Function Maximization

Samuel Wilson


Simple Example

Bayesian Optimization seek the global maximum of any user defined function. As a simple example, let’s define a simple function:

simpleFunction <- function(x) dnorm(x,3,2)*1.5 + dnorm(x,7,1) + dnorm(x,10,2)
maximized <- optim(8,simpleFunction,method = "L-BFGS-B",lower = 0, upper = 15,control = list(fnscale = -1))$par
ggplot(data = data.frame(x=c(0,15)),aes(x=x)) + 
  stat_function(fun = simpleFunction) +
  geom_vline(xintercept = maximized,linetype="dashed")

We can see that this function is maximized around x~7.023. We can use bayesOpt to find the global maximum of this function. We just need to define the bounds, and the initial parameters we want to sample:

bounds <- list(x=c(0,15))
initGrid <- data.frame(x=c(0,5,10))

Here, we run bayesOpt. The function begins by running simpleFunction 3 times, and then fits a Gaussian process to the results in a process called Kriging. We then calculate the x which maximizes our expected improvement, and run simpleFunction at this x. We then go through 1 more iteration of this:

FUN <- function(x) list(Score = simpleFunction(x))
optObj <- bayesOpt(
  , bounds = bounds
  , initGrid = initGrid
  , acq = "ei"
  , iters.n = 2
  , gsPoints = 25

Let’s see how close the algorithm got to the global maximum:

#> $x
#> [1] 6.440633

The process is getting pretty close! We were only about 12% shy of the global optimum:

#> [1] 1.120063

Let’s run the process for a little longer:

optObj <- addIterations(optObj,iters.n=2,verbose=0)
#> [1] 1.002453

We have now found an x very close to the global optimum.