## Tuning Parameters

Some statistical and machine learning models contain tuning parameters (also known as hyperparameters), which are parameters that cannot be directly estimated by the model. An example would be the number of neighbors in a K-nearest neighbors model. To determine reasonable values of these elements, some indirect method is used such as resampling or profile likelihood. Search methods, such as genetic algorithms or Bayesian search can also be used to determine good values.

In any case, some information is needed to create a grid or to validate whether a candidate value is appropriate (e.g. the number of neighbors should be a positive integer). dials is designed to:

• Create an easy to use framework for describing and querying tuning parameters. This can include getting sequences or random tuning values, validating current values, transforming parameters, and other tasks.
• Standardize the names of different parameters. Different packages in R use different argument names for the same quantities. dials proposes some standardized names so that the user doesn’t need to memorize the syntactical minutiae of every package.
• Work with the other tidymodels packages for modeling and machine learning using tidyverse principles.

## Parameter Objects

Parameter objects contain information about possible values, ranges, types, and other aspects. They have two classes: the general param class and a more specific subclass related to the type of variable. Double and integer valued data have the subclass quant_param while character and logicals have qual_param. There are some common elements for each:

• Labels are strings that describe the parameter (e.g. “Number of Components”).
• Defaults are optional single values that can be set when one non-random value is requested.

Otherwise, the information contained in parameter objects are different for different data types.

### Numeric Parameters

An example of a numeric tuning parameter is the cost-complexity parameter of CART trees, otherwise known as $$C_p$$. A parameter object for $$C_p$$ can be created in dials using:

library(dials)
cost_complexity()
#> Cost-Complexity Parameter (quantitative)
#> Transformer: log-10 [1e-100, Inf]
#> Range (transformed scale): [-10, -1]

Note that this parameter is handled in log units and the default range of values is between 10^-10 and 0.1. The range of possible values can be returned and changed based on some utility functions. We’ll use the pipe operator here:

library(dplyr)
cost_complexity() %>% range_get()
#> $lower #> [1] 1e-10 #> #>$upper
#> [1] 0.1
cost_complexity() %>% range_set(c(-5, 1))
#> Cost-Complexity Parameter (quantitative)
#> Transformer: log-10 [1e-100, Inf]
#> Range (transformed scale): [-5, 1]

# Or using the range argument
# during creation
cost_complexity(range = c(-5, 1))
#> Cost-Complexity Parameter (quantitative)
#> Transformer: log-10 [1e-100, Inf]
#> Range (transformed scale): [-5, 1]

Values for this parameter can be obtained in a few different ways. To get a sequence of values that span the range:

# Natural units:
cost_complexity() %>% value_seq(n = 4)
#> [1] 1e-10 1e-07 1e-04 1e-01

# Stay in the transformed space:
cost_complexity() %>% value_seq(n = 4, original = FALSE)
#> [1] -10  -7  -4  -1

Random values can be sampled too. A random uniform distribution is used (between the range values). Since this parameter has a transformation associated with it, the values are simulated in the transformed scale and then returned in the natural units (although the original argument can be used here):

set.seed(5473)
cost_complexity() %>% value_sample(n = 4)
#> [1] 6.91e-09 8.46e-04 3.45e-06 5.90e-10

For CART trees, there is a discrete set of values that exist for a given data set. It may be a good idea to assign these possible values to the object. We can get them by fitting an initial rpart model and then adding the values to the object. For mtcars, there are only three values:

library(rpart)
cart_mod <- rpart(mpg ~ ., data = mtcars, control = rpart.control(cp = 0.000001))
cart_mod$cptable #> CP nsplit rel error xerror xstd #> 1 0.643125 0 1.000 1.064 0.258 #> 2 0.097484 1 0.357 0.687 0.180 #> 3 0.000001 2 0.259 0.576 0.126 cp_vals <- cart_mod$cptable[, "CP"]

# We should only keep values associated with at least one split:
cp_vals <- cp_vals[ cart_mod\$cptable[, "nsplit"] > 0 ]

# Here the specific Cp values, on their natural scale, are added:
mtcars_cp <- cost_complexity() %>% value_set(cp_vals)
#> Error in new_quant_param():
#> ! Some values are not valid: 0.09748...

The error occurs because the values are not in the transformed scale:

mtcars_cp <- cost_complexity() %>% value_set(log10(cp_vals))
mtcars_cp
#> Cost-Complexity Parameter (quantitative)
#> Transformer: log-10 [1e-100, Inf]
#> Range (transformed scale): [-10, -1]
#> Values: 2

Now, if a sequence or random sample is requested, it uses the set values:

mtcars_cp %>% value_seq(2)
#> [1] 0.097484 0.000001
# Sampling specific values is done with replacement
mtcars_cp %>%
value_sample(20) %>%
table()
#> .
#>              1e-06 0.0974840733898344
#>                  9                 11

Any transformations from the scales package can be used with the numeric parameters, or a custom transformation generated with scales::trans_new().

trans_raise <- scales::trans_new(
"raise",
transform = function(x) 2^x ,
inverse = function(x) -log2(x)
)
custom_cost <- cost(range = c(1, 10), trans = trans_raise)
custom_cost
#> Cost (quantitative)
#> Transformer: raise [-Inf, Inf]
#> Range (transformed scale): [1, 10]

Note that if a transformation is used, the range argument specifies the parameter range on the transformed scale. For this version of cost(), parameter values are sampled between 1 and 10 and then transformed back to the original scale by the inverse -log2(). So on the original scale, the sampled values are between -log2(10) and -log2(1).

-log2(c(10, 1))
#> [1] -3.32  0.00
value_sample(custom_cost, 100) %>% range()
#> [1] -3.3172 -0.0314

### Discrete Parameters

In the discrete case there is no notion of a range. The parameter objects are defined by their discrete values. For example, consider a parameter for the types of kernel functions that is used with distance functions:

weight_func()
#> Distance Weighting Function  (qualitative)
#> 'rectangular', 'triangular', 'epanechnikov', 'biweight', 'triweight', '...

The helper functions are analogues to the quantitative parameters:

# redefine values
weight_func() %>% value_set(c("rectangular", "triangular"))
#> Distance Weighting Function  (qualitative)
#> 'rectangular' and 'triangular'
weight_func() %>% value_sample(3)
#> [1] "triangular" "inv"        "triweight"

# the sequence is returned in the order of the levels
weight_func() %>% value_seq(3)
#> [1] "rectangular"  "triangular"   "epanechnikov"

## Creating Novel Parameters

The package contains two constructors that can be used to create new quantitative and qualitative parameters, new_quant_param() and new_qual_param(). The How to create a tuning parameter function article walks you through a detailed example.

## Unknown Values

There are some cases where the range of parameter values are data dependent. For example, the upper bound on the number of neighbors cannot be known if the number of data points in the training set is not known. For that reason, some parameters have an unknown placeholder:

mtry()
#> # Randomly Selected Predictors (quantitative)
#> Range: [1, ?]
sample_size()
#> # Observations Sampled (quantitative)
#> Range: [?, ?]
num_terms()
#> # Model Terms (quantitative)
#> Range: [1, ?]
num_comp()
#> # Components (quantitative)
#> Range: [1, ?]
# and so on

These values must be initialized prior to generating parameter values. The finalize() methods can be used to help remove the unknowns:

finalize(mtry(), x = mtcars[, -1])
#> # Randomly Selected Predictors (quantitative)
#> Range: [1, 10]

## Parameter Sets

These are collection of parameters used in a model, recipe, or other object. They can also be created manually and can have alternate identification fields:

glmnet_set <- parameters(list(lambda = penalty(), alpha = mixture()))
glmnet_set
#> Collection of 2 parameters for tuning
#>
#>  identifier    type    object
#>      lambda penalty nparam[+]
#>       alpha mixture nparam[+]

# can be updated too
update(glmnet_set, alpha = mixture(c(.3, .6)))
#> Collection of 2 parameters for tuning
#>
#>  identifier    type    object
#>      lambda penalty nparam[+]
#>       alpha mixture nparam[+]

These objects can be very helpful when creating tuning grids.

## Parameter Grids

Sets or combinations of parameters can be created for use in grid search. grid_regular(), grid_random(), grid_max_entropy(), and grid_latin_hypercube() take any number of param objects or a parameter set.

For example, for a glmnet model, a regular grid might be:

grid_regular(
mixture(),
penalty(),
levels = 3 # or c(3, 4), etc
)
#> # A tibble: 9 × 2
#>   mixture      penalty
#>     <dbl>        <dbl>
#> 1     0   0.0000000001
#> 2     0.5 0.0000000001
#> 3     1   0.0000000001
#> 4     0   0.00001
#> 5     0.5 0.00001
#> 6     1   0.00001
#> 7     0   1
#> 8     0.5 1
#> 9     1   1

and, similarly, a random grid is created using

set.seed(1041)
grid_random(
mixture(),
penalty(),
size = 6
)
#> # A tibble: 6 × 2
#>   mixture     penalty
#>     <dbl>       <dbl>
#> 1   0.200 0.0176
#> 2   0.750 0.000388
#> 3   0.191 0.000000159
#> 4   0.929 0.00000176
#> 5   0.143 0.0442
#> 6   0.973 0.0110