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.

- 01 - Installation
- 02 - Package Process
- 03 - Bayesian Optimization Intuition
- 04 - Simple Example
- 05 - Hyperparameter Tuning
- 06 - Running In Parallel
- 07 - Sampling Multiple Promising Points at Once
- 08 - How Long Should it Run For?
- 09 - Setting Stopping Criteria

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")
```

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

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

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:

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.

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
```

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
```

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.

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.

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"
```

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