# In Depth Tutorial

## Parameters (using paradox)

The `paradox` package offers a language for the description of parameter spaces, as well as tools for useful operations on these parameter spaces. A parameter space is often useful when describing:

- A set of sensible input values for an R function
- The set of possible values that slots of a configuration object can take
- The search space of an optimization process

The tools provided by `paradox` therefore relate to:

- Parameter checking: Verifying that a set of parameters satisfies the conditions of a parameter space
- Parameter sampling: Generating parameter values that lie in the parameter space for systematic exploration of program behavior depending on these parameters

`paradox` is, by nature, an auxiliary package that derives its usefulness from other packages that make use of it. It is heavily utilized in other mlr-org packages such as `mlr3`, `mlr3pipelines`, `mlr3tuning` and `miesmuschel`.

### Reference Based Objects

`paradox` is the spiritual successor to the `ParamHelpers` package and was written from scratch. The most important consequence of this is that some objects created in `paradox` are “reference-based”, unlike most other objects in R. When a change is made to a `ParamSet` object, for example by changing the `$values` field, all variables that point to this `ParamSet` will contain the changed object. To create an independent copy of a `ParamSet`, the `$clone(deep = TRUE)` method needs to be used:

    library("paradox")

    ps1 = ps(a = p_int(init = 1))
    ps2 = ps1
    ps3 = ps1$clone(deep = TRUE)
    print(ps1) # the same for ps2 and ps3
    #> <ParamSet(1)>
    #>        id    class lower upper nlevels        default  value
    #>    <char>   <char> <num> <num>   <num>         <list> <list>
    #> 1:      a ParamInt  -Inf   Inf     Inf <NoDefault[0]>      1

    ps1$values$a = 2

    print(ps1) # ps1 value of 'a' was changed
    #> <ParamSet(1)>
    #>        id    class lower upper nlevels        default  value
    #>    <char>   <char> <num> <num>   <num>         <list> <list>
    #> 1:      a ParamInt  -Inf   Inf     Inf <NoDefault[0]>      2

    print(ps2) # contains the same reference as ps1, so also changed
    #> <ParamSet(1)>
    #>        id    class lower upper nlevels        default  value
    #>    <char>   <char> <num> <num>   <num>         <list> <list>
    #> 1:      a ParamInt  -Inf   Inf     Inf <NoDefault[0]>      2

    print(ps3) # is a "clone" of the old ps1 with 'a' == 1
    #> <ParamSet(1)>
    #>        id    class lower upper nlevels        default  value
    #>    <char>   <char> <num> <num>   <num>         <list> <list>
    #> 1:      a ParamInt  -Inf   Inf     Inf <NoDefault[0]>      1

### Defining a Parameter Space

#### `Domain` Representing Single Parameters

Parameter spaces are made up of individual parameters, which usually can take a single atomic value. Consider, for example, trying to configure the `rpart` package’s `rpart.control` object. It has various components (`minsplit`, `cp`, …) that all take a single value.

These components are represented by `Domain` objects, which are constructed by calls of the form `p_xxx()`:

- `p_int()` for integer numbers
- `p_dbl()` for real numbers
- `p_fct()` for categorical values, similar to R `factor`s
- `p_lgl()` for truth values (`TRUE` / `FALSE`), as `logical`s in R
- `p_uty()` for parameters that can take any value

A `ParamSet` that represent a given set of parameters is created by calling `ps()` with named arguments that are `Domain` objects. While `Domain` themselves are R objects that can in principle be handled and manipulated, they should not be changed after construction.

    library("paradox")
    param_set = ps(
      parA = p_lgl(init = FALSE),
      parB = p_int(lower = 0, upper = 10, tags = c("tag1", "tag2")),
      parC = p_dbl(lower = 0, upper = 4, special_vals = list(NULL)),
      parD = p_fct(levels = c("x", "y", "z"), default = "y"),
      parE = p_uty(custom_check = function(x) checkmate::checkFunction(x))
    )
    param_set
    #> <ParamSet(5)>
    #>        id    class lower upper nlevels        default  value
    #>    <char>   <char> <num> <num>   <num>         <list> <list>
    #> 1:   parA ParamLgl    NA    NA       2 <NoDefault[0]>  FALSE
    #> 2:   parB ParamInt     0    10      11 <NoDefault[0]>
    #> 3:   parC ParamDbl     0     4     Inf <NoDefault[0]>
    #> 4:   parD ParamFct    NA    NA       3              y
    #> 5:   parE ParamUty    NA    NA     Inf <NoDefault[0]>

Every parameter can have:

- default - A default value, indicating the behaviour of something if the specific value is not given.
- init - An initial value, which is set in `$values` when the `ParamSet` is created. Note that this is not the same as `default`: `default` is used when a parameter is not present in `$values`, while `init` is the value that is set upon creation.
- special_vals - A list of values that are accepted even if they do not conform to the type
- tags - Tags that can be used to organize parameters
- trafo - A transformation function that is applied to the parameter value after it has been sampled. It is for example used through the `Design$transpose()` function after a `Design` was created by `generate_design_random()` or similar functions.

The numeric (`p_int()` and `p_dbl()`) parameters furthermore allow for specification of a lower and upper bound. Meanwhile, the `p_fct()` parameter must be given a vector of levels that define the possible states its parameter can take. The `p_uty` parameter can also have a `custom_check` function that must return `TRUE` when a value is acceptable and may return a `character(1)` error description otherwise. The example above defines `parE` as a parameter that only accepts functions.

All values which are given to the constructor are then accessible from the `ParamSet` for inspection using `$`. The `ParamSet` should be considered immutable, except for some fields such as `$values`, `$deps`, `$tags`. Bounds and levels should not be changed after construction. Instead, a new `ParamSet` should be constructed.

Besides the possible values that can be given to a constructor, there are also the `$class`, `$nlevels`, `$is_bounded`, `$has_default`, `$storage_type`, `$is_number` and `$is_categ` slots that give information about a parameter.

A list of all slots can be found in `?Param`.

    param_set$lower
    #> parA parB parC parD parE
    #>   NA    0    0   NA   NA

    param_set$parD$levels
    #> NULL

    param_set$class
    #>       parA       parB       parC       parD       parE
    #> "ParamLgl" "ParamInt" "ParamDbl" "ParamFct" "ParamUty"

It is also possible to get all information of a `ParamSet` as `data.table` by calling `as.data.table()`.

    as.data.table(param_set)
    #>        id    class lower upper      levels nlevels is_bounded special_vals
    #>    <char>   <char> <num> <num>      <list>   <num>     <lgcl>       <list>
    #> 1:   parA ParamLgl    NA    NA  TRUE,FALSE       2       TRUE    <list[0]>
    #> 2:   parB ParamInt     0    10                  11       TRUE    <list[0]>
    #> 3:   parC ParamDbl     0     4                 Inf       TRUE    <list[1]>
    #> 4:   parD ParamFct    NA    NA       x,y,z       3       TRUE    <list[0]>
    #> 5:   parE ParamUty    NA    NA                 Inf      FALSE    <list[0]>
    #>           default storage_type      tags
    #>            <list>       <char>    <list>
    #> 1: <NoDefault[0]>      logical
    #> 2: <NoDefault[0]>      integer tag1,tag2
    #> 3: <NoDefault[0]>      numeric
    #> 4:              y    character
    #> 5: <NoDefault[0]>         list

##### Type / Range Checking

The `ParamSet` object offers the possibility to check whether a value satisfies its condition, i.e. is of the right type, and also falls within the range of allowed values, using the `$test()`, `$check()`, and `$assert()` functions. Their argument must be a named list with values that are checked against the respective parameters, and it is possible to check only a subset of parameters. `test()` should be used within conditional checks and returns `TRUE` or `FALSE`, while `check()` returns an error description when a value does not conform to the parameter (and thus plays well with the `"checkmate::assert()"` function). `assert()` will throw an error whenever a value does not fit.

    param_set$test(list(parA = FALSE, parB = 0))
    #> [1] TRUE

    param_set$test(list(parA = "FALSE"))
    #> [1] FALSE

    param_set$check(list(parA = "FALSE"))
    #> [1] "parA: Must be of type 'logical flag', not 'character'"

#### Parameter Sets

The ordered collection of parameters is handled in a `ParamSet`. It is typically created by calling `ps()`, but can also be initialized using the `ParamSet$new()` function. The main difference is that `ps()` takes named arguments, whereas `ParamSet$new()` takes a named list. The latter makes it easier to construct a `ParamSet` programmatically, but is slightly more verbose.

`ParamSet`s can be combined using `c()` or `ps_union` (the latter of which takes a list), and they have a `$subset()` method that allows for subsetting. All of these functions return a new, cloned `ParamSet` object, and do not modify the original `ParamSet`.

    ps1 = ParamSet$new(list(x = p_int(), y = p_dbl()))
    ps2 = ParamSet$new(list(z = p_fct(levels = c("a", "b", "c"))))
    ps_all = c(ps1, ps2)
    print(ps_all)
    #> <ParamSet(3)>
    #>        id    class lower upper nlevels        default  value
    #>    <char>   <char> <num> <num>   <num>         <list> <list>
    #> 1:      x ParamInt  -Inf   Inf     Inf <NoDefault[0]>
    #> 2:      y ParamDbl  -Inf   Inf     Inf <NoDefault[0]>
    #> 3:      z ParamFct    NA    NA       3 <NoDefault[0]>

    ps_all$subset(c("x", "z"))
    #> <ParamSet(2)>
    #>        id    class lower upper nlevels        default  value
    #>    <char>   <char> <num> <num>   <num>         <list> <list>
    #> 1:      x ParamInt  -Inf   Inf     Inf <NoDefault[0]>
    #> 2:      z ParamFct    NA    NA       3 <NoDefault[0]>

`ParamSet`s of each individual parameters can be accessed through the `$subspaces()` function.

It is possible to get the `ParamSet` as a `data.table` using `as.data.table()`. This makes it easy to subset parameters on certain conditions and aggregate information about them, using the variety of methods provided by `data.table`.

    as.data.table(ps_all)
    #>        id    class lower upper levels nlevels is_bounded special_vals
    #>    <char>   <char> <num> <num> <list>   <num>     <lgcl>       <list>
    #> 1:      x ParamInt  -Inf   Inf            Inf      FALSE    <list[0]>
    #> 2:      y ParamDbl  -Inf   Inf            Inf      FALSE    <list[0]>
    #> 3:      z ParamFct    NA    NA  a,b,c       3       TRUE    <list[0]>
    #>           default storage_type   tags
    #>            <list>       <char> <list>
    #> 1: <NoDefault[0]>      integer
    #> 2: <NoDefault[0]>      numeric
    #> 3: <NoDefault[0]>    character

##### Values in a `ParamSet`

Although a `ParamSet` fundamentally represents a value space, it also has a slot `$values` that can contain a point within that space. This is useful because many things that define a parameter space need similar operations (like parameter checking) that can be simplified. The `$values` slot contains a named list that is always checked against parameter constraints. When trying to set parameter values, e.g. for `mlr3` `Learner`s, it is the `$values` slot of its `$param_set` that needs to be used.

    ps1$values = list(x = 1, y = 1.5)
    ps1$values$y = 2.5
    print(ps1$values)
    #> $x
    #> [1] 1
    #>
    #> $y
    #> [1] 2.5

The parameter constraints are automatically checked:

    ps1$values$x = 1.5
    #> Error in self$assert(xs, sanitize = TRUE): Assertion on 'xs' failed: x: Must be of type 'single integerish value', not 'double'.

##### Dependencies

It is often the case that certain parameters are irrelevant or should not be given depending on values of other parameters. An example would be a parameter that switches a certain algorithm feature (for example regularization) on or off, combined with another parameter that controls the behavior of that feature (e.g. a regularization parameter). The second parameter would be said to depend on the first parameter having the value `TRUE`.

A dependency can be added using the `$add_dep` method, which takes both the ids of the “depender” and “dependee” parameters as well as a `Condition` object. The `Condition` object represents the check to be performed on the “dependee”. Currently it can be created using `CondEqual()` and `CondAnyOf()`. Multiple dependencies can be added, and parameters that depend on others can again be depended on, as long as no cyclic dependencies are introduced.

The consequence of dependencies are twofold: For one, the `$check()`, `$test()` and `$assert()` tests will not accept the presence of a parameter if its dependency is not met, when the `check_strict` argument is given as `TRUE`. Furthermore, when sampling or creating grid designs from a `ParamSet`, the dependencies will be respected.

The easiest way to set dependencies is to give the `depends` argument to the `Domain` constructor.

The following example makes parameter `D` depend on parameter `A` being `FALSE`, and parameter `B` depend on parameter `D` being one of `"x"` or `"y"`. This introduces an implicit dependency of `B` on `A` being `FALSE` as well, because `D` does not take any value if `A` is `TRUE`.

    p = ps(
      A = p_lgl(init = FALSE),
      B = p_int(lower = 0, upper = 10, depends = D %in% c("x", "y")),
      C = p_dbl(lower = 0, upper = 4),
      D = p_fct(levels = c("x", "y", "z"), depends = A == FALSE)
    )

Note that the `depends` argument is limited to operators `==` and `%in%`, so `D = p_fct(..., depends = !A)` would not work.

    p$check(list(A = FALSE, D = "x", B = 1), check_strict = TRUE)  # OK: all dependencies met
    #> [1] TRUE

    p$check(list(A = FALSE, D = "z", B = 1), check_strict = TRUE)  # B's dependency is not met
    #> [1] "B: can only be set if the following condition is met 'D %in% {x, y}'. Instead the current parameter value is: D == z"

    p$check(list(A = FALSE, B = 1), check_strict = TRUE)           # B's dependency is not met
    #> [1] "B: can only be set if the following condition is met 'D %in% {x, y}'. Instead the parameter value for 'D' is not set at all. Try setting 'D' to a value that satisfies the condition"

    p$check(list(A = FALSE, D = "z"), check_strict = TRUE)         # OK: B is absent
    #> [1] TRUE

    p$check(list(A = TRUE), check_strict = TRUE)                   # OK: neither B nor D present
    #> [1] TRUE

    p$check(list(A = TRUE, D = "x", B = 1), check_strict = TRUE)   # D's dependency is not met
    #> [1] "D: can only be set if the following condition is met 'A == FALSE'. Instead the current parameter value is: A == TRUE"

    p$check(list(A = TRUE, B = 1), check_strict = TRUE)            # B's dependency is not met
    #> [1] "B: can only be set if the following condition is met 'D %in% {x, y}'. Instead the parameter value for 'D' is not set at all. Try setting 'D' to a value that satisfies the condition"

Internally, the dependencies are represented as a `data.table`, which can be accessed listed in the `$deps` slot. This `data.table` can even be mutated, to e.g. remove dependencies. There are no sanity checks done when the `$deps` slot is changed this way. Therefore it is advised to be cautious.

    p$deps
    #> Index: <id>
    #>        id     on                  cond
    #>    <char> <char>                <list>
    #> 1:      B      D <Condition:CondAnyOf>
    #> 2:      D      A <Condition:CondEqual>

#### Vector Parameters

Unlike in the old `ParamHelpers` package, there are no more vectorial parameters in `paradox`. Instead, it is now possible to create multiple copies of a single parameter using the `ps_replicate` function. This creates a `ParamSet` consisting of multiple copies of the parameter, which can then (optionally) be added to another `ParamSet`.

    ps2d = ps_replicate(ps(x = p_dbl(lower = 0, upper = 1)), 2)
    print(ps2d)
    #> <ParamSet(2)>
    #>        id    class lower upper nlevels        default  value
    #>    <char>   <char> <num> <num>   <num>         <list> <list>
    #> 1: rep1.x ParamDbl     0     1     Inf <NoDefault[0]>
    #> 2: rep2.x ParamDbl     0     1     Inf <NoDefault[0]>

It is also possible to use a `ParamUty` to accept vectorial parameters, which also works for parameters of variable length. A `ParamSet` containing a `ParamUty` can be used for parameter checking, but not for sampling. To sample values for a method that needs a vectorial parameter, it is advised to use an `$extra_trafo` transformation function that creates a vector from atomic values.

Assembling a vector from repeated parameters is aided by the parameter’s `$tags`: Parameters that were generated by the `pr_replicate()` command can be tagged as belonging to a group of repeated parameters.

    ps2d = ps_replicate(ps(x = p_dbl(0, 1), y = p_int(0, 10)), 2, tag_params = TRUE)
    ps2d$values = list(rep1.x = 0.2, rep2.x = 0.4, rep1.y = 3, rep2.y = 4)
    ps2d$tags
    #> $rep1.x
    #> [1] "param_x"
    #>
    #> $rep1.y
    #> [1] "param_y"
    #>
    #> $rep2.x
    #> [1] "param_x"
    #>
    #> $rep2.y
    #> [1] "param_y"

    ps2d$get_values(tags = "param_x")
    #> $rep1.x
    #> [1] 0.2
    #>
    #> $rep2.x
    #> [1] 0.4

### Parameter Sampling

It is often useful to have a list of possible parameter values that can be systematically iterated through, for example to find parameter values for which an algorithm performs particularly well (tuning). `paradox` offers a variety of functions that allow creating evenly-spaced parameter values in a “grid” design as well as random sampling. In the latter case, it is possible to influence the sampling distribution in more or less fine detail.

A point to always keep in mind while sampling is that only numerical and factorial parameters that are bounded can be sampled from, i.e. not `ParamUty`. Furthermore, for most samplers `p_int()` and `p_dbl()` must have finite lower and upper bounds.

#### Parameter Designs

Functions that sample the parameter space fundamentally return an object of the `Design` class. These objects contain the sampled data as a `data.table` under the `$data` slot, and also offer conversion to a list of parameter-values using the `$transpose()` function.

#### Grid Design

The `generate_design_grid()` function is used to create grid designs that contain all combinations of parameter values: All possible values for `ParamLgl` and `ParamFct`, and values with a given resolution for `ParamInt` and `ParamDbl`. The resolution can be given for all numeric parameters, or for specific named parameters through the `param_resolutions` parameter.

    ps_small = ps(A = p_dbl(0, 1), B = p_dbl(0, 1))
    design = generate_design_grid(ps_small, 2)
    print(design)
    #> <Design> with 4 rows:
    #>        A     B
    #>    <num> <num>
    #> 1:     0     0
    #> 2:     0     1
    #> 3:     1     0
    #> 4:     1     1

    generate_design_grid(ps_small, param_resolutions = c(A = 3, B = 2))
    #> <Design> with 6 rows:
    #>        A     B
    #>    <num> <num>
    #> 1:   0.0     0
    #> 2:   0.0     1
    #> 3:   0.5     0
    #> 4:   0.5     1
    #> 5:   1.0     0
    #> 6:   1.0     1

#### Random Sampling

`paradox` offers different methods for random sampling, which vary in the degree to which they can be configured. The easiest way to get a uniformly random sample of parameters is `generate_design_random()`. It is also possible to create “latin hypercube” sampled parameter values using `generate_design_lhs()`, which utilizes the `lhs` package. LHS-sampling creates low-discrepancy sampled values that cover the parameter space more evenly than purely random values. `generate_design_sobol()` can be used to sample using the Sobol sequence.

    pvrand = generate_design_random(ps_small, 500)
    pvlhs = generate_design_lhs(ps_small, 500)
    pvsobol = generate_design_sobol(ps_small, 500)

#### Generalized Sampling: The `Sampler` Class

It may sometimes be desirable to configure parameter sampling in more detail. `paradox` uses the `Sampler` abstract base class for sampling, which has many different sub-classes that can be parameterized and combined to control the sampling process. It is even possible to create further sub-classes of the `Sampler` class (or of any of its subclasses) for even more possibilities.

Every `Sampler` object has a `sample()` function, which takes one argument, the number of instances to sample, and returns a `Design` object.

##### 1D-Samplers

There is a variety of samplers that sample values for a single parameter. These are `Sampler1DUnif` (uniform sampling), `Sampler1DCateg` (sampling for categorical parameters), `Sampler1DNormal` (normally distributed sampling, truncated at parameter bounds), and `Sampler1DRfun` (arbitrary 1D sampling, given a random-function). These are initialized with a one-dimensional `ParamSet`, and can then be used to sample values.

    sampA = Sampler1DCateg$new(ps(x = p_fct(letters)))
    sampA$sample(5)
    #> <Design> with 5 rows:
    #>         x
    #>    <char>
    #> 1:      t
    #> 2:      q
    #> 3:      s
    #> 4:      r
    #> 5:      r

##### Hierarchical Sampler

The `SamplerHierarchical` sampler is an auxiliary sampler that combines many 1D-Samplers to get a combined distribution. Its name “hierarchical” implies that it is able to respect parameter dependencies. This suggests that parameters only get sampled when their dependencies are met.

The following example shows how this works: The `Int` parameter `B` depends on the `Lgl` parameter `A` being `TRUE`. `A` is sampled to be `TRUE` in about half the cases, in which case `B` takes a value between 0 and 10. In the cases where `A` is `FALSE`, `B` is set to `NA`.

    p = ps(
      A = p_lgl(),
      B = p_int(0, 10, depends = A == TRUE)
    )

    p_subspaces = p$subspaces()

    sampH = SamplerHierarchical$new(p,
      list(Sampler1DCateg$new(p_subspaces$A),
        Sampler1DUnif$new(p_subspaces$B))
    )
    sampled = sampH$sample(1000)
    head(sampled$data)
    #>         A     B
    #>    <lgcl> <int>
    #> 1:   TRUE     0
    #> 2:  FALSE    NA
    #> 3:   TRUE    10
    #> 4:   TRUE     1
    #> 5:  FALSE    NA
    #> 6:  FALSE    NA

    table(sampled$data[, c("A", "B")], useNA = "ifany")
    #>        B
    #> A         0   1   2   3   4   5   6   7   8   9  10 <NA>
    #>   FALSE   0   0   0   0   0   0   0   0   0   0   0  490
    #>   TRUE   47  55  49  51  42  44  48  43  49  50  32    0

##### Joint Sampler

Another way of combining samplers is the `SamplerJointIndep`. `SamplerJointIndep` also makes it possible to combine `Sampler`s that are not 1D. However, `SamplerJointIndep` currently can not handle `ParamSet`s with dependencies.

    sampJ = SamplerJointIndep$new(
      list(Sampler1DUnif$new(ps(x = p_dbl(0, 1))),
        Sampler1DUnif$new(ps(y = p_dbl(0, 1))))
    )
    sampJ$sample(5)
    #> <Design> with 5 rows:
    #>             x         y
    #>         <num>     <num>
    #> 1: 0.13875892 0.4809255
    #> 2: 0.10818250 0.3024228
    #> 3: 0.08687645 0.8289372
    #> 4: 0.48591974 0.8695142
    #> 5: 0.38264215 0.6523572

##### SamplerUnif

The `Sampler` used in `generate_design_random()` is the `SamplerUnif` sampler, which corresponds to a `HierarchicalSampler` of `Sampler1DUnif` for all parameters.

### Parameter Transformation

While the different `Sampler`s allow for a wide specification of parameter distributions, there are cases where the simplest way of getting a desired distribution is to sample parameters from a simple distribution (such as the uniform distribution) and then transform them. This can be done by constructing a `Domain` with a `trafo` argument, or assigning a function to the `$extra_trafo` field of a `ParamSet`. The latter can also be done by passing an `.extra_trafo` argument to the `ps()` shorthand constructor.

A `trafo` function in a `Domain` is called with a single parameter, the value to be transformed. It can only operate on the dimension of a single parameter.

The `$extra_trafo` function is called with two parameters:

- The list of parameter values to be transformed as `x`. Unlike the `Domain`’s `trafo`, the `$extra_trafo` handles the whole parameter set and can even model “interactions” between parameters.
- The `ParamSet` itself as `param_set`

The `$extra_trafo` function must return a list of transformed parameter values.

The transformation is performed when calling the `$transpose` function of the `Design` object returned by a `Sampler` with the `trafo` ParamSet to `TRUE` (the default). The following, for example, creates a parameter that is exponentially distributed:

    psexp = ps(par = p_dbl(0, 1, trafo = function(x) -log(x)))

    design = generate_design_random(psexp, 3)
    print(design)  # not transformed: between 0 and 1
    #> <Design> with 3 rows:
    #>            par
    #>          <num>
    #> 1: 0.027439239
    #> 2: 0.712310770
    #> 3: 0.002020327

    design$transpose()  # trafo is TRUE
    #> [[1]]
    #> [[1]]$par
    #> [1] 3.595781
    #>
    #>
    #> [[2]]
    #> [[2]]$par
    #> [1] 0.339241
    #>
    #>
    #> [[3]]
    #> [[3]]$par
    #> [1] 6.204496

Compare this to `$transpose()` without transformation:

    design$transpose(trafo = FALSE)
    #> [[1]]
    #> [[1]]$par
    #> [1] 0.02743924
    #>
    #>
    #> [[2]]
    #> [[2]]$par
    #> [1] 0.7123108
    #>
    #>
    #> [[3]]
    #> [[3]]$par
    #> [1] 0.002020327

Another way to get tihs effect, using `$extra_trafo`, would be:

    psexp = ps(par = p_dbl(0, 1))
    psexp$extra_trafo = function(x, param_set) {
      x$par = -log(x$par)
      x
    }

However, the `trafo` way is more recommended when transforming parameters independently. `$extra_trafo` is more useful when transforming parameters that interact in some way, or when new parameters should be generated.

#### Transformation between Types

Usually the design created with one `ParamSet` is then used to configure other objects that themselves have a `ParamSet` which defines the values they take. The `ParamSet`s which can be used for random sampling, however, are restricted in some ways: They must have finite bounds, and they may not contain “untyped” (`ParamUty`) parameters. `$trafo` provides the glue for these situations. There is relatively little constraint on the trafo function’s return value, so it is possible to return values that have different bounds or even types than the original `ParamSet`. It is even possible to remove some parameters and add new ones.

Suppose, for example, that a certain method requires a function as a parameter. Let’s say a function that summarizes its data in a certain way. The user can pass functions like `median()` or `mean()`, but could also pass quantiles or something completely different. This method would probably use the following `ParamSet`:

    methodPS = ps(fun = p_uty(custom_check = function(x) checkmate::checkFunction(x, nargs = 1)))

    print(methodPS)
    #> <ParamSet(1)>
    #>        id    class lower upper nlevels        default  value
    #>    <char>   <char> <num> <num>   <num>         <list> <list>
    #> 1:    fun ParamUty    NA    NA     Inf <NoDefault[0]>

If one wanted to sample this method, using one of four functions, a way to do this would be:

    samplingPS = ps(
      fun = p_fct(c("mean", "median", "min", "max"),
        trafo = function(x) get(x, mode = "function"))
    )

    design = generate_design_random(samplingPS, 2)
    print(design)
    #> <Design> with 2 rows:
    #>       fun
    #>    <char>
    #> 1:    max
    #> 2: median

Note that the `Design` only contains the column “`fun`” as a `character` column. To get a single value as a function, the `$transpose` function is used.

    xvals = design$transpose()
    print(xvals[[1]])
    #> $fun
    #> function (..., na.rm = FALSE)  .Primitive("max")

We can now check that it fits the requirements set by `methodPS`, and that `fun` it is in fact a function:

    methodPS$check(xvals[[1]])
    #> [1] TRUE

    xvals[[1]]$fun(1:10)
    #> [1] 10

`p_fct()` has a shortcut for this kind of transformation, where a `character` is transformed into a specific set of (typically non-scalar) values. When its `levels` argument is given as a named `list` (or named non-`character` vector), it constructs a `Domain` that does the trafo automatically. A way to perform the above would therefore be:

    samplingPS = ps(
      fun = p_fct(list("mean" = mean, "median" = median, "min" = min, "max" = max))
    )

    generate_design_random(samplingPS, 1)$transpose()
    #> [[1]]
    #> [[1]]$fun
    #> function (x, na.rm = FALSE, ...)
    #> UseMethod("median")
    #> <bytecode: 0x5574b0c0c1c0>
    #> <environment: namespace:stats>

Imagine now that a different kind of parametrization of the function is desired: The user wants to give a function that selects a certain quantile, where the quantile is set by a parameter. In that case the `$transpose` function could generate a function in a different way.

For interpretability, the parameter should be called “`quantile`” before transformation, and the “`fun`” parameter is generated on the fly. We therefore use an `extra_trafo` here, given as a function to the `ps()` call.

    samplingPS2 = ps(quantile = p_dbl(0, 1),
      .extra_trafo = function(x, param_set) {
        # x$quantile is a `numeric(1)` between 0 and 1.
        # We want to turn it into a function!
        list(fun = function(input) quantile(input, x$quantile))
      }
    )

    design = generate_design_random(samplingPS2, 2)
    print(design)
    #> <Design> with 2 rows:
    #>     quantile
    #>        <num>
    #> 1: 0.6471182
    #> 2: 0.5637477

The `Design` now contains the column “`quantile`” that will be used by the `$transpose` function to create the `fun` parameter. We also check that it fits the requirement set by `methodPS`, and that it is a function.

    xvals = design$transpose()
    print(xvals[[1]])
    #> $fun
    #> function(input) quantile(input, x$quantile)
    #> <environment: 0x5574b4095300>

    methodPS$check(xvals[[1]])
    #> [1] TRUE

    xvals[[1]]$fun(1:10)
    #> 64.71182%
    #>  6.824064

### Defining a Tuning Spaces

When running an optimization, it is important to inform the tuning algorithm about what hyperparameters are valid. Here the names, types, and valid ranges of each hyperparameter are important. All this information is communicated with objects of the class `ParamSet`, which is defined in `paradox`.

Note, that `ParamSet` objects exist in two contexts. First, `ParamSet`-objects are used to define the space of valid parameter settings for a learner (and other objects). Second, they are used to define a search space for tuning. We are mainly interested in the latter. For example we can consider the `minsplit` parameter of the `mlr_learners_classif.rpart", "classif.rpart Learner`. The `ParamSet` associated with the learner has a lower but no upper bound. However, for tuning the value, a lower and upper bound must be given because tuning search spaces need to be bounded. For `Learner` or `PipeOp` objects, typically “unbounded” `ParamSet`s are used. Here, however, we will mainly focus on creating “bounded” `ParamSet`s that can be used for tuning.

#### Creating `ParamSet`s

An empty `"ParamSet` – not yet very useful – can be constructed using just the `"ps"` call:

    search_space = ps()
    print(search_space)
    #> <ParamSet(0)>
    #> Empty.

`ps` takes named `Domain` arguments that are turned into parameters. A possible search space for the `"classif.svm"` learner could for example be:

    search_space = ps(
      cost = p_dbl(lower = 0.1, upper = 10),
      kernel = p_fct(levels = c("polynomial", "radial"))
    )
    print(search_space)
    #> <ParamSet(2)>
    #>        id    class lower upper nlevels        default  value
    #>    <char>   <char> <num> <num>   <num>         <list> <list>
    #> 1:   cost ParamDbl   0.1    10     Inf <NoDefault[0]>
    #> 2: kernel ParamFct    NA    NA       2 <NoDefault[0]>

There are five domain constructors that produce a parameters when given to `ps`:

Constructor | Description | Is bounded?
--- | --- | ---
`p_dbl` | Real valued parameter (“double”) | When `upper` and `lower` are given
`p_int` | Integer parameter | When `upper` and `lower` are given
`p_fct` | Discrete valued parameter (“factor”) | Always
`p_lgl` | Logical / Boolean parameter | Always
`p_uty` | Untyped parameter | Never

These domain constructors each take some of the following arguments:

- `lower`, `upper`: lower and upper bound of numerical parameters (`p_dbl` and `p_int`). These need to be given to get bounded parameter spaces valid for tuning.
- `levels`: Allowed categorical values for `p_fct` parameters. Required argument for `p_fct`. See below for more details on this parameter.
- `trafo`: transformation function, see below.
- `depends`: dependencies, see below.
- `tags`: Further information about a parameter, used for example by the `hyperband` tuner.
- `init`: . Not used for tuning search spaces.
- `default`: Value corresponding to default behavior when the parameter is not given. Not used for tuning search spaces.
- `special_vals`: Valid values besides the normally accepted values for a parameter. Not used for tuning search spaces.
- `custom_check`: Function that checks whether a value given to `p_uty` is valid. Not used for tuning search spaces.

The `lower` and `upper` parameters are always in the first and second position respectively, except for `p_fct` where `levels` is in the first position. It is preferred to omit the labels (ex: `upper = 0.1` becomes just `0.1`). This way of defining a `ParamSet` is more concise than the equivalent definition above. Preferred:

    search_space = ps(cost = p_dbl(0.1, 10), kernel = p_fct(c("polynomial", "radial")))

#### Transformations (`trafo`)

We can use the `paradox` function `generate_design_grid` to look at the values that would be evaluated by grid search. (We are using `rbindlist()` here because the result of `$transpose()` is a list that is harder to read. If we didn’t use `$transpose()`, on the other hand, the transformations that we investigate here are not applied.) In `generate_design_grid(search_space, 3)`, `search_space` is the `ParamSet` argument and 3 is the specified resolution in the parameter space. The resolution for categorical parameters is ignored; these parameters always produce a grid over all of their valid levels. For numerical parameters the endpoints of the params are always included in the grid, so if there were 3 levels for the kernel instead of 2 there would be 9 rows, or if the resolution was 4 in this example there would be 8 rows in the resulting table.

    library("data.table")
    rbindlist(generate_design_grid(search_space, 3)$transpose())
    #>     cost     kernel
    #>    <num>     <char>
    #> 1:  0.10 polynomial
    #> 2:  0.10     radial
    #> 3:  5.05 polynomial
    #> 4:  5.05     radial
    #> 5: 10.00 polynomial
    #> 6: 10.00     radial

We notice that the `cost` parameter is taken on a linear scale. We assume, however, that the difference of cost between `0.1` and `1` should have a similar effect as the difference between `1` and `10`. Therefore it makes more sense to tune it on a logarithmic scale. This is done by using a transformation (`trafo`). This is a function that is applied to a parameter after it has been sampled by the tuner. We can tune `cost` on a logarithmic scale by sampling on the linear scale `[-1, 1]` and computing `10^x` from that value.

    search_space = ps(
      cost = p_dbl(-1, 1, trafo = function(x) 10^x),
      kernel = p_fct(c("polynomial", "radial"))
    )
    rbindlist(generate_design_grid(search_space, 3)$transpose())
    #>     cost     kernel
    #>    <num>     <char>
    #> 1:   0.1 polynomial
    #> 2:   0.1     radial
    #> 3:   1.0 polynomial
    #> 4:   1.0     radial
    #> 5:  10.0 polynomial
    #> 6:  10.0     radial

It is even possible to attach another transformation to the `ParamSet` as a whole that gets executed after individual parameter’s transformations were performed. It is given through the `.extra_trafo` argument and should be a function with parameters `x` and `param_set` that takes a list of parameter values in `x` and returns a modified list. This transformation can access all parameter values of an evaluation and modify them with interactions. It is even possible to add or remove parameters. (The following is a bit of a silly example.)

    search_space = ps(
      cost = p_dbl(-1, 1, trafo = function(x) 10^x),
      kernel = p_fct(c("polynomial", "radial")),
      .extra_trafo = function(x, param_set) {
        if (x$kernel == "polynomial") {
          x$cost = x$cost * 2
        }
        x
      }
    )
    rbindlist(generate_design_grid(search_space, 3)$transpose())
    #>     cost     kernel
    #>    <num>     <char>
    #> 1:   0.2 polynomial
    #> 2:   0.1     radial
    #> 3:   2.0 polynomial
    #> 4:   1.0     radial
    #> 5:  20.0 polynomial
    #> 6:  10.0     radial

The available types of search space parameters are limited: continuous, integer, discrete, and logical scalars. There are many machine learning algorithms, however, that take parameters of other types, for example vectors or functions. These can not be defined in a search space `ParamSet`, and they are often given as `p_uty()` in the `Learner`’s `ParamSet`. When trying to tune over these hyperparameters, it is necessary to perform a Transformation that changes the type of a parameter.

An example is the `class.weights` parameter of the Support Vector Machine (SVM), which takes a named vector of class weights with one entry for each target class. The trafo that would tune `class.weights` for the `tsk("spam")` dataset could be:

    search_space = ps(
      class.weights = p_dbl(0.1, 0.9, trafo = function(x) c(spam = x, nonspam = 1 - x))
    )
    generate_design_grid(search_space, 3)$transpose()
    #> [[1]]
    #> [[1]]$class.weights
    #>    spam nonspam
    #>     0.1     0.9
    #>
    #>
    #> [[2]]
    #> [[2]]$class.weights
    #>    spam nonspam
    #>     0.5     0.5
    #>
    #>
    #> [[3]]
    #> [[3]]$class.weights
    #>    spam nonspam
    #>     0.9     0.1

(We are omitting `rbindlist()` in this example because it breaks the vector valued return elements.)

### Automatic Factor Level Transformation

A common use-case is the necessity to specify a list of values that should all be tried (or sampled from). It may be the case that a hyperparameter accepts function objects as values and a certain list of functions should be tried. Or it may be that a choice of special numeric values should be tried. For this, the `p_fct` constructor’s `level` argument may be a value that is not a `character` vector, but something else. If, for example, only the values `0.1`, `3`, and `10` should be tried for the `cost` parameter, even when doing random search, then the following search space would achieve that:

    search_space = ps(
      cost = p_fct(c(0.1, 3, 10)),
      kernel = p_fct(c("polynomial", "radial"))
    )
    rbindlist(generate_design_grid(search_space, 3)$transpose())
    #>     cost     kernel
    #>    <num>     <char>
    #> 1:   0.1 polynomial
    #> 2:   0.1     radial
    #> 3:   3.0 polynomial
    #> 4:   3.0     radial
    #> 5:  10.0 polynomial
    #> 6:  10.0     radial

This is equivalent to the following:

    search_space = ps(
      cost = p_fct(c("0.1", "3", "10"),
        trafo = function(x) list(`0.1` = 0.1, `3` = 3, `10` = 10)[[x]]),
      kernel = p_fct(c("polynomial", "radial"))
    )
    rbindlist(generate_design_grid(search_space, 3)$transpose())
    #>     cost     kernel
    #>    <num>     <char>
    #> 1:   0.1 polynomial
    #> 2:   0.1     radial
    #> 3:   3.0 polynomial
    #> 4:   3.0     radial
    #> 5:  10.0 polynomial
    #> 6:  10.0     radial

Note: Though the resolution is 3 here, in this case it doesn’t matter because both `cost` and `kernel` are factors (the resolution for categorical variables is ignored, these parameters always produce a grid over all their valid levels).

This may seem silly, but makes sense when considering that factorial tuning parameters are always `character` values:

    search_space = ps(
      cost = p_fct(c(0.1, 3, 10)),
      kernel = p_fct(c("polynomial", "radial"))
    )
    typeof(search_space$params$cost$levels)
    #> [1] "NULL"

Be aware that this results in an “unordered” hyperparameter, however. Tuning algorithms that make use of ordering information of parameters, like genetic algorithms or model based optimization, will perform worse when this is done. For these algorithms, it may make more sense to define a `p_dbl` or `p_int` with a more fitting trafo.

The `class.weights` case from above can also be implemented like this, if there are only a few candidates of `class.weights` vectors that should be tried. Note that the `levels` argument of `p_fct` must be named if there is no easy way for `as.character()` to create names:

    search_space = ps(
      class.weights = p_fct(
        list(
          candidate_a = c(spam = 0.5, nonspam = 0.5),
          candidate_b = c(spam = 0.3, nonspam = 0.7)
        )
      )
    )
    generate_design_grid(search_space)$transpose()
    #> [[1]]
    #> [[1]]$class.weights
    #>    spam nonspam
    #>     0.5     0.5
    #>
    #>
    #> [[2]]
    #> [[2]]$class.weights
    #>    spam nonspam
    #>     0.3     0.7

#### Parameter Dependencies (`depends`)

Some parameters are only relevant when another parameter has a certain value, or one of several values. The Support Vector Machine (SVM), for example, has the `degree` parameter that is only valid when `kernel` is `"polynomial"`. This can be specified using the `depends` argument. It is an expression that must involve other parameters and be of the form `<param> == <scalar>`, `<param> %in% <vector>`, or multiple of these chained by `&&`. To tune the `degree` parameter, one would need to do the following:

    search_space = ps(
      cost = p_dbl(-1, 1, trafo = function(x) 10^x),
      kernel = p_fct(c("polynomial", "radial")),
      degree = p_int(1, 3, depends = kernel == "polynomial")
    )
    rbindlist(generate_design_grid(search_space, 3)$transpose(), fill = TRUE)
    #>      cost     kernel degree
    #>     <num>     <char>  <int>
    #>  1:   0.1 polynomial      1
    #>  2:   0.1 polynomial      2
    #>  3:   0.1 polynomial      3
    #>  4:   0.1     radial     NA
    #>  5:   1.0 polynomial      1
    #>  6:   1.0 polynomial      2
    #>  7:   1.0 polynomial      3
    #>  8:   1.0     radial     NA
    #>  9:  10.0 polynomial      1
    #> 10:  10.0 polynomial      2
    #> 11:  10.0 polynomial      3
    #> 12:  10.0     radial     NA

#### Creating Tuning ParamSets from other ParamSets

Having to define a tuning `ParamSet` for a `Learner` that already has parameter set information may seem unnecessarily tedious, and there is indeed a way to create tuning `ParamSet`s from a `Learner`’s `ParamSet`, making use of as much information as already available.

This is done by setting values of a `Learner`’s `ParamSet` to so-called `TuneToken`s, constructed with a `to_tune` call. This can be done in the same way that other hyperparameters are set to specific values. It can be understood as the hyperparameters being tagged for later tuning. The resulting `ParamSet` used for tuning can be retrieved using the `$search_space()` method.

    library("mlr3learners")
    learner = lrn("classif.svm")
    learner$param_set$values$kernel = "polynomial" # for example
    learner$param_set$values$degree = to_tune(lower = 1, upper = 3)

    print(learner$param_set$search_space())

    rbindlist(generate_design_grid(
      learner$param_set$search_space(), 3)$transpose()
    )

It is possible to omit `lower` here, because it can be inferred from the lower bound of the `degree` parameter itself. For other parameters, that are already bounded, it is possible to not give any bounds at all, because their ranges are already bounded. An example is the logical `shrinking` hyperparameter:

    learner$param_set$values$shrinking = to_tune()

    print(learner$param_set$search_space())

    rbindlist(generate_design_grid(
      learner$param_set$search_space(), 3)$transpose()
    )

`"to_tune"` can also be constructed with a `Domain` object, i.e. something constructed with a `p_***` call. This way it is possible to tune continuous parameters with discrete values, or to give trafos or dependencies. One could, for example, tune the `cost` as above on three given special values, and introduce a dependency of `shrinking` on it. Notice that a short form for `to_tune(<levels>)` is a short form of `to_tune(p_fct(<levels>))`. When introducing the dependency, we need to use the `degree` value from before the implicit trafo, which is the name or `as.character()` of the respective value, here `"val2"`!

    learner$param_set$values$type = "C-classification" # needs to be set because of a bug in paradox
    learner$param_set$values$cost = to_tune(c(val1 = 0.3, val2 = 0.7))
    learner$param_set$values$shrinking = to_tune(p_lgl(depends = cost == "val2"))

    print(learner$param_set$search_space())

    rbindlist(generate_design_grid(learner$param_set$search_space(), 3)$transpose(), fill = TRUE)

The `search_space()` picks up dependencies from the underlying `ParamSet` automatically. So if the `kernel` is tuned, then `degree` automatically gets the dependency on it, without us having to specify that. (Here we reset `cost` and `shrinking` to `NULL` for the sake of clarity of the generated output.)

    learner$param_set$values$cost = NULL
    learner$param_set$values$shrinking = NULL
    learner$param_set$values$kernel = to_tune(c("polynomial", "radial"))

    print(learner$param_set$search_space())

    rbindlist(generate_design_grid(learner$param_set$search_space(), 3)$transpose(), fill = TRUE)

It is even possible to define whole `ParamSet`s that get tuned over for a single parameter. This may be especially useful for vector hyperparameters that should be searched along multiple dimensions. This `ParamSet` must, however, have an `.extra_trafo` that returns a list with a single element, because it corresponds to a single hyperparameter that is being tuned. Suppose the `class.weights` hyperparameter should be tuned along two dimensions:

    learner$param_set$values$class.weights = to_tune(
      ps(spam = p_dbl(0.1, 0.9), nonspam = p_dbl(0.1, 0.9),
        .extra_trafo = function(x, param_set) list(c(spam = x$spam, nonspam = x$nonspam))
    ))
    head(generate_design_grid(learner$param_set$search_space(), 3)$transpose(), 3)