Accelerating co-ordinate ascent updates for linear regression using DAAREM

Analysis settings

These variables specify how the data are generated: n is the number of simulated samples, p is the number of simulated predictors, na is the number of simulated predictors that have a nonzero effect, se is the variance of the residual.

n  <- 200 
p  <- 500
na <- 10
se <- 4

This specifies the prior on the regression coefficients: it is normal with zero mean and variance s0.

s0 <- 1/se

Set up environment

Load some packages and function definitions used in the example below.

library(MASS)
library(daarem)
library(ggplot2)
library(cowplot)
source("../code/misc.R")
source("../code/datasim.R")
source("../code/ridge.R")
source("../code/mr_ash.R")

Initialize the sequence of pseudorandom numbers.

set.seed(1)

Simulate a data set

Simulate predictors with “decaying” correlations.

X <- simulate_predictors_decaying_corr(n,p,s = 0.5)
X <- scale(X,center = TRUE,scale = FALSE)

Generate additive effects for the markers so that exactly na of them have a nonzero effect on the trait.

i    <- sample(p,na)
b    <- rep(0,p)
b[i] <- rnorm(na)

Simulate the continuous outcomes, and center them.

y <- drop(X %*% b + sqrt(se)*rnorm(n))
y <- y - mean(y)

Run ridge regression co-ordinate ascent updates

Set the initial estimate of the coefficients.

b0 <- rep(0,p)

Fit the ridge regression model by running 100 iterations of the basic co-ordinate ascent updates. Note that the co-ordinate ascent updates are very simple, and are easily implemented in a single line of R code; see the code for the ridge.update function.

out <- system.time(fit1 <- ridge(X,y,b0,s0,numiter = 100))
f1  <- ridge.objective(X,y,fit1$b,s0)
cat(sprintf("Computation took %0.2f seconds.\n",out["elapsed"]))
cat(sprintf("Objective value at solution is %0.12f.\n",f1))
# Computation took 0.46 seconds.
# Objective value at solution is -20.573760535831.

Run accelerated co-ordinate ascent updates

Fit the ridge regression model again, this time using DAAREM to speed up the co-ordinate ascent algorithm.

out <- system.time(fit2 <- daarridge(X,y,b0,s0,numiter = 100))
f2  <- ridge.objective(X,y,fit2$b,s0)
cat(sprintf("Computation took %0.2f seconds.\n",out["elapsed"]))
cat(sprintf("Objective value at solution is %0.12f.\n",f2))
# Computation took 0.49 seconds.
# Objective value at solution is -20.332771749786.

We see that the DAAREM solution is better (it has a higher posterior value).

Plot improvement in solution over time

Since the ridge estimate as a closed-form solution, we can easily compare the above estimates obtained via co-ordinate ascent against the actual solution.

bhat <- drop(solve(t(X) %*% X + diag(rep(1/s0,p)),t(X) %*% y))
f    <- ridge.objective(X,y,bhat,s0)

This plot shows the improvement in the solution over time for the two co-ordinate ascent algorithms: the vertical axis (“distance to best solution”) shows the difference between the largest log-posterior obtained, and the log-posterior at the actual ridge solution (bhat).

pdat <-
  rbind(data.frame(iter = 1:100,dist = f - fit1$value,method = "basic"),
        data.frame(iter = 1:100,dist = f - fit2$value,method = "accelerated"))
p <- ggplot(pdat,aes(x = iter,y = dist,col = method)) +
  geom_line(size = 1) +
  scale_y_continuous(trans = "log10",breaks = 10^seq(-8,4)) +
  scale_color_manual(values = c("darkorange","dodgerblue")) +
  labs(x = "iteration",y = "distance from solution")
print(p)

Past versions of plot-iter-vs-objective-1.png

From this plot, we see that the accelerated algorithm progresses much more rapidly toward the solution; after 100 iterations, it nearly recovers the actual ridge estimates, whereas the unaccelerated version is still very far away.

Linear regression with mixture-of-normals priors

Next, we consider a less simple case in which the regression coefficients are independently and identically drawn from mixture of zero-centered normals; this can be seen as a multivariate extension to the adaptive shrinkage model, so we call this “multivariate regression adaptive shrinkage” (mr-ash). Although posterior computations with this model are more difficult than with ridge regression, we can nonetheless obtain simple co-ordinate ascent updates for computing posterior expectations of the coefficients if we introduce a variational approximation to the posterior distribution. The full derivation is omitted here; see the code in the mr_ash_update function for details. (Note that the co-ordinate ascent updates, unlike the ridge regression updates, are only guaranteed to recover a local maximum of the objective function being optimized.)

These two variables specify the variances and mixture weights for the mixture-of-normals priors. Here we illustrate mr-ash with a prior that is a mixture of three normals.

s0 <- c(0.1,1,10)^2/se
w  <- c(0.5,0.25,0.25)

Run mr-ash co-ordinate ascent updates

Fit the mr-ash model by running 200 iterations of the basic co-ordinate ascent updates.

out <- system.time(fit3 <- mr_ash(X,y,b0,se,s0,w,numiter = 100))
cat(sprintf("Computation took %0.2f seconds.\n",out["elapsed"]))
# Computation took 1.11 seconds.

Run accelerated mr-ash co-ordinate ascent updates

Fit the mr-ash model again, this time using DAAREM to speed up the co-ordinate ascent updates.

out <- system.time(fit4 <- daar_mr_ash(X,y,b0,se,s0,w,numiter = 100))
cat(sprintf("Computation took %0.2f seconds.\n",out["elapsed"]))
# Computation took 1.19 seconds.

Like the plot above, plot shows the improvement in the solution over time for the basic and accelated co-ordinate ascent algorithms for mr-ash. Both algorithms end up at the same solution, but the “accelerated” version indeed arrives at the solution much more quickly.

f    <- max(c(fit3$value,fit4$value)) + 1e-8
pdat <- 
  rbind(data.frame(iter = 1:100,dist = f - fit3$value,method = "basic"),
        data.frame(iter = 1:100,dist = f - fit4$value,method = "accelerated"))
p <- ggplot(pdat,aes(x = iter,y = dist,col = method)) +
  geom_line(size = 1) +
  scale_y_continuous(trans = "log10",breaks = 10^seq(-8,4)) +
  scale_color_manual(values = c("darkorange","dodgerblue")) +
  labs(x = "iteration",y = "distance from best solution")
print(p)

Past versions of plot-iter-vs-mr-ash-objective-1.png

Session information

sessionInfo()
# R version 3.4.3 (2017-11-30)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS High Sierra 10.13.6
# 
# Matrix products: default
# BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
# LAPACK: /Library/Frameworks/R.framework/Versions/3.4/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] cowplot_0.9.4 ggplot2_3.1.0 daarem_0.3    MASS_7.3-48  
# 
# loaded via a namespace (and not attached):
#  [1] Rcpp_1.0.1        compiler_3.4.3    pillar_1.3.1     
#  [4] git2r_0.25.2.9008 plyr_1.8.4        workflowr_1.4.0  
#  [7] tools_3.4.3       digest_0.6.18     evaluate_0.13    
# [10] tibble_2.1.1      gtable_0.2.0      pkgconfig_2.0.2  
# [13] rlang_0.3.1       yaml_2.2.0        xfun_0.7         
# [16] withr_2.1.2.9000  stringr_1.4.0     dplyr_0.8.0.1    
# [19] knitr_1.23        fs_1.2.7          rprojroot_1.3-2  
# [22] grid_3.4.3        tidyselect_0.2.5  glue_1.3.1       
# [25] R6_2.4.0          rmarkdown_1.13    purrr_0.2.5      
# [28] magrittr_1.5      whisker_0.3-2     backports_1.1.2  
# [31] scales_0.5.0      htmltools_0.3.6   assertthat_0.2.0 
# [34] colorspace_1.4-0  labeling_0.3      stringi_1.4.3    
# [37] lazyeval_0.2.1    munsell_0.4.3     crayon_1.3.4