Accelerating co-ordinate ascent updates for linear regression using DAAREM

Last updated: 2019-06-04

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Rmd	a79dd49	Peter Carbonetto	2019-06-04	wflow_publish(“mixem.Rmd”)
html	52e323d	Peter Carbonetto	2019-06-04	Made some revisions to the mixem analysis.
Rmd	4743aea	Peter Carbonetto	2019-06-04	wflow_publish(“mixem.Rmd”)
html	936cdbb	Peter Carbonetto	2019-06-04	Built initial draft of “mixem” analysis.
Rmd	7048a42	Peter Carbonetto	2019-06-04	wflow_publish(“mixem.Rmd”)
Rmd	1636e77	Peter Carbonetto	2019-06-04	wflow_publish(“index.Rmd”)

Here we illustrate the use of DAAREM to accelerate a very simple EM algorithm—the E and M steps are implemented in three lines of R code—for computing maximum-likelihood estimates of mixture proportions in a mixture model.

Set up environment

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

library(ggplot2)
library(cowplot)
library(daarem)
source("../code/misc.R")
source("../code/mixem.R")

Load data set

Load the 100,000 x 100 conditional likelihood matrix computed from a simulated data set.

load("../data/mixdata.RData")
n <- nrow(L)
m <- ncol(L)
cat(sprintf("Loaded %d x %d data matrix.\n",n,m))
# Loaded 100000 x 10 data matrix.

Set the initial estimate of the mixture proportions.

x0 <- rep(1/m,m)

Run basic EM updates

Compute maximum-likelihood estimates of the mixture proportions by running 200 iterations of the standard EM updates. Note that the E and M steps are very simple, and easy to implement in R; in particular, in function mixem.update, the E step is implemented in 2 lines of R code, and the M step requires only one more line of code.

out <- system.time(fit1 <- mixem(L,x0,numiter = 200))
f1  <- mixobjective(L,fit1$x)
cat(sprintf("Computation took %0.2f seconds.\n",out["elapsed"]))
cat(sprintf("Log-likelihood at EM estimate is %0.12f.\n",f1))
# Computation took 11.01 seconds.
# Log-likelihood at EM estimate is -59912.068371303445.

Run accelerated EM

Re-run the EM updates, this time using DAAREM to accelerate convergence toward the solution.

out <- system.time(fit2 <- mixdaarem(L,x0,numiter = 200,order = 4))
f2  <- mixobjective(L,fit2$x)
cat(sprintf("Computation took %0.2f seconds.\n",out["elapsed"]))
cat(sprintf("Objective value at DAAREM estimate is %0.12f.\n",f2))
# Computation took 6.58 seconds.
# Objective value at DAAREM estimate is -59895.960281994718.

Observe that the this second estimate has a much higher likelihood.

Plot improvement in solution over time

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-likelihood obtained, and the log-likelihood at the “gold-standard” solution. The gold-standard solution was computed using mixsqp.

f    <- mixobjective(L,x)
pdat <-
  rbind(data.frame(iter = 1:200,dist = f - fit1$value,method = "EM"),
        data.frame(iter = 1:200,dist = f - fit2$value,method = "DAAREM"))
p <- ggplot(pdat,aes(x = iter,y = dist,col = method)) +
  geom_line(size = 1) +
  scale_y_continuous(trans = "log10",breaks = 10^seq(-4,4)) +
  scale_color_manual(values = c("darkorange","dodgerblue")) +
  labs(x = "iteration",y = "distance from solution")
print(p)

Version	Author	Date
52e323d	Peter Carbonetto	2019-06-04
936cdbb	Peter Carbonetto	2019-06-04

From this plot, we see that the accelerated EM methods gets much closer to the solution, although it seems to “plateau” after about 100 iterations. Nonetheless, it is much improved over the basic EM algorithm.