Last updated: 2020-06-22

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Rmd ab27ea5 Matthew Stephens 2020-06-22 workflowr::wflow_publish(“mr_ash_vs_lasso_02.Rmd”)

library("mr.ash.alpha")
library("mr.mash.alpha")
library("glmnet")
Loading required package: Matrix
Loaded glmnet 3.0-2

Introduction

This is a follow-up to my previous investigation where it looks like mr.ash may have convergence issues on cases with dense variables and high PVE.

Here I want to try to check that this really is a convergence issue by checking the objective function from different initialization strategies. I use the mr.mash implementation here since we believe it computes objective correctly even when prior is fixed, which at time of writing was not true for mr.ash.

I run mr.mash in different ways:

  1. Initialized just b from lasso solution
  2. Fix g and V to true g and true V, init from lasso solution.
  3. Fix g and V to true g and true V, init from true b
  set.seed(123)
  n <- 500
  p <- 1000
  p_causal <- 500 # number of causal variables (simulated effects N(0,1))
  pve <- 0.95
  nrep = 10
  rmse = list(mr_mash=rep(0,nrep),lasso = rep(0,nrep),ridge=rep(0,nrep),mr_ash=rep(0,nrep))
  obj = list(mr_mash=rep(0,nrep),lasso = rep(0,nrep),ridge=rep(0,nrep),mr_ash=rep(0,nrep))
  
  for(i in 1:nrep){
    sim=list()
    sim$X =  matrix(rnorm(n*p,sd=1),nrow=n)
    B <- rep(0,p)
    causal_variables <- sample(x=(1:p), size=p_causal)
    B[causal_variables] <- rnorm(n=p_causal, mean=0, sd=1)
    sim$B = B
    sim$Y = sim$X %*% sim$B
    
    E = rnorm(n,sd = sqrt((1-pve)/(pve))*sd(sim$Y))
    sim$Y = sim$Y + E
    sim$E = E
    
    fit_lasso <- cv.glmnet(x=sim$X, y=sim$Y, family="gaussian", alpha=1, standardize=FALSE)
    fit_ridge <- cv.glmnet(x=sim$X, y=sim$Y, family="gaussian", alpha=0, standardize=FALSE)
    fit_mrash <- mr.ash.alpha::mr.ash(sim$X, sim$Y, beta.init=coef(fit_lasso)[-1], standardize = FALSE)
     
    ###Fit mr.mash univariate
    grid <- fit_mrash$data$sa2 * fit_mrash$sigma2
    s0 <- vector("list", length(grid)+1)
    for(j in 1:(length(grid)+1)){
      s0[[j]] <- matrix(c(0, grid)[j], ncol=1, nrow=1)
    }
    fit_mrmash <- mr.mash(sim$X, cbind(sim$Y), s0, tol=1e-8, convergence_criterion="ELBO", update_w0=TRUE,
                          update_w0_method="EM", compute_ELBO=TRUE, standardize=FALSE, verbose=FALSE, update_V=TRUE,
                          mu1_init=matrix(coef(fit_lasso)[-1], nrow=p, ncol=1), w0_threshold=0)
    
    ###Fit mr.mash univariate using true g etc
    s2 = (sqrt((1-pve)/(pve))*sd(sim$Y))^2
    grid = c(1)
    s0 <- vector("list", length(grid)+1)
    for(j in 1:(length(grid)+1)){
      s0[[j]] <- matrix(c(0, grid)[j], ncol=1, nrow=1)
    }
    fit_mrmash_trueg_trueV <- mr.mash(sim$X, cbind(sim$Y), s0, w0 = c(0.5,0.5), V=matrix(s2,nrow=1,ncol=1),tol=1e-8, convergence_criterion="ELBO", update_w0=FALSE, compute_ELBO=TRUE, standardize=FALSE, verbose=FALSE, update_V=FALSE,mu1_init=matrix(coef(fit_lasso)[-1], nrow=p, ncol=1), w0_threshold=0)
    
    fit_mrmash_trueg_trueV_trueb <- mr.mash(sim$X, cbind(sim$Y), s0, w0 = c(0.5,0.5), V=matrix(s2,nrow=1,ncol=1),tol=1e-8, convergence_criterion="ELBO", update_w0=FALSE, compute_ELBO=TRUE, standardize=FALSE, verbose=FALSE, update_V=FALSE,mu1_init=matrix(sim$B, nrow=p, ncol=1), w0_threshold=0)
    
    
    rmse$mr_ash[i] = sqrt(mean((sim$B-fit_mrash$beta)^2))
    rmse$mr_mash[i] = sqrt(mean((sim$B-fit_mrmash$mu1)^2))
    rmse$lasso[i] = sqrt(mean((sim$B-coef(fit_lasso)[-1])^2))
    rmse$ridge[i] = sqrt(mean((sim$B-coef(fit_ridge)[-1])^2))
    rmse$mr_mash_trueg_trueV[i] = sqrt(mean((sim$B-fit_mrmash_trueg_trueV$mu1)^2))
    rmse$mr_mash_trueg_trueV_trueb[i] = sqrt(mean((sim$B-fit_mrmash_trueg_trueV_trueb$mu1)^2))
    
    
    obj$mr_ash[i] = min(fit_mrash$varobj)
    obj$mr_mash[i] = fit_mrmash$ELBO
    obj$mr_mash_trueg_trueV[i] = fit_mrmash_trueg_trueV$ELBO
    obj$mr_mash_trueg_trueV_trueb[i] = fit_mrmash_trueg_trueV_trueb$ELBO
  }
Processing the inputs... Done!
Fitting the optimization algorithm... 
Warning in mr.mash(sim$X, cbind(sim$Y), s0, tol = 1e-08, convergence_criterion =
"ELBO", : Max number of iterations reached. Try increasing max_iter.
Done!
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mr.mash vs mr.ash

First compare RMSE and objective of mr.ash vs mr.mash from default settings (same grid for both). Note the obective in mr.ash is the negative ELBO.

  plot(rmse$mr_ash,rmse$mr_mash, main="RMSE: mr.ash vs mr.mash", ylab="mr.mash", xlab="mr.ash")
  abline(a=0,b=1)

  plot(obj$mr_ash,obj$mr_mash, main="objective: mr.ash vs mr.mash", ylab="mr.mash", xlab="mr.ash")
  abline(a=0,b=-1)

mr.mash vs ridge/lasso

Now compare RMSE against lasso and ridge; as we know lasso is better here.

 plot(rmse$mr_mash,rmse$lasso, xlim=c(0.5,0.7), ylim=c(0.5,0.7), main="RMSE, mr_mash vs lasso (black) and ridge (red)")
  points(rmse$mr_mash,rmse$ridge,col=2)
  abline(a=0,b=1)

mr.mash objective with true g and true V

Now compare objective with default initialization vs fix g and V to true values. We see the objective is consistently better when g and V are estimated. (Red shows initialization from true b)

  plot(obj$mr_mash,obj$mr_mash_trueg_trueV, xlim=c(-2500,-2200),ylim=c(-2500,-2200), main="Compare objective: g,V estimated vs fixed")
  points(obj$mr_mash,obj$mr_mash_trueg_trueV_trueb, col=2)
  abline(a=0,b=1)

mr.mash objective with true g and true V

Now compare rmse. We confirm the rmse performance is better for true (g,V), as it should be. But, of course, this is cheating…

  plot(rmse$mr_mash,rmse$mr_mash_trueg_trueV,xlim=c(0.5,0.7),ylim=c(0.5,0.7))
  points(rmse$mr_mash,rmse$mr_mash_trueg_trueV_trueb, col=2)
  abline(a=0,b=1)

My explanation for this behaviour is that the gap between the variational approximation and true posterior is smaller for (g,V) that correspond to less signal. So it tends to favor a solution with less signal than it should.


sessionInfo()
R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.6

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] glmnet_3.0-2         Matrix_1.2-18        mr.mash.alpha_0.1-79
[4] mr.ash.alpha_0.1-34 

loaded via a namespace (and not attached):
 [1] MBSP_1.0           Rcpp_1.0.4.6       compiler_3.6.0     later_1.0.0       
 [5] git2r_0.26.1       workflowr_1.6.1    iterators_1.0.12   tools_3.6.0       
 [9] digest_0.6.25      evaluate_0.14      lattice_0.20-40    GIGrvg_0.5        
[13] rlang_0.4.5        foreach_1.4.8      yaml_2.2.1         mvtnorm_1.1-1     
[17] SparseM_1.78       xfun_0.12          coda_0.19-3        stringr_1.4.0     
[21] knitr_1.28         fs_1.3.2           MatrixModels_0.4-1 rprojroot_1.3-2   
[25] grid_3.6.0         glue_1.4.0         R6_2.4.1           rmarkdown_2.1     
[29] mixsqp_0.3-43      irlba_2.3.3        magrittr_1.5       whisker_0.4       
[33] codetools_0.2-16   backports_1.1.5    promises_1.1.0     htmltools_0.4.0   
[37] matrixStats_0.56.0 mcmc_0.9-7         MASS_7.3-51.5      shape_1.4.4       
[41] httpuv_1.5.2       quantreg_5.54      stringi_1.4.6      MCMCpack_1.4-8