Last updated: 2023-10-13

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File Version Author Date Message
Rmd 08156f9 Matthew Stephens 2023-10-13 workflowr::wflow_publish("analysis/sloppy_admm.Rmd") 
library(ebmr.alpha)
library(ebnm)

Introduction

I wanted to try fitting the following model:

\[Y = Xu + e\]

where \[e \sim N(0,s^2)\] and \[u = b + v\] \[v \sim N(0,s_u^2)\] \[b \sim g()\] where \(g\) is a (potentially-sparse) prior to be estimated by Empirical Bayes.

One motivation here is that \(v\) is a set of “dense” effects, and \(b\) is a set of (potentially) “sparse” effects. If \(g\) is a point mass at 0 then this is ridge regression. If \(s_u =0\) then this is a (potentially) sparse regression model. So this model generalizes sparse regression and ridge regression. (If we set \(g\) as a point-normal prior then this is the BSLMM model of Zhou, Carbonetto and Stephens).

But the real motivation is that I think this model might be easy to fit by using a variational approximation and an “ADMM-like” algorithm.

If we integrate out \(v\) then we can rewrite the prior as: \[u|b \sim N(b, s_u^2)\] and \[b \sim g()\].

If we make the variational approximation \(q(b,u) = q_b(b)q_u(u)\) then the update for \(q_u\) is: \[q_u = \bar{b} + Ridge(y-X\bar{b},X,s_u^2,s^2)\] where \(\bar{b}\) denotes the expectation of \(q_b\) and \(Ridge(y,X,s^2_u,s^2)\) denotes the computation of the posterior for a ridge regression with response \(y\), covariates \(X\), prior variance \(s^2_u\) and error variance \(s^2\).

The update for \(g,q_b\) is \[(g,q_b) = EBNM(\bar{u}, s^2_u)\]

Finally, the update for \(s^2_u\) is \[s^2_u = (1/p)\sum_j [E(b_j^2) + E(u_j^2) - 2\bar{b}_j\bar{u}_j] = (1/p)\sum_j [Var(b_j)+Var(u_j) + (\bar{b}_j-\bar{u}_j)^2].\]

Example data

First I simulate some example data for testing, using a (0th order) trendfiltering example. This is a somewhat tricky example because the columns of the \(X\) matrix are so correlated.

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 + 0.1*rnorm(n)

norm = mean(Y^2) # normalize Y because it makes it easier to compare with glmnet
Y = Y/norm
btrue = btrue/norm
plot(Y)
lines(X %*% btrue)

Ridge regression

To implement these updates I need a function to perform ridge regression. So here I implement and test this function.

Here is code to fit ridge regression with fixed prior and residual variance. It returns the posterior mean (Eb) and the marginal posterior variances (Vb).

ridge = function(y,A,prior_variance,prior_mean=rep(0,ncol(A)),residual_variance=1){
  n = length(y)
  p = ncol(A)
  L = chol(t(A) %*% A + (residual_variance/prior_variance)*diag(p))
  b = backsolve(L, t(A) %*% y + (residual_variance/prior_variance)*prior_mean, transpose=TRUE)
  b = backsolve(L, b)
  #b = chol2inv(L) %*% (t(A) %*% y + (residual_variance/prior_variance)*prior_mean)
  Sigma = residual_variance * chol2inv(L) # posterior variance
  return(list(Eb = b, Vb=diag(Sigma)))
}

Here I check this code gives me the same answer as ebmr (which does empirical Bayes, so estimates prior and residual variance). Looks good.

y.fit.ebr = ebmr(X,Y, maxiter = 200, ebnv_fn = ebnv.pm)
plot(Y)
lines(X %*% btrue)
lines(X %*% y.fit.ebr$mu,col=2)

Emu = y.fit.ebr$mu # posterior mean
Vmu = y.fit.ebr$residual_variance * y.fit.ebr$Sigma_diag # variance 
prior_var = y.fit.ebr$sb2 * y.fit.ebr$residual_variance   # prior variance 
residual_var = y.fit.ebr$residual_variance # residual variance
temp = ridge(Y, X, prior_variance= prior_var, residual_variance = residual_var)
plot(temp$Eb, Emu)
abline(a=0,b=1)

plot(temp$Vb, Vmu)
abline(a=0,b=1)

The algorithm

sloppy_admm = function(X,y,maxiter=100){
  y.fit.ridge = ebmr(X,y, maxiter = 100, ebnv_fn = ebnv.pm) # fit a ridge regression
  n = nrow(X)
  p = ncol(X)
  
  Eb = rep(0,p)
  Vb = rep(0,p)
  Eu = y.fit.ebr$mu # posterior mean
  Vu= 0 #Vu = y.fit.ebr$residual_variance * y.fit.ebr$Sigma_diag # variance 
  #su2 = y.fit.ebr$sb2 * y.fit.ebr$residual_variance   # prior variance 
  s2 = y.fit.ebr$residual_variance # residual variance
  
  for(i in 1:maxiter){
    su2 = mean(Vb + Vu + (Eb-Eu)^2)
    res.ebnm = ebnm::ebnm_ash(Eu,sqrt(su2))
    Eb = res.ebnm$posterior$mean
    Vb = res.ebnm$posterior$sd^2
    
    fit.rr = ridge(y,X,su2,Eb,s2)
    Eu = fit.rr$Eb
    Vu = fit.rr$Vb
  }
  return(list(Eu=Eu,Eu.ridge = y.fit.ebr$mu))
}

Here I compare the sloppy admm fit (red) with ridge(green):

plot(Y)
lines(X %*% btrue)
res = sloppy_admm(X,Y)
lines(X %*% res$Eu ,col=2)
lines(X %*% res$Eu.ridge,col=3)


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] ebnm_1.0-55      ebmr.alpha_0.2.8

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.11       horseshoe_0.2.0   invgamma_1.1      mvtnorm_1.2-3    
 [5] lattice_0.20-45   rprojroot_2.0.3   digest_0.6.31     utf8_1.2.3       
 [9] truncnorm_1.0-9   R6_2.5.1          evaluate_0.20     ggplot2_3.4.3    
[13] highr_0.10        pillar_1.9.0      rlang_1.1.1       rstudioapi_0.14  
[17] irlba_2.3.5.1     whisker_0.4.1     jquerylib_0.1.4   R.oo_1.25.0      
[21] R.utils_2.12.2    Matrix_1.5-3      rmarkdown_2.20    splines_4.2.1    
[25] stringr_1.5.0     munsell_0.5.0     mixsqp_0.3-48     compiler_4.2.1   
[29] httpuv_1.6.9      xfun_0.37         pkgconfig_2.0.3   SQUAREM_2021.1   
[33] htmltools_0.5.4   tidyselect_1.2.0  tibble_3.2.1      workflowr_1.7.0  
[37] fansi_1.0.4       withr_2.5.0       dplyr_1.1.3       later_1.3.0      
[41] R.methodsS3_1.8.2 grid_4.2.1        jsonlite_1.8.4    gtable_0.3.4     
[45] lifecycle_1.0.3   git2r_0.31.0      magrittr_2.0.3    scales_1.2.1     
[49] cli_3.6.1         stringi_1.7.12    cachem_1.0.7      fs_1.6.1         
[53] promises_1.2.0.1  bslib_0.4.2       generics_0.1.3    vctrs_0.6.3      
[57] trust_0.1-8       tools_4.2.1       glue_1.6.2        fastmap_1.1.1    
[61] yaml_2.3.7        colorspace_2.1-0  ashr_2.2-63       deconvolveR_1.2-1
[65] knitr_1.42        sass_0.4.5