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Parallelizable Bayesian Optimization

This README contains a thorough walkthrough of Bayesian optimization and the syntax needed to use this package, with simple and complex examples. More information can be found in the package vignettes and manual.

Installation

You can install the most recent stable version of ParBayesianOptimization from CRAN with:

install.packages("ParBayesianOptimization")

You can also install the most recent development version from github using devtools:

# install.packages("devtools")
devtools::install_github("AnotherSamWilson/ParBayesianOptimization")

Package Process

Machine learning projects will commonly require a user to “tune” a model’s hyperparameters to find a good balance between bias and variance. Several tools are available in a data scientist’s toolbox to handle this task, the most blunt of which is a grid search. A grid search gauges the model performance over a pre-defined set of hyperparameters without regard for past performance. As models increase in complexity and training time, grid searches become unwieldly.

Idealy, we would use the information from prior model evaluations to guide us in our future parameter searches. This is precisely the idea behind Bayesian Optimization, in which our prior response distribution is iteratively updated based on our best guess of where the best parameters are. The ParBayesianOptimization package does exactly this in the following process:

Initial parameters are scored
Gaussian Process is fit/updated
Parameter is found which maximizes an acquisition function
This parameter is scored
Repeat steps 2-4 until some stopping criteria is met

Bayesian Optimization Intuition

As an example, let’s say we are only tuning 1 hyperparameter in an random forest model, the number of trees, within the bounds [1,15000]. We have initialized the process by randomly sampling the scoring function 7 times, and get the following results:

Trees.In.Forest	Score
1000	0.30
3000	0.31
5000	0.14
9000	0.40
11000	0.40
15000	0.16

In this example, Score can be generalized to any error metric that we want to maximize (negative RMSE, AUC, etc.). Keep in mind, Bayesian optimization can be used to maximize any black box function, hyperparameter tuning is just a common use case. Given these scores, how do we go about determining the best number of trees to try next? As it turns out, Gaussian processes can give us a very good definition of our assumption about how the Score (model performance) is distributed over the hyperparameters. Fitting a Gaussian process to the data above, we can see the expected value of Score across our parameter bounds, as well as the uncertainty bands:

Before we can select our next candidate parameter to run the scoring function on, we need to determine how we define a “good” parameter inside this prior distribution. This is done by maximizing different acquisition functions within the Gaussian process. The acquisition function tells is how much utility there is at a certain unexplored space. In the chart above, the lower 3 graphs show examples different acquisition functions.

Our expected improvement in the graph above is maximized at \~10000. If we run our process with the new Trees in Forest = 10000, we can update our Gaussian process for a new prediction about which would be best to sample next.

The utility functions that are maximized in this package are defined as follows:

Simple Example

In this example, we are optimizing a simple function with 1 input and 1 output. We, the user, need to define the function that we want to optimize. This function should return, at a minimum, a list with a Score element. You can also return other elements that you want to keep track of in each run of the scoring function, which we show in the section Hyperparameter Tuning.

simpleFunction <- function(x) dnorm(x,3,2)*1.5 + dnorm(x,7,1) + dnorm(x,10,2)

# Find the x that maximizes our simpleFunction
xmax <- optim(8,simpleFunction,method = "L-BFGS-B",lower = 0, upper = 15,control = list(fnscale = -1))$par

# Get a visual
library(ggplot2)
ggplot(data = data.frame(x=c(0,15)),aes(x=x)) + 
  stat_function(fun = simpleFunction) +
  geom_vline(xintercept = xmax,linetype="dashed") +
  ggtitle("simpleFunction") +
  theme_bw()

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:

library(ParBayesianOptimization)

FUN <- function(x) list(Score = simpleFunction(x))

set.seed(6)
optObjSimp <- bayesOpt(
  FUN = FUN
  , bounds = bounds
  , initGrid = initGrid
  , iters.n = 2
)

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

getBestPars(optObjSimp)
#> $x
#> [1] 6.718184

The process is getting pretty close! We were only about 3% shy of the global optimum:

simpleFunction(getBestPars(optObjSimp)$x)/simpleFunction(7.023)
#> [1] 0.968611

Let’s run the process for a little longer:

optObjSimp <- addIterations(optObjSimp,iters.n=3,verbose=0)
simpleFunction(getBestPars(optObjSimp)$x)/simpleFunction(7.023)
#> [1] 0.9958626

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

Hyperparameter Tuning

In this example, we will be using the agaricus.train dataset provided in the XGBoost package. Here, we load the packages, data, and create a folds object to be used in the scoring function.

library("xgboost")

data(agaricus.train, package = "xgboost")

Folds <- list(
    Fold1 = as.integer(seq(1,nrow(agaricus.train$data),by = 3))
  , Fold2 = as.integer(seq(2,nrow(agaricus.train$data),by = 3))
  , Fold3 = as.integer(seq(3,nrow(agaricus.train$data),by = 3))
)

Now we need to define the scoring function. This function should, at a minimum, return a list with a Score element, which is the model evaluation metric we want to maximize. We can also retain other pieces of information created by the scoring function by including them as named elements of the returned list. In this case, we want to retain the optimal number of rounds determined by the xgb.cv:

scoringFunction <- function(max_depth, min_child_weight, subsample) {

  dtrain <- xgb.DMatrix(agaricus.train$data,label = agaricus.train$label)

  Pars <- list( 
      booster = "gbtree"
    , eta = 0.001
    , max_depth = max_depth
    , min_child_weight = min_child_weight
    , subsample = subsample
    , objective = "binary:logistic"
    , eval_metric = "auc"
  )

  xgbcv <- xgb.cv(
      params = Pars
    , data = dtrain
    , nround = 100
    , folds = Folds
    , early_stopping_rounds = 5
    , maximize = TRUE
    , verbose = 0
  )

  return(list(Score = max(xgbcv$evaluation_log$test_auc_mean)
             , nrounds = xgbcv$best_iteration
             )
         )
}

We also need to tell our process the bounds it is allowed to search within:

bounds <- list( 
    max_depth = c(1L, 5L)
  , min_child_weight = c(0, 25)
  , subsample = c(0.25, 1)
)

We are now ready to put this all into the bayesOpt function.

set.seed(0)

tNoPar <- system.time(
  optObj <- bayesOpt(
      FUN = scoringFunction
    , bounds = bounds
    , initPoints = 4
    , iters.n = 4
    , iters.k = 1
  )
)

The console informs us that the process initialized by running scoringFunction 4 times. It then fit a Gaussian process to the parameter-score pairs, found the global optimum of the acquisition function, and ran scoringFunction again. This process continued until we had 6 parameter-score pairs. You can interrogate the optObj object to see the results:

optObj$scoreSummary
#>    Epoch Iteration max_depth min_child_weight subsample gpUtility acqOptimum inBounds Elapsed     Score nrounds errorMessage
#> 1:     0         1         2         1.670129 0.7880670        NA      FALSE     TRUE    0.14 0.9777163       2           NA
#> 2:     0         2         2        14.913213 0.8763154        NA      FALSE     TRUE    0.33 0.9763760      15           NA
#> 3:     0         3         4        18.833690 0.3403900        NA      FALSE     TRUE    0.43 0.9931657      18           NA
#> 4:     0         4         4         8.639925 0.5499186        NA      FALSE     TRUE    0.23 0.9981437       7           NA
#> 5:     1         5         4        21.871937 1.0000000 0.5857961       TRUE     TRUE    0.12 0.9945933       1           NA
#> 6:     2         6         4         0.000000 0.9439879 0.6668303       TRUE     TRUE    0.25 0.9990567       7           NA
#> 7:     3         7         5         1.395119 0.7071802 0.2973497       TRUE     TRUE    0.18 0.9984577       4           NA
#> 8:     4         8         5         0.000000 0.2500000 0.3221660       TRUE     TRUE    0.32 0.9994020      10           NA

getBestPars(optObj)
#> $max_depth
#> [1] 5
#> 
#> $min_child_weight
#> [1] 0
#> 
#> $subsample
#> [1] 0.25

Running In Parallel

The process that the package uses to run in parallel is explained above. Actually setting the process up to run in parallel is relatively simple, we just need to take two extra steps. We need to load any packages and objects required by FUN into the back ends, after registering our cluster:

library(doParallel)
cl <- makeCluster(2)
registerDoParallel(cl)
clusterExport(cl,c('Folds','agaricus.train'))
clusterEvalQ(cl,expr= {
  library(xgboost)
})

We can now run our process in paralel! Make sure you set iters.k to some sensible value to take advantage of the parallelization setup. Since we have registered 2 cores, we set iters.k to 2:

tWithPar <- system.time(
  optObj <- bayesOpt(
      FUN = scoringFunction
    , bounds = bounds
    , initPoints = 4
    , iters.n = 4
    , iters.k = 2
    , parallel = TRUE
  )
)
stopCluster(cl)
registerDoSEQ()

We managed to massively cut the process time by running the process on 2 cores in parallel. However, keep in mind we only performed 2 optimization steps, versus the 4 performed in the sequential example:

tWithPar
#>    user  system elapsed 
#>    0.99    0.03    7.91
tNoPar
#>    user  system elapsed 
#>   24.13    2.40   21.70

Sampling Multiple Promising Points at Once

Sometimes we may want to sample multiple promising points at the same optimization step (Epoch). This is especially effective if the process is being run in parallel. The bayesOpt function always samples the global optimum of the acquisition function, however it is also possible to tell it to sample local optimums of the acquisition function at the same time.

Using the acqThresh parameter, you can specify the minimum percentage utility of the global optimum required for a different local optimum to be considered. As an example, let’s say we are optimizing 1 input x, which is bounded between [0,1]. Our acquisition function may look like the following:

In this case, there are 3 promising candidate parameters: x \~ [0.318,0.541,0.782] with corresponding upper confidence bounds of y \~ [1.195,1.304,1.029], respectively. We may want to run our scoring function on several of the local maximums. If acqThresh is set to be below 1.029/1.304 \~ 0.789 and iters.k is set to at least 3, the process would use all 3 of the local maximums as candidate parameter sets in the next round of scoring function runs.

How Long Should it Run For?

Going back to the example in Simple Example, (if you let this run for a few more iterations and set plotProgress = TRUE) you will notice this chart is updated at each iteration:

optObjSimp <- addIterations(optObjSimp,2,verbose=FALSE)
plot(optObjSimp)

As you thoroughly explore the parameter space, you reduce the uncertainty in the unexplored areas. As you reduce uncertainty, you tend to reduce utility, which can be thought of as the potential to find a better parameter set than the one you already have. Notice that the expected improvement converged to 0 after iteration 5. If you see a similar pattern, you can be fairly certain that you have found an (approximately) global optimum.

Setting Time Limits and Other Halting Criteria

Many times the scoring function can vary in its completion time. It may be difficult for the user to forecast how long a single run will take, let alone X sequential runs. For this reason, you can set a time limit. You can also set a minimum utility limit, or you can set both, in which case the process stops when either condition is met. You can see how the process stopped by viewing the stopStatus element in the returned object:

set.seed(0)

tNoPar <- system.time(
  optObj <- bayesOpt(
      FUN = scoringFunction
    , bounds = bounds
    , initPoints = 4
    , iters.n = 400
    , iters.k = 1
    , otherHalting = list(timeLimit = 5)
  )
)

optObj$stopStatus
#> [1] "Time Limit - 5 seconds."
#> attr(,"class")
#> [1] "stopEarlyMsg"

AnotherSamWilson / ParBayesianOptimization

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