Last updated: 2019-10-21

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Rmd e35052f Matthew Stephens 2019-10-21 workflowr::wflow_publish(“ridge_conjugate_gradient.Rmd”)

Introduction

The goal here is for me to experiment with using conjugate gradient to solve ridge regression, mostly as a learning experience for me.

Conjugate gradient code

This code was modified from the matlab code given on Wikipedia.

# A an n by n PSD matrix; x and b are column vectors of length n
conjgrad = function(A, b, x){
    r = b - A %*% x
    p = r
    rsold = t(r) %*% r

    for(i in 1:length(b)){
        Ap = A %*% p
        alpha = as.numeric(rsold / (t(p) %*% Ap))
        x = x + alpha * p
        r = r - alpha * Ap
        rsnew = t(r) %*% r
        if(sqrt(rsnew) < 1e-10){
          break
        }
        p = r + as.numeric(rsnew / rsold) * p
        rsold = rsnew
    }
    return(list(x=x,niter = i))
}

Apply to changepoint

Here we simulate some data from a changepoint model

set.seed(100)
n = 100
p = n
X = matrix(0,nrow=n,ncol=n)
for(i in 1:n){
  X[i:n,i] = 1
}
btrue = rep(0,n)
btrue[40] = 8
Y = X %*% btrue + rnorm(n)
plot(Y)
lines(X %*% btrue)

Now apply the conjugate gradient method to solve the ridge regression problem, with prior variance and residual variance both = 1, so \(A = X'X + I\) and \(b=X'Y\).

A = t(X) %*% X + diag(n)
b = t(X) %*% Y
res = conjgrad(A, b, x = cbind(rep(0,100))) 
res$niter
[1] 60
plot(Y)
lines(X %*% btrue)
lines(X %*% res$x,col=2)

Preconditioning

Now we try preconditioning by \(T=(X'X)^{-1}\), and solving \(TAx = Tb\). Obviously we would not want to compute \(T\) explicitly in practice. I’m just doing it here to see how it affects iterations. It roughly halves the iterations here.

T = solve(t(X) %*% X)
A2 = T %*% A
b2 = T %*% b
res2 = conjgrad(A2, b2, x = cbind(rep(0,100))) 
res2$niter
[1] 27
plot(Y)
lines(X %*% btrue)
lines(X %*% res$x,col=2)


sessionInfo()
R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.4

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/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     

loaded via a namespace (and not attached):
 [1] workflowr_1.4.0 Rcpp_1.0.2      digest_0.6.20   rprojroot_1.3-2
 [5] backports_1.1.4 git2r_0.26.1    magrittr_1.5    evaluate_0.14  
 [9] stringi_1.4.3   fs_1.3.1        whisker_0.3-2   rmarkdown_1.14 
[13] tools_3.6.0     stringr_1.4.0   glue_1.3.1      xfun_0.8       
[17] yaml_2.2.0      compiler_3.6.0  htmltools_0.3.6 knitr_1.23