S\left(\sum_ix_{i,j}[y_i-\widehat{y}_i^{(j)}],n\lambda\right)
Again, if there are weights \mathbf{\omega}=(\omega_i), the coordinate-wise update becomes
\beta_{j,\lambda,{\color{red}{\omega}}} = \frac{1}{\sum_i {\color{red}{\omega_i}}x_{ij}^2}S\left(\sum_i{\color{red}{\omega_i}}x_{i,j}[y_i-\widehat{y}_i^{(j)}],n\lambda\right)
The code to compute this componentwise descent is

lasso_coord_desc = function(X,y,beta,lambda,tol=1e-6,maxiter=1000){
  beta = as.matrix(beta)
  X = as.matrix(X)
  omega = rep(1/length(y),length(y))
  obj = numeric(length=(maxiter+1))
  betalist = list(length(maxiter+1))
  betalist[[1]] = beta
  beta0list = numeric(length(maxiter+1))
  beta0 = sum(y-X%*%beta)/(length(y))
  beta0list[1] = beta0
  for (j in 1:maxiter){
    for (k in 1:length(beta)){
      r = y - X[,-k]%*%beta[-k] - beta0*rep(1,length(y))
      beta[k] = (1/sum(omega*X[,k]^2))*
        soft_thresholding(t(omega*r)%*%X[,k],length(y)*lambda)
    }
    beta0 = sum(y-X%*%beta)/(length(y))
    beta0list[j+1] = beta0
    betalist[[j+1]] = beta
    obj[j] = (1/2)*(1/length(y))*norm(omega*(y - X%*%beta - 
           beta0*rep(1,length(y))),'F')^2 + lambda*sum(abs(beta))
    if (norm(rbind(beta0list[j],betalist[[j]]) - 
             rbind(beta0,beta),'F') &lt; tol) { break } 
  } 
  return(list(obj=obj[1:j],beta=beta,intercept=beta0)) }

For instance, consider the following (simple) dataset, with three covariates

chicago = read.table("http://freakonometrics.free.fr/chicago.txt",header=TRUE,sep=";")

that we can “normalize” (or “standardize“)

X = model.matrix(lm(Fire~.,data=chicago))[,2:4]
for(j in 1:3) X[,j] = (X[,j]-mean(X[,j]))/sd(X[,j])
y = chicago$Fire
y = (y-mean(y))/sd(y)

To initialize the algorithm, use the OLS estimate

beta_init = lm(Fire~0+.,data=chicago)$coef

For instance

lasso_coord_desc(X,y,beta_init,lambda=.001)
$obj
[1] 0.001014426 0.001008009 0.001009558 0.001011094 0.001011119 0.001011119
 
$beta
          [,1]
X_1  0.0000000
X_2  0.3836087
X_3 -0.5026137
 
$intercept
[1] 2.060999e-16

and we can get the standard Lasso plot by looping,


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