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Introduction

I wanted to use IRLS to fit a simple logistic regression. The following code does so for all columns of X simultaneously. That is, it fits y ~ mu + b Xj for j= 1…p.

Single vector x

I start by writing a function to fit a single vector x. The code is based on the IRLS algorithm for logistic regression, as described in the book “Elements of Statistical Learning” by Hastie, Tibshirani and Friedman (2009), section 4.4.1, written in a way that it makes it easy to vectorize/repeat over the columns of a matrix X via matrix multiplications. (See below)

logistic_IRLS_simple <- function(x, y, max_iter = 100, tolerance = 1e-6, lambda=0) {
  # Initialize coefficients
  mu <- 0
  beta <- 0
  converged <- FALSE
  
  for (iter in 1:max_iter) {
    eta <- mu + x * beta  # Linear predictor
    pi <- exp(eta) / (1 + exp(eta))  # Predicted probabilities

    # Weights for IRLS 
    w <- pi * (1 - pi)

    # Working response variable
    z <- eta + (y - pi) / (pi * (1 - pi)) 

    # Weighted least squares update
    #These are the elements of the X'X matrix (a b) (c d)
    a = sum(w)
    b = sum(w*x)
    c = sum(w*x)
    d = sum(w*x^2) + lambda
    
    wz = sum(w*z)
    wxz = sum(w*x*z)
    
    new_mu <- (d*wz - b*wxz) / (a*d - b*c)
    new_beta <- (-c*wz + a*wxz) /  (a*d - b*c)

    # Check for convergence
    if (all(abs(new_beta - beta) < tolerance)) {
      converged = TRUE
      break
    }

    beta <- new_beta
    mu <- new_mu
  }

  # Return fitted coefficients
  return(list(mu = mu, beta = beta, converged = converged, iter=iter))
}

Now I want to test the function with some simulated data. Simulate a simple logistic regression:

set.seed(1)
n <- 10000
x <- rnorm(n)
eta <- 5 + 2*x
pi <- exp(eta) / (1 + exp(eta))
y <- rbinom(n, 1, pi)
logistic_IRLS_simple(x, y)
$mu
[1] 5.043687

$beta
[1] 2.046827

$converged
[1] TRUE

$iter
[1] 9
glm(y~x, family = binomial)

Call:  glm(formula = y ~ x, family = binomial)

Coefficients:
(Intercept)            x  
      5.044        2.047  

Degrees of Freedom: 9999 Total (i.e. Null);  9998 Residual
Null Deviance:      2994 
Residual Deviance: 1998     AIC: 2002

And compare with glmnet. Note that glmnet requires two variables so I add an intercept. One thing I don’t understand: I tried setting “intercept=FALSE” but and not penalizing the intercept but it produced different results, which was unexpected.

library(glmnet)
Loading required package: Matrix
Loaded glmnet 4.1-8
fit = glmnet(cbind(rep(1,n),x), y, family="binomial",alpha=0, lambda=1/n)
fit$beta
2 x 1 sparse Matrix of class "dgCMatrix"
        s0
  .       
x 2.031983
logistic_IRLS_simple(x, y, lambda=1)
$mu
[1] 5.024606

$beta
[1] 2.033247

$converged
[1] TRUE

$iter
[1] 9
fit = glmnet(cbind(rep(1,n),x), y, family="binomial",alpha=0, lambda=10/n)
fit$beta
2 x 1 sparse Matrix of class "dgCMatrix"
        s0
  .       
x 1.920453
logistic_IRLS_simple(x, y, lambda=10)
$mu
[1] 4.872961

$beta
[1] 1.923797

$converged
[1] TRUE

$iter
[1] 9
fit2 = glmnet(cbind(rep(1,n),x), y, family="binomial",alpha=0, lambda=10/n, penalty.factor = c(0,1), intercept = FALSE)

coef(fit2)
3 x 1 sparse Matrix of class "dgCMatrix"
                   s0
(Intercept) .        
            .        
x           0.2065842
coef(fit)
3 x 1 sparse Matrix of class "dgCMatrix"
                  s0
(Intercept) 4.868390
            .       
x           1.920453

Matrix version

I now write a function that repeats the above for all columns of X simultaneously via vector/matrix operations.

logistic_IRLS_simple_matrix <- function(X, y, max_iter = 100, tolerance = 1e-6, lambda = 0) {
  # Initialize coefficients
  p <- ncol(X)
  mu <- rep(0,p)
  beta <- rep(0,p)
  converged <- FALSE
  
  for (iter in 1:max_iter) {
    eta <- t(mu + t(X) * beta)  # Linear predictor
    pi <- exp(eta) / (1 + exp(eta))  # Predicted probabilities

    # Weights for IRLS 
    w <- pi * (1 - pi)

    # Working response variable
    z <- eta + (y - pi) / (pi * (1 - pi)) 

    # Weighted least squares update
    #These are the elements of the X'X matrix (a b) (c d)
    a = colSums(w)
    b = colSums(w*X)
    c = b
    d = colSums(w*X^2) + lambda
    
    wz = colSums(w*z)
    wxz = colSums(w*X*z)
    
    new_mu <- (d*wz - b*wxz) / (a*d - b*c)
    new_beta <- (-c*wz + a*wxz) /  (a*d - b*c)

    # Check for convergence
    if (all(abs(new_beta - beta) < tolerance)) {
      converged = TRUE
      break
    }

    beta <- new_beta
    mu <- new_mu
  }

  # Return fitted coefficients
  return(list(mu = mu, beta = beta, converged = converged, iter=iter))
}

Now I test the function with some simulated data. It seems to work.

set.seed(1)
n <- 10000
X <- cbind(rnorm(n), rnorm(n))
eta <- 10 + 2*X[,1] + 3*X[,2]
pi <- exp(eta) / (1 + exp(eta))
y <- rbinom(n, 1, pi)

glm(y~X[,1], family = binomial)

Call:  glm(formula = y ~ X[, 1], family = binomial)

Coefficients:
(Intercept)       X[, 1]  
      6.084        1.427  

Degrees of Freedom: 9999 Total (i.e. Null);  9998 Residual
Null Deviance:      764.1 
Residual Deviance: 639.4    AIC: 643.4
glm(y~X[,2], family = binomial)

Call:  glm(formula = y ~ X[, 2], family = binomial)

Coefficients:
(Intercept)       X[, 2]  
      7.214        2.211  

Degrees of Freedom: 9999 Total (i.e. Null);  9998 Residual
Null Deviance:      764.1 
Residual Deviance: 521.9    AIC: 525.9
logistic_IRLS_simple_matrix(X, y)
$mu
[1] 6.083537 7.213586

$beta
[1] 1.427305 2.211099

$converged
[1] TRUE

$iter
[1] 10

sessionInfo()
R version 4.2.1 (2022-06-23)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur ... 10.16

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] glmnet_4.1-8 Matrix_1.6-4

loaded via a namespace (and not attached):
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[25] stringr_1.5.1     knitr_1.45        fs_1.6.3          vctrs_0.6.5      
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[33] R6_2.5.1          fansi_1.0.6       survival_3.5-7    rmarkdown_2.25   
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