multiple regression with ash, parallel coordinate ascent

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Last updated: 2019-10-12

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library("ashr")

Introduction

The idea here is to investigate a parallel approach to updating bhat in multiple regression with ash.

The basic idea is that the optimal b should be the fixed point of the following iterations: 1. r = (Y-Xb) 2. bhat = b + d^{-1}X’r 3. shat = sigma/sqrt(d) 4. b = ash(bhat,shat)

More accurately, I believe that if b is a fixed point of this then it will also be a fixed point of the regular coordinate ascent (at least, something like this should be true.)

To start with we fix g to N(0,1) to keep thing simple….

(Note that the rescaling step was supposed to allow scaling prior and posterior by some constant, but results suggest I may not have the details right here. Ignore this for now.)

mr_ash_parallel_ca_fix = function(X,Y,b_init=NULL,max_iter=100,sigma=1,tol=1e-5,rescale=FALSE){
  if(is.null(b_init)){b_init = rep(0,ncol(X))}
  b = b_init
  d = Matrix::colSums(X * X)
  for(i in 1:max_iter){
    r = Y- X %*% b
    bhat = as.vector(b + (1/d)*(t(X) %*% r))
    s = sigma/sqrt(d)
    b2 = get_pm(ash(bhat,s,g=normalmix(1,0,1),fixg=TRUE))
    if(sum((b2-b)^2)<tol){break;}
    b=b2
    
    if(rescale){
      fitted = X %*% b  
      c = (1/sum(fitted^2)) * sum(fitted*Y) # regress Y on fitted values
      b = c*b
    }
    
  }
  
  print(paste0("niter = ",i))
  return(b)
}

mr_ash_ca_fix = function(X,Y,b_init=NULL,max_iter=100,sigma=1,tol=1e-3,rescale=FALSE){
  if(is.null(b_init)){b_init = rep(0,ncol(X))}
  b = b_init
  p = ncol(X)
  d = Matrix::colSums(X * X)
  c = 1
  r = Y - X %*%  b
  for(i in 1:max_iter){
    err = 0
    for(j in 1:p){
      r = r + b[j]*X[,j] 
      bhat = (1/d[j])*sum(X[,j]* r)
      s = sigma/sqrt(d[j])
      bj_new = get_pm(ash(bhat,s,g=normalmix(1,0,c),fixg=TRUE))
      err = err + (b[j]-bj_new)^2
      b[j] = bj_new
      r = r - b[j]*X[,j] # recompute residuals
    }
    if(rescale){
      fitted = Y - r  #so fitted = Xb
      c = (1/sum(fitted^2)) * sum(fitted*Y) # regress Y on fitted values
      b = c*b
      r = Y - c * fitted
    }
    if(err<tol){break;}
  }
  
  print(paste0("niter = ",i))
  return(b)
}



ridge = function(X,Y,sigma=1){
  p = ncol(X)
  S = sigma^2*diag(p) + t(X) %*% X
  bhat = solve(S, t(X) %*% Y)
  return(bhat)
}

A toy example to check:

set.seed(123)
n= 100
p=10
X = matrix(rnorm(n*p),ncol=p,nrow=n)
btrue = rnorm(p)
Y = X %*% btrue + rnorm(n)

b.pca = mr_ash_parallel_ca_fix(X,Y)

[1] "niter = 10"

b.ca = mr_ash_ca_fix(X,Y)

[1] "niter = 4"

b.ridge = ridge(X,Y)
plot(btrue,b.ca)
points(btrue,b.pca,col=2,pch=2)
points(btrue,b.ridge,col=3,pch=3)

Past versions of unnamed-chunk-3-1.png

Version	Author	Date
721b34f	Matthew Stephens	2019-10-07

And a sparse example

btrue[1:5]=0
Y = X %*% btrue + rnorm(n)
b.pca = mr_ash_parallel_ca_fix(X,Y)

[1] "niter = 9"

b.ca = mr_ash_ca_fix(X,Y)

[1] "niter = 4"

b.ridge = ridge(X,Y)
plot(btrue,b.ca)
points(btrue,b.pca,col=2,pch=2)
points(btrue,b.ridge,col=3,pch=3)

Past versions of unnamed-chunk-4-1.png

Version	Author	Date
721b34f	Matthew Stephens	2019-10-07

Now try example with X duplicated. As might have been anticipated, it fails to converge and returns a ridiculous solution that seems to be diverging off to +-infinity.

set.seed(123)
n= 100
p=10
X = matrix(rnorm(n*p),ncol=p,nrow=n)
X = cbind(X ,X)
btrue = rnorm(2*p)
Y = X %*% btrue + rnorm(n)

b.pca = mr_ash_parallel_ca_fix(X,Y)

[1] "niter = 100"

b.ca = mr_ash_ca_fix(X,Y)

[1] "niter = 48"

b.ridge = ridge(X,Y)
plot(btrue,b.ca)
points(btrue,b.pca,col=2,pch=2)
points(btrue,b.ridge,col=3,pch=3)

Past versions of unnamed-chunk-5-1.png

Version	Author	Date
721b34f	Matthew Stephens	2019-10-07

print(b.pca)

 [1] -8.204435e+27  1.079512e+27  4.606275e+27 -7.740797e+27  7.422769e+27
 [6]  5.492114e+27  5.070576e+26 -8.111340e+27  2.752684e+27 -5.756308e+27
[11] -8.204435e+27  1.079512e+27  4.606275e+27 -7.740797e+27  7.422769e+27
[16]  5.492114e+27  5.070576e+26 -8.111340e+27  2.752684e+27 -5.756308e+27

Try same thing initializing from truth - it still diverges.

b.pca = mr_ash_parallel_ca_fix(X,Y,b_init = btrue)

[1] "niter = 100"

plot(btrue,b.pca)

Past versions of unnamed-chunk-6-1.png

Version	Author	Date
721b34f	Matthew Stephens	2019-10-07

Note that the fitted values do not fit Y at all for the parallel case. The others do.

plot(Y,X %*% b.pca)

plot(Y,X %*% b.ca, col=, pch=2, main="fitted values for ridge and CA")
points(Y,X %*% b.ridge, col=3,pch=3)

See if CA matches ridge with more stringent convergence tolerance:

b.ca = mr_ash_ca_fix(X,Y,tol = 1e-8,max_iter=1000)

[1] "niter = 370"

plot(btrue,b.ca)
points(btrue,b.ridge,col=3,pch=3)

Past versions of unnamed-chunk-8-1.png

Version	Author	Date
721b34f	Matthew Stephens	2019-10-07

Rescaling

Next I tried rescaling the fitted values and prior each iteration by a constant c. This might seem ad hoc, but I think something like this can be justified as scaling both the prior and the posterior approximation (although results later suggest I might have the details wrong…)

In this example, rescaling definitely stabilizes the estimates…

b.pca = mr_ash_parallel_ca_fix(X,Y,b_init = btrue,max_iter = 100,rescale=TRUE)

[1] "niter = 100"

plot(Y,X %*% b.pca)
points(Y,X %*% b.ca, col=2, pch=2)
points(Y,X %*% b.ridge, col=3,pch=3)

I poked around and found that in fact it was flipping between two different solutions. Here I run it for 99 and 98 iterations to illustrate.

b.pca.99= mr_ash_parallel_ca_fix(X,Y,b_init = btrue,max_iter = 99,rescale=TRUE)

[1] "niter = 99"

b.pca.98= mr_ash_parallel_ca_fix(X,Y,b_init = btrue,max_iter = 98,rescale=TRUE)

[1] "niter = 98"

plot(b.pca,b.pca.98)

Past versions of unnamed-chunk-10-1.png

Version	Author	Date
721b34f	Matthew Stephens	2019-10-07

plot(b.pca,b.pca.99)

Past versions of unnamed-chunk-10-2.png

Version	Author	Date
721b34f	Matthew Stephens	2019-10-07

trend filtering example

This example the X will be highly correlated, but not completely so. It is designed to be challenging but easy to visualize what is going on.

set.seed(100)
n = 100
p = n
X = matrix(0,nrow=n,ncol=n)
for(i in 1:n){
  X[i:n,i] = 1:(n-i+1)
}
btrue = rep(0,n)
btrue[40] = 8
btrue[41] = -8
Y = X %*% btrue + rnorm(n)
plot(Y)
lines(X %*% btrue)

The ridge solution is much better than ca, presumably due to very slow convergence. (The solution looks better with 1000 iterations, but has still not converged; not shown here to keep runtime down…)

bhat_ca = mr_ash_ca_fix(X,Y,max_iter = 100,tol=1-8)

[1] "niter = 100"

bhat_r = ridge(X,Y)
plot(Y)
lines(X %*% bhat_ca,col=2)
lines(X %*% bhat_r, col=3)

Past versions of unnamed-chunk-12-1.png

Version	Author	Date
721b34f	Matthew Stephens	2019-10-07

Parallel version goes crazy; again rescaling helps stabilize but not really working.

bhat_pca = mr_ash_parallel_ca_fix(X,Y,max_iter= 10) 

[1] "niter = 10"

plot(X %*% bhat_pca,col=4)

Past versions of unnamed-chunk-13-1.png

Version	Author	Date
721b34f	Matthew Stephens	2019-10-07

bhat_pca = mr_ash_parallel_ca_fix(X,Y,max_iter= 100, rescale=TRUE) 

[1] "niter = 100"

plot(Y)
lines(X %*% bhat_pca,col=4)

Past versions of unnamed-chunk-13-2.png

Version	Author	Date
721b34f	Matthew Stephens	2019-10-07

Session information

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     

other attached packages:
[1] ashr_2.2-38

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.2        knitr_1.23        whisker_0.3-2    
 [4] magrittr_1.5      workflowr_1.4.0   MASS_7.3-51.4    
 [7] pscl_1.5.2        doParallel_1.0.14 SQUAREM_2017.10-1
[10] lattice_0.20-38   foreach_1.4.7     stringr_1.4.0    
[13] tools_3.6.0       parallel_3.6.0    grid_3.6.0       
[16] xfun_0.8          git2r_0.26.1      htmltools_0.3.6  
[19] iterators_1.0.12  yaml_2.2.0        rprojroot_1.3-2  
[22] digest_0.6.20     mixsqp_0.1-97     Matrix_1.2-17    
[25] fs_1.3.1          codetools_0.2-16  glue_1.3.1       
[28] evaluate_0.14     rmarkdown_1.14    stringi_1.4.3    
[31] compiler_3.6.0    backports_1.1.4   truncnorm_1.0-8