Last updated: 2020-06-22
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library("mr.ash.alpha")
library("mr.mash.alpha")
library("glmnet")
Loading required package: Matrix
Loaded glmnet 3.0-2
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:
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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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)
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)
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)
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