Author: Alex Zwanenburg
Date: 2026-04-22

# Learning algorithms and hyperparameter optimisation

#### Alex Zwanenburg

#### 2026-04-22

Learning algorithms create models that relate input data to the outcome. Development data is used to train models, after which they can be used to predict outcomes for new data. Familiar implements several commonly used algorithms. This vignette first describes the learners and their hyperparameters. Then, hyperparameter optimisation is described in more detail.

Overview of learners implemented in familiar. a These learners test multiple distributions or linking families and attempt to find the best option. b k-nearest neighbours learners allow for setting the distance metric using `*`. If omitted (e.g. `knn`), the kernel is either `euclidean` (numeric features) or `gower` (mixed features). c The SVM kernel is indicated by `*`. If omitted (e.g. `svm_c`), the radial basis function is used as kernel. d SVM type is indicated by `*`, and must be one of `c`, `nu`, or `epsilon`.
learner | tag | binomial | multinomial | continuous | survival |
--- | --- | --- | --- | --- | --- | ---
generalised linear models |  |  |  |  |  |
generala | `glm` | × | × | × | × |
logistic | `glm_logistic` | × |  |  |  |
cauchy | `glm_cauchy` | × |  |  |  |
complementary log-log | `glm_loglog` | × |  |  |  |
normal | `glm_probit` | × |  |  |  |
multinomial | `glm_multinomial` |  | × |  |  |
log-normal | `glm_log`, `glm_log_gaussian` |  |  | × |  |
normal (gaussian) | `glm_gaussian` |  |  | × |  |
inverse gaussian | `glm_inv_gaussian` |  |  | × |  |
log-poisson | `glm_log_poisson` |  |  | x |  |
poisson | `glm_poisson` |  |  | x |  |
cox | `cox` |  |  |  | × |
exponential | `survival_regr_exponential` |  |  |  | × |
gaussian | `survival_regr_gaussian` |  |  |  | × |
logistic | `survival_regr_logistic` |  |  |  | × |
log-logistic | `survival_regr_loglogistic` |  |  |  | × |
log-normal | `survival_regr_lognormal` |  |  |  | × |
survival regressiona | `survival_regr` |  |  |  | × |
weibull | `survival_regr_weibull` |  |  |  | × |
lasso regression models |  |  |  |  |  |
generala | `lasso` | × | × | × | × |
logistic | `lasso_binomial` | × |  |  |  |
multi-logistic | `lasso_multinomial` |  | × |  |  |
normal (gaussian) | `lasso_gaussian` |  |  | × |  |
poisson | `lasso_poisson` |  |  |  |  |
cox | `lasso_cox` |  |  |  | × |
ridge regression models |  |  |  |  |  |
generala | `ridge` | × | × | × | × |
logistic | `ridge_binomial` | × |  |  |  |
multi-logistic | `ridge_multinomial` |  | × |  |  |
normal (gaussian) | `ridge_gaussian` |  |  | × |  |
poisson | `ridge_poisson` |  |  |  |  |
cox | `ridge_cox` |  |  |  | × |
elastic net regression models |  |  |  |  |  |
generala | `elastic_net` | × | × | × | × |
logistic | `elastic_net_binomial` | × |  |  |  |
multi-logistic | `elastic_net_multinomial` |  | × |  |  |
normal (gaussian) | `elastic_net_gaussian` |  |  | × |  |
poisson | `elastic_net_poisson` |  |  |  |  |
cox | `elastic_net_cox` |  |  |  | × |
extreme gradient boosted linear models |  |  |  |  |  |
generala | `xgboost_lm` | × | × | × | × |
logistic | `xgboost_lm_logistic` | × | × |  |  |
gamma | `xgboost_lm_gamma` |  |  | × |  |
gaussian | `xgboost_lm_gaussian` |  |  | × |  |
poisson | `xgboost_lm_poisson` |  |  | × |  |
cox | `xgboost_lm_cox` |  |  |  | × |
random forests |  |  |  |  |  |
random forest (RFSRC) | `random_forest_rfsrc` | × | × | × | × |
random forest (ranger) | `random_forest_ranger` | × | × | × | × |
extreme gradient boosted trees |  |  |  |  |  |
generala | `xgboost_tree` | × | × | × | × |
logistic | `xgboost_tree_logistic` | × | × |  |  |
gamma | `xgboost_tree_gamma` |  |  | × |  |
gaussian | `xgboost_tree_gaussian` |  |  | × |  |
poisson | `xgboost_tree_poisson` |  |  | × |  |
cox | `xgboost_tree_cox` |  |  |  | × |
extreme gradient boosted DART trees |  |  |  |  |  |
generala | `xgboost_dart` | × | × | × | × |
logistic | `xgboost_dart_logistic` | × | × |  |  |
gamma | `xgboost_dart_gamma` |  |  | × |  |
gaussian | `xgboost_dart_gaussian` |  |  | × |  |
poisson | `xgboost_dart_poisson` |  |  | × |  |
cox | `xgboost_dart_cox` |  |  |  | × |
bayesian models |  |  |  |  |  |
naive bayes | `naive_bayes` | × | × |  |  |
nearest neighbour models |  |  |  |  |  |
k-nearest neighboursb | `k_nearest_neighbours_*`, `knn_*` | × | × | x |  |
support vector machines |  |  |  |  |  |
\(C\)-classificationc | `svm_c_*` | × | × |  |  |
\(\nu\)-classification/ regressionc | `svm_nu_*` | × | × | × |  |
\(\epsilon\)-regressionc | `svm_eps_*` |  |  | × |  |
linear kerneld | `svm_*_linear` | × | × | × |  |
polynomial kerneld | `svm_*_polynomial` | × | × | × |  |
radial kerneld | `svm_*_radial` | × | × | × |  |
sigmoid kerneld | `svm_*_sigmoid` | × | × | × |  |

## Configuration options

Learners, their hyperparameters, and parallelisation options can be specified using the tags or arguments below:

Configuration options for model development.
tag / argument | description | default
--- | --- | ---
`learner` | The desired learner. Multiple learners may be provided at the same time. This setting has no default and must be provided. | – (required)
`hyperparameter` | Each learner has one or more hyperparameters which can be specified. Sequential model-based optimisation is used to identify hyperparameter values from the data unless these are specifically specified here. See the section on hyperparameter optimisation for more details. | – (optional)
`parallel_model_development` | Enables parallel processing for model development. Ignored if `parallel = FALSE`. Overrides `parallel = TRUE`. | `TRUE`

## Generalised linear models

Generalised linear models (GLM) are easy to understand, use and share. In many situations, GLM may be preferred over more complex models as they are easier to report and validate. The most basic GLM is the linear model. The linear model for an outcome variable \(y_j\) and a single predictor variable \(x_j\) for sample \(j\) is:

\[y_j=\beta_0 + \beta_1 x_j + \epsilon_j\] Here, \(\beta_0\) and \(\beta_1\) are both model coefficients. \(\beta_0\) is also called the model intercept. \(\epsilon_j\) is an error term that describes the difference between the predicted and the observed outcome for sample \(j\). When a linear model is developed, the \(\beta\)-coefficients are fitted so that the mean-squared error over the sample population is minimised.

The above model is a univariate model as it includes only a single predictor. A multivariate linear model includes multiple predictors \(\mathbf{X_j}\) and is denoted as:

\[y_j= \mathbf{\beta}\mathbf{X_j} + \epsilon_j\]

\(\mathbf{\beta}\mathbf{X_j}\) is the linear predictor. The GLM generalises this linear predictor through a linking function \(g\) (Nelder and Wedderburn 1972):

\[y_j=g\left(\mathbf{\beta}\mathbf{X}\right) + \epsilon_j\]

For example, binomial outcomes are commonly modelled using logistic regression with the `logit` linking function. Multiple linking functions are available in familiar and are detailed below.

### Linear models for binomial outcomes

Linear models for binomial outcomes derive from the `stats` package which is part of the R core distribution (R Core Team 2024). Hyperparameters for these models include linkage functions and are shown in the table below.

parameter | tag | values | optimised | comments
--- | --- | --- | --- | ---
signature size | `sign_size` | \(\mathbb{Z} \in \left[1,n\right]\) | yes | –
linking function | `family` | `logistic`, `probit`, `loglog`, `cauchy` | `glm` only | The linking function is not optimised when it is specified, e.g. `glm_logistic`.
sample weighting | `sample_weighting` | `inverse_number_of_samples`, `effective_number_of_samples`, `none` | no | Sample weighting allows for mitigating class imbalances. The default is `inverse_number_of_samples`. Instances with the majority class receive less weight.
beta | `sample_weighting_beta` | \(\mathbb{Z} \in \left[-6,-1\right]\) | `effective_number_of_samples` only | Specifies the \(\beta\) parameter for effective number of samples weighting (Cui et al. 2019). It is expressed on the \(\log_{10}\) scale: \(\beta=1-10^\texttt{beta}\).

### Linear models for multinomial outcomes

The linear model for multinomial outcomes is implemented using `nnet::multinom` (Venables and Ripley 2002). Hyperparameters for this model are shown in the table below.

parameter | tag | values | optimised | comments
--- | --- | --- | --- | ---
signature size | `sign_size` | \(\mathbb{Z} \in \left[1,n\right]\) | yes | –
linking function | `family` | `multinomial` | no | There is only one linking function available.
sample weighting | `sample_weighting` | `inverse_number_of_samples`, `effective_number_of_samples`, `none` | no | Sample weighting allows for mitigating class imbalances. The default is `inverse_number_of_samples`. Instances with the majority class receive less weight.
beta | `sample_weighting_beta` | \(\mathbb{Z} \in \left[-6,-1\right]\) | `effective_number_of_samples` only | Specifies the \(\beta\) parameter for effective number of samples weighting (Cui et al. 2019). It is expressed on the \(\log_{10}\) scale: \(\beta=1-10^\texttt{beta}\).

### Linear models for continuous outcomes

Linear models for continuous outcomes derive from the `stats` package of the R core distribution (R Core Team 2024). Hyperparameters for these models include linkage functions and are shown in the table below.

parameter | tag | values | optimised | comments
--- | --- | --- | --- | ---
signature size | `sign_size` | \(\mathbb{Z} \in \left[1,n\right]\) | yes | –
linking function | `family` | `gaussian`, `log_gaussian`, `inv_gaussian`, `poisson`, `log_poisson` | `glm` only | The linking function is not optimised when it is specified, e.g. `glm_poisson`.

### Linear models for survival outcomes

Linear models for survival outcomes are divided into semi-parametric and parametric models. The semi-parametric Cox proportional hazards model (Cox 1972) is based on the `survival::coxph` function (Therneau and Grambsch 2000). Tied survival times are resolved using the default method by Efron (1977).

Various fully parametric models are also available, which differ in the assumed distribution of the outcome, i.e. the linking function. The parametric models are all based on `survival::survreg` (Therneau and Grambsch 2000).

Hyperparameters for semi-parametric and parametric survival models are shown in the table below.

parameter | tag | values | optimised | comments
--- | --- | --- | --- | ---
signature size | `sign_size` | \(\mathbb{Z} \in \left[1,n\right]\) | yes | –
linking function | `family` | `weibull`, `exponential`, `gaussian`, `logistic`, `lognormal`, `loglogistic` | `survival_regr` only | The linking function is not optimised when it is specified, e.g. `survival_regr_weibull`. The non-parametric `cox` learner does not have a linking function.

## Lasso, ridge and elastic net regression

Generalised linear models can be problematic as there is no inherent limit to model complexity, which can easily lead to overfitting. Penalised regression, or shrinkage, methods address this issue by penalising model complexity.

Three shrinkage methods are implemented in `familiar`, namely ridge regression, lasso and elastic net, which are all implemented using the `glmnet` package (Hastie et al. 2009; Simon et al. 2011). The set of hyperparameters is shown in the table below. The optimal `lambda` parameter is determined by cross-validation as part of the `glmnet::cv.glmnet` function, and is not directly determined using hyperparameter optimisation.

parameter | tag | values | optimised | comments
--- | --- | --- | --- | ---
signature size | `sign_size` | \(\mathbb{Z} \in \left[1,n\right]\) | yes | –
elastic net penalty | `alpha` | \(\mathbb{R} \in \left[0,1\right]\) | elastic net | This penalty is fixed for ridge regression (`alpha = 0`) and lasso (`alpha = 1`).
optimal lambda | `lambda_min` | `lambda.1se`, `lambda.min` | no | Default is `lambda.min`.
number of CV folds | `n_folds` | \(\mathbb{Z} \in \left[3,n\right]\) | no | Default is \(3\) if \(n<30\), \(\lfloor n/10\rfloor\) if \(30\leq n \leq 200\) and \(20\) if \(n>200\).
sample weighting | `sample_weighting` | `inverse_number_of_samples`, `effective_number_of_samples`, `none` | no | Sample weighting allows for mitigating class imbalances. The default is `inverse_number_of_samples` for `binomial` and `multinomial` outcomes, and `none` otherwise. Instances with the majority class receive less weight.
beta | `sample_weighting_beta` | \(\mathbb{Z} \in \left[-6,-1\right]\) | `effective_number_of_samples` only | Specifies the \(\beta\) parameter for effective number of samples weighting (Cui et al. 2019). It is expressed on the \(\log_{10}\) scale: \(\beta=1-10^\texttt{beta}\).
normalisation | `normalise` | `FALSE`, `TRUE` | no | Default is `FALSE`, as normalisation is part of pre-processing in familiar.

## Extreme gradient boosted linear models and trees

Boosting is a procedure which combines many weak learners to form a more powerful panel (Schapire 1990; Hastie et al. 2009). Extreme gradient boosting is a gradient boosting implementation that was highly successful in several machine learning competitions. Learners are implemented using the `xgboost` package (Chen and Guestrin 2016). Three types are implemented: regression based boosting (`xgboost_lm`), regression-tree based boosting (`xgboost_tree`) and regression-tree based boosting with drop-outs (`xgboost_dart`).

parameter | tag | values | optimised | comments
--- | --- | --- | --- | ---
signature size | `sign_size` | \(\mathbb{Z} \in \left[1,n\right]\) | yes | –
family | `learn_objective` | `gaussian`, `multinomial_logistic`, `binomial_logistic`, `poisson`, `gamma`, `cox` | general tags | The family is not optimised when it is specified, e.g. `xgboost_lm_linear`.
boosting iterations | `n_boost` | \(\mathbb{R} \in \left[0,\infty \right)\) | yes | This parameter is expressed on the \(\log_{10}\) scale, i.e. the actual value will be \(10^\texttt{n_boost}\). The default range is \(\left[0, 3 \right]\).
learning rate | `learning_rate` | \(\mathbb{R} \in \left(-\infty,0\right]\) | yes | This parameter is expressed on the \(\log_{10}\) scale. The default range is \(\left[-5, 0 \right]\).
L1 regularisation | `alpha` | \(\mathbb{R} \in \left[-6, \infty\right)\) | yes | This parameter is expressed on the \(\log_{10}\) scale with a \(10^{-6}\) offset. The default range is \(\left[-6, 3\right]\).
L2 regularisation | `lambda` | \(\mathbb{R} \in \left[-6, \infty\right)\) | yes | This parameter is expressed on the \(\log_{10}\) scale with a \(10^{-6}\) offset. The default range is \(\left[-6, 3\right]\).
sample weighting | `sample_weighting` | `inverse_number_of_samples`, `effective_number_of_samples`, `none` | no | Sample weighting allows for mitigating class imbalances. The default is `inverse_number_of_samples` for `binomial` and `multinomial` outcomes, and `none` otherwise. Instances with the majority class receive less weight.
beta | `sample_weighting_beta` | \(\mathbb{Z} \in \left[-6,-1\right]\) | `effective_number_of_samples` only | Specifies the \(\beta\) parameter for effective number of samples weighting (Cui et al. 2019). It is expressed on the \(\log_{10}\) scale: \(\beta=1-10^\texttt{beta}\).
maximum tree depth | `tree_depth` | \(\mathbb{Z} \in \left[1,\infty\right)\) | `gbtree`, `dart` | Maximum depth to which trees are allowed to grow. The default range is \(\left[1, 10\right]\).
subsampling fraction | `sample_size` | \(\mathbb{R} \in \left(0, 1.0\right]\) | `gbtree`, `dart` | Fraction of available data that is used for to create a single tree. The default range is \(\left[2 / m, 1.0\right]\), with \(m\) the number of samples.
minimum sum of instance weight | `min_child_weight` | \(\mathbb{R} \in \left[0,\infty\right)\) | `gbtree`, `dart` | Minimal instance weight required for further branch partitioning, or the number of instances required in each node. This parameter is expressed on the \(\log_{10}\) scale with a \(-1\) offset. The default range is \(\left[0, 2\right]\).
min. splitting error reduction | `gamma` | \(\mathbb{R} \in \left[-6, \infty\right)\) | `gbtree`, `dart` | Minimum error reduction required to allow splitting. This parameter is expressed on the \(\log_{10}\) scale with a \(10^{-6}\) offset. The default range is \(\left[-6, 3\right]\). `continuous`-type outcomes are normalised to the \(\left[0, 1\right]\) range prior to model fitting to deal with scaling issues. Values are converted back to the original scale after prediction.
DART sampling algorithm | `sample_type` | `uniform`, `weighted` | `dart` | –
drop-out rate | `rate_drop` | \(\mathbb{R} \in \left[0,1\right)\) | `dart` | –

## Random forest (RFSRC)

Random forests (explain) (Breiman 2001). The random forest learner is implemented through the `randomForestSRC` package, which provides a unified interface for different types of forests (Ishwaran et al. 2008, 2011).

Hyperparameters for random forest learners are shown in the table below.

parameter | tag | values | optimised | comments
--- | --- | --- | --- | ---
signature size | `sign_size` | \(\mathbb{Z} \in \left[1,n\right]\) | yes | –
number of trees | `n_tree` | \(\mathbb{Z} \in \left[0,\infty\right)\) | yes | This parameter is expressed on the \(\log_{2}\) scale, i.e. the actual input value will be \(2^\texttt{n_tree}\) (Oshiro et al. 2012). The default range is \(\left[4, 10\right]\).
subsampling fraction | `sample_size` | \(\mathbb{R} \in \left(0, 1.0\right]\) | yes | Fraction of available data that is used for to create a single tree. The default range is \(\left[2 / m, 1.0\right]\), with \(m\) the number of samples.
number of features at each node | `m_try` | \(\mathbb{R} \in \left[0.0, 1.0\right]\) | yes | Familiar ensures that there is always at least one candidate feature.
node size | `node_size` | \(\mathbb{Z} \in \left[1, \infty\right)\) | yes | Minimum number of unique samples in terminal nodes. The default range is \(\left[5, \lfloor m / 3\rfloor\right]\), with \(m\) the number of samples.
maximum tree depth | `tree_depth` | \(\mathbb{Z} \in \left[1,\infty\right)\) | yes | Maximum depth to which trees are allowed to grow. The default range is \(\left[1, 10\right]\).
number of split points | `n_split` | \(\mathbb{Z} \in \left[0, \infty\right)\) | no | By default, splitting is deterministic and has one split point (\(0\)).
splitting rule | `split_rule` | `gini`, `auc`, `entropy`, `mse`, `quantile.regr`, `la.quantile.regr`, `logrank`, `logrankscore`, `bs.gradient` | no | Default splitting rules are `gini` for `binomial` and `multinonial` outcomes, `mse` for `continuous` outcomes and `logrank` for `survival` outcomes.
sample weighting | `sample_weighting` | `inverse_number_of_samples`, `effective_number_of_samples`, `none` | no | Sample weighting allows for mitigating class imbalances. The default is `inverse_number_of_samples` for `binomial` and `multinomial` outcomes, and `none` otherwise. Instances with the majority class receive less weight.
beta | `sample_weighting_beta` | \(\mathbb{Z} \in \left[-6,-1\right]\) | `effective_number_of_samples` only | Specifies the \(\beta\) parameter for effective number of samples weighting (Cui et al. 2019). It is expressed on the \(\log_{10}\) scale: \(\beta=1-10^\texttt{beta}\).

Note that optimising that optimising the number of trees can be slow, as big forests take longer to construct and perform more computations for predictions. Hence hyperparameter optimisation may be sped up by limiting the range of the `n_tree` parameter, or setting a single value.

## Random forest (ranger)

The second implementation of random forests comes from the `ranger` package (Wright and Ziegler 2017). It is generally faster than the implementation in `randomForestSRC` and allows for different splitting rules, such as maximally selected rank statistics (Lausen and Schumacher 1992; Wright et al. 2017) and concordance index-based splitting (Schmid et al. 2016). Hyperparameters for the random forest are shown in the table below.

parameter | tag | values | optimised | comments
--- | --- | --- | --- | ---
signature size | `sign_size` | \(\mathbb{Z} \in \left[1,n\right]\) | yes | –
number of trees | `n_tree` | \(\mathbb{Z} \in \left[0,\infty\right)\) | yes | This parameter is expressed on the \(\log_{2}\) scale, i.e. the actual input value will be \(2^\texttt{n_tree}\) (Oshiro et al. 2012). The default range is \(\left[4, 10\right]\).
subsampling fraction | `sample_size` | \(\mathbb{R} \in \left(0, 1.0\right]\) | yes | Fraction of available data that is used for to create a single tree. The default range is \(\left[2 / m, 1.0\right]\), with \(m\) the number of samples.
number of features at each node | `m_try` | \(\mathbb{R} \in \left[0.0, 1.0\right]\) | yes | Familiar ensures that there is always at least one candidate feature.
node size | `node_size` | \(\mathbb{Z} \in \left[1, \infty\right)\) | yes | Minimum number of unique samples in terminal nodes. The default range is \(\left[5, \lfloor m / 3\rfloor\right]\), with \(m\) the number of samples.
maximum tree depth | `tree_depth` | \(\mathbb{Z} \in \left[1,\infty\right)\) | yes | Maximum depth to which trees are allowed to grow. The default range is \(\left[1, 10\right]\).
splitting rule | `split_rule` | `gini`, `extratrees`, `variance`, `logrank`, `C`, `maxstat` | no | Default splitting rules are `gini` for `binomial` and `multinomial` outcomes and `maxstat` for `continuous` and `survival` outcomes.
significance split threshold | `alpha` | \(\mathbb{R} \in \left(0.0, 1.0\right]\) | `maxstat` | Minimum significance level for further splitting. The default range is \(\left[10^{-6}, 1.0\right]\)
sample weighting | `sample_weighting` | `inverse_number_of_samples`, `effective_number_of_samples`, `none` | no | Sample weighting allows for mitigating class imbalances. The default is `inverse_number_of_samples` for `binomial` and `multinomial` outcomes, and `none` otherwise. Instances with the majority class receive less weight.
beta | `sample_weighting_beta` | \(\mathbb{Z} \in \left[-6,-1\right]\) | `effective_number_of_samples` only | Specifies the \(\beta\) parameter for effective number of samples weighting (Cui et al. 2019). It is expressed on the \(\log_{10}\) scale: \(\beta=1-10^\texttt{beta}\).

Note that optimising the number of trees can be slow, as big forests take longer to construct and perform more computations for predictions. Hence hyperparameter optimisation may be sped up by limiting the range of the `n_tree` parameter or setting a single value.

## Naive Bayes

The naive Bayes classifier uses Bayes rule to predict posterior probabilities. The naive Bayes classifier uses the `e1071::naiveBayes` function (Meyer et al. 2021). The hyperparameters of the classifier are shown in the table below.

parameter | tag | values | optimised | comments
--- | --- | --- | --- | ---
signature size | `sign_size` | \(\mathbb{Z} \in \left[1,n\right]\) | yes | –
laplace smoothing | `laplace` | \(\mathbb{R} \in \left[0.0, \infty \right)\) | yes | The default range is \(\left[0, 10\right]\).

## k-nearest neighbours

k-nearest neighbours is a simple clustering algorithm that classifies samples based on the classes of their k nearest neighbours in feature space. The `e1071::gknn` function is used to implement k-nearest neighbours (Meyer et al. 2021). The hyperparameters are shown in the table below.

parameter | tag | values | optimised | comments
--- | --- | --- | --- | ---
signature size | `sign_size` | \(\mathbb{Z} \in \left[1,n\right]\) | yes | –
number of nearest neighbours | `k` | \(\mathbb{Z} \in \left[1,m\right]\) | yes | The default range is \(\left[1, \lceil 2 m^{1/3} \rceil \right]\), with \(m\) the number of samples.
distance metric | `distance_metric` | all metrics supported by `proxy::dist` | yes | The default set of metrics is `gower`, `euclidean` and `manhattan`.

## Support vector machines

Support vector machines were originally defined to find optimal margins between classes for classification problems. Support vector machines were popularized after the use of the kernel trick was described. Using the kernel trick the calculations are performed in an implicit high-dimensional feature space, which is considerably more efficient than explicit calculations (Boser et al. 1992).

Familiar implements SVM using `e1071::svm`. By default, both features and outcome data are scaled internally. This was used to derive the default hyperparameter ranges specified in the table below.

parameter | tag | values | optimised | comments
--- | --- | --- | --- | ---
signature size | `sign_size` | \(\mathbb{Z} \in \left[1,n\right]\) | yes | –
SVM kernel | `kernel` | `linear`, `polynomial`, `radial`, `sigmoid` | no | Default is `radial`, unless specified as part of the learner’s name.
\(C\) | `c` | \(\mathbb{R}\) | yes | The cost for violating constraints is expressed on a \(\log_{10}\) scale, i.e. the actual value is \(10^\texttt{c}\). The default range is \(\left[-5, 3\right]\).
\(\epsilon\) | `epsilon` | \(\mathbb{R}\) | `continuous` outcomes | The error tolerance \(\epsilon\) is only used for regression SVM. \(\epsilon\) is expressed on a \(\log_{10}\) scale. The default range is \(\left[-5, 1\right]\).
\(\nu\) | `nu` | \(\mathbb{R}\) | `nu` SVM type | `nu` provides upper bounds for the fraction of training errors and lower bounds for the fraction of support vectors (Chang and Lin 2011). It is expressed on a \(\log_{10}\) scale. The default range is \(\left[-5, 1\right]\).
inverse kernel width | `gamma` | \(\mathbb{R}\) | non-linear kernels | This parameter specifies the inverse of the kernel width. It is expressed on the \(\log_{10}\) scale. The default range is \(\left[-9, 3\right]\).
polynomial degree | `degree` | \(\mathbb{Z} \in \left[1, \infty\right)\) | polynomial kernel | The default range is \(\left[1, 5\right]\).
kernel offset | `offset` | \(\mathbb{R} \in \left[0, \infty\right)\) | polynomial and sigmoid kernels | Negative values are not allowed. The default range is \(\left[0, 1 \right]\).

Several types of SVM algorithms exist for classification and regression. The following SVM types are implemented:

- \(C\)-classification: `c`
- \(\nu\)-classification: `nu`
- \(\nu\)-regression: `nu`
- \(\epsilon\)-regression: `eps`

The following kernels are implemented:

- Linear kernel: `linear`
- Polynomial kernel: `polynomial`
- Radial basis function kernel: `radial`
- Hyperbolic tangent (sigmoid) kernel: `sigmoid`

# Hyperparameter optimization

Hyperparameter optimisation is conducted to select model parameters lead to good, generalisable models. The main hyperparameter optimisation framework used by familiar is based on sequential model-based optimisation (SMBO) (Hutter et al. 2011), but with significant updates and extensions.

Overall, the following steps are conducted to optimise hyperparameters for a given learner and dataset:

- A set of bootstraps of the development dataset is made. This allows for training the models using the in-bag data and assessing performance using out-of-bag data. This is independent of any bootstraps that may have been defined as part of the experimental design. In case a subsampling method is specified in the experimental design, the corresponding internal development dataset is bootstrapped.

- Variable importance is determined for each of the in-bag datasets to avoid positively biasing the optimisation process through information leakage.

- An initial exploration of the hyperparameter space is conducted. A model is trained using each hyperparameter set and a bootstrap subsample. It is then assessed by computing an optimisation score based on model performance in the out-of-bag, and optionally, in-bag data.

- Exploration of the hyperparameter space then continues by iteratively comparing the best known hyperparameter set against challenger hyperparameter sets in so-called intensify steps. The challengers are selected by predicting the expected model performance for a new hyperparameter set and computing its utility (Shahriari et al. 2016). In addition, we predict model run times, and actively prune potential slow challengers. Utility is computed using an acquisition function, of which several are available in familiar. The set of challenger hyperparameter sets may moreover be iteratively pruned, dependent on the exploration method.

- Exploration stops if no significant improvement to model performance could be established by further exploring the hyperparameter space, the total number of iterations has been reached, or bootstraps samples have been exhausted.

- If the best set of hyperparameters does not lead to a model that exceeds performance of a naive model, e.g. one that always predicts the majority class or median value, a naive model is trained instead.

## Predicting run time of model

Models take a certain time to train. Familiar is actively measuring this time during hyperparameter optimisation for two reasons: first, to optimise assignment of jobs to parallel nodes; and secondly, to prune potential challenger sets that produce models that tend to run longer than the best-known model. The strictness increases with increased exploration.

Let \(m\) be the number of bootstraps used to quantify the most visited hyperparameter sets, and \(M\) the total number of bootstrap samples that can be visited. Then let \(t_{\texttt{opt}}\) be the runtime of the best known model. The maximum time that a challenger hyperparameter set is allowed for training is then empirically set to:

\[t_{\texttt{max}} = \left(5 - 4 \frac{m}{M} \right) t_{\texttt{opt}} \]

Hence, \(t_{\texttt{max}}\) converges to \(t_{\texttt{opt}}\) for \(m \rightarrow M\).

If maximum runtime is relatively insignificant, i.e. \(t_{\texttt{max}} < 10.0\) seconds, a threshold of \(t_{\texttt{max}}= 10.0\) is used.

If the maximum runtime is not known, i.e. none of the hyperparameter sets evaluated so far produced a valid model, the maximum time threshold is set to infinite. In effect, no hyperparameter sets are pruned based on expected run time.

A random forest is trained to infer runtimes for (new) hyperparameter sets, based on the runtimes observed for visited hyperparameter sets. The random forest subsequently infers runtime for a challenger hyperparameter set. The runtime estimate is compared against \(t_{\texttt{max}}\), and if it exceeds the threshold, it is rejected and not evaluated.

## Assessing goodness of hyperparameter sets

A summary score \(s\) determines how good a set of hyperparameters \(\mathbf{x}\) is. This score is computed in two steps. First an objective score is computed to assess model performance for a hyperparameter set in a bootstrap subsample. Subsequently an optimisation score is computed within the subsample. Finally a summary score is determined from the optimisation score over all subsamples.

An objective score \(s''\) is computed from the performance metric value for a specific hyperparameter set for in-bag (IB; \(s''_{\textrm{IB}}\)) and out-of-bag data (OOB; \(s''_{\textrm{OOB}}\)). The objective score always lies in the interval \([-1.0, 1.0]\). An objective score of \(1.0\) always indicates the best possible score. A score of \(0.0\) indicates that the hyperparameter set leads to the same degree of performance as the best educated guess, i.e. the majority class (for binomial and multinomial outcomes), the median value (for continuous outcomes), or tied risk or survival times (for survival outcomes). Objective scores that are either missing or below \(-1.0\) are truncated to the value of \(-1.0\). In case multiple metrics are used to assess model performance, the mean of the respective objective scores is used.

The optimisation scores \(s'\) and summary score \(s\) of each hyperparameter set are subsequently computed using one of the following functions, which can be set using the `optimisation_function` parameter:

1. `validation`, `max_validation` (default): The optimisation score for each bootstrap subsample is \(s'=s''_{\textrm{OOB}}\). The summary score \(s\) is then the average of optimisation scores \(s'\). This is a commonly used criterion that tries to maximise the score on the OOB validation data.

2. `balanced`: The optimisation score for each bootstrap subsample is \(s'=s''_{\textrm{OOB}} - \left| s''_{\textrm{OOB}} - s''_{\textrm{IB}} \right|\). The summary score \(s\) is then the average of optimisation scores \(s'\). A variation on `max_validation` with a penalty for differences in performance between the IB and OOB data. The underlying idea is that a good set of hyperparameters should lead to models that perform well on both development and validation data.

3. `strong_balance`: The optimisation score for each bootstrap subsample is \(s'=s''_{\textrm{OOB}} - 2\left| s''_{\textrm{OOB}}-s''_{\textrm{IB}} \right|\). The summary score \(s\) is then the average of optimisation scores \(s'\). A variant of `balanced` with a stronger penalty term.

4. `validation_minus_sd`: The optimisation score for each bootstrap subsample is \(s'=s''_{\textrm{OOB}}\). The summary score \(s\) is then the average of optimisation scores \(s'\) minus the standard deviation of \(s'\). This penalises hyperparameter sets that lead to wide variance on OOB data.

5. `validation_25th_percentile`: The optimisation score for each bootstrap subsample is \(s'=s''_{\textrm{OOB}}\). The summary score \(s\) is the 25th percentile of optimisation scores \(s'\). Like `validation_minus_sd` this penalises hyperparameter sets that lead to wide variance on OOB data.

6. `model_estimate`: The optimisation score for each bootstrap subsample is \(s'=s''_{\textrm{OOB}}\). The model used to infer utility of new hyperparameter sets (see the next section Predicting optimisation score for new hyperparameter sets) is used to predict the expected optimisation score \(\mu(\mathbf{x})\). This score is used as summary score \(s\). Not available for random search.

7. `model_estimate_minus_sd`: The optimisation score for each bootstrap subsample is \(s'=s''_{\textrm{OOB}}\). The model used to infer utility of new hyperparameter sets (see the next section Predicting optimisation score for new hyperparameter sets) is used to predict the expected optimisation score \(\mu(\mathbf{x})\) and its standard deviation \(\sigma(\mathbf{x})\). Then \(s = \mu(\mathbf{x}) - \sigma(\mathbf{x})\). This penalises hyperparameter sets for which a large variance is expected. Not available for random search.

8. `model_balanced_estimate`: The optimisation score for each bootstrap subsample is \(s'=s''_{\textrm{OOB}} - \left| s''_{\textrm{OOB}} - s''_{\textrm{IB}} \right|\). The model used to infer utility of new hyperparameter sets (see the next section Predicting optimisation score for new hyperparameter sets) is used to predict the expected optimisation score \(\mu(\mathbf{x})\). This score is used as summary score \(s\). The method is similar to `model_estimate`, but predicts the balanced score instead of the validation score. Not available for random search.

9. `model_balanced_estimate_minus_sd`: The optimisation score for each bootstrap subsample is \(s'=s''_{\textrm{OOB}} - \left| s''_{\textrm{OOB}} - s''_{\textrm{IB}} \right|\). The model used to infer utility of new hyperparameter sets (see the next section Predicting optimisation score for new hyperparameter sets) is used to predict the expected optimisation score \(\mu(\mathbf{x})\) and its standard deviation \(\sigma(\mathbf{x})\). Then \(s = \mu(\mathbf{x}) - \sigma(\mathbf{x})\). This penalises hyperparameter sets for which a large variance is expected. The method is similar to `model_estimate_minus_sd`, but predicts the balanced score and its variance instead. Not available for random search.

The summary score is then used to select the best known hyperparameter set, i.e. the set that maximises the summary score. Moreover, the optimisation scores are used to identify candidate hyperparameter sets as described in the following section. The optimisation function is set using the `optimisation_function` argument.

## Predicting optimisation score for new hyperparameter sets

Any point in the hyperparameter space has a single, scalar, optimisation score value that is a priori unknown. During the optimisation process, the algorithm samples from the hyperparameter space by selecting hyperparameter sets and computing the optimisation score for one or more bootstraps. For each hyperparameter set the resulting scores are distributed around the actual value.

A key point of Bayesian optimisation is the ability to estimate the usefulness or utility of new hyperparameter sets. This ability is facilitated by modelling the optimisation score of new hyperparameter sets using the optimisation scores of observed hyperparameter sets.

The following models can be used for this purpose:

- `gaussian_process` (default): Creates a localised approximate Gaussian deterministic Gaussian Process. This is implemented using the `laGP` package (Gramacy 2016).

- `bayesian_additive_regression_trees` or `bart`: Uses Bayesian Additive Regression Trees for inference. Unlike standard random forests, BART allows for estimating posterior distributions directly and can extrapolate. BART is implemented using the `BART` package (Sparapani et al. 2021).

- `random_forest`: Creates a random forest for inference. A random forest was originally used in the SMBO algorithm by Hutter et al. (Hutter et al. 2011). A weakness of random forests is their lack of extrapolation beyond observed values, which limits their usefulness in exploiting promising areas of hyperparameter space somewhat.

In addition, familiar can perform random search (`random` or `random_search`). This forgoes the use of models to steer optimisation. Instead, the hyperparameter space is sampled at random.

The learner used to predict optimisation scores can be specified using the `hyperparameter_learner` parameter.

## Acquisition functions for utility of hyperparameter sets

The expected values and the posterior values of optimisation scores from the models are used to compute utility of new hyperparameter sets using an acquisition function. The following acquisition functions are available in familiar (Shahriari et al. 2016). Let \(\nu=f(\mathbf{x})\) be the posterior distribution of hyperparameter set \(\mathbf{x}\), and \(\tau\) the best observed optimisation score. Let also \(\mu(\mathbf{x})\) and \(\sigma(\mathbf{x})\) be the sample mean and sample standard deviation for set \(\mathbf{x}\) (implicitly for round \(m\)):

- `improvement_probability`: The probability of improvement quantifies the probability that the expected optimisation score for a set \(\mathbf{x}\) is better than the best observed optimisation score \(\tau\):

  \[\alpha(\mathbf{x}) = P(\nu > \tau) = \Phi \left(\frac{\mu(\mathbf{x}) - \tau} {\sigma(\mathbf{x})}\right)\]

  Here \(\Phi\) is the cumulative distribution function of a normal distribution. Note that this acquisition function is typically prone to convergence to local minima, although in our algorithm this may be mitigated by the alternating random search strategy.

- `improvement_empirical_probability`: Similar to `improvement_probability`, but based directly on optimisation scores predicted by the individual decision trees:

  \[\alpha(\mathbf{x}) = P(\nu > \tau) = \frac{1}{N} \sum_{i=1}^N \left[\nu_i > \tau\right]\]

  with \(\left[\ldots\right]\) denoting an Iverson bracket, which takes the value 1 if the condition specified by \(\ldots\) is true, and 0 otherwise.

- `expected_improvement` (default): Expected improvement is based on an improvement function, and is be computed as follows:

  \[\alpha(\mathbf{x})=(\mu(\mathbf{x})-\tau) \Phi \left(\frac{\mu(\mathbf{x}) - \tau}{\sigma(\mathbf{x})}\right) + \sigma(\mathbf{x}) \phi\left(\frac{\mu(\mathbf{x}) - \tau}{\sigma(\mathbf{x})}\right)\]

  with \(\phi\) denoting the normal probability density function. If \(\sigma(\mathbf{x}) = 0\), \(\alpha(\mathbf{x})=0\).

- `upper_confidence_bound`: This acquisition function is based on the upper confidence bound of the distribution, and is defined as (Srinivas et al. 2012):

  \[\alpha(\mathbf{x}, t)=\mu(\mathbf{x}) + \beta_t \sigma(\mathbf{x})\]

  Here the parameter \(\beta_t\) depends on the current round \(t\), which in our implementation would be roughly equivalent to the maximum number of bootstraps performed for any of the hyperparameter sets, minus \(1\). The \(\beta_t\) parameter is then defined as \(\beta_t= 2 \log(|D|(t+1)^2\pi^2/6\delta)\) (Srinivas et al. 2012). Here, \(\delta \in [0,1]\) and \(|D|\) the dimensionality of the hyperparameter space. For their experiments Srinivas et al. used \(\delta=0.1\), and noted that scaling down \(\beta_t\) by a factor \(5\) improves the algorithm. Hence, we use \(\beta_t= \frac{2}{5} \log\left(\frac{5}{3} |D|(t+1)^2\pi^2\right)\).

- `bayes_upper_confidence_bound`: Proposed by Kaufmann et al. (Kaufmann et al. 2012), the Bayesian upper confidence bound is defined as:

  \[\alpha(\mathbf{x}, t)=Q \left(1-\frac{1}{(t + 1) \log(n)^c}, \nu_t\right)\]

  with \(Q\) being a quantile function. Note that we here deviate from the original definition by substituting the original \(t\) by \(t+1\), and not constraining the posterior distribution \(\nu\) to a known function. In the original, \(n\) defines the number of rounds, with \(t\) the current round. In our algorithm, this is interpreted as \(n\) being the total number of bootstraps, and \(t\) the maximum number of bootstraps already performed for any of the hyperparameter sets. Hence our implementation is the same, aside from the lack of constraint on the posterior distribution. Based on the simulations and recommendations by Kaufmann et al. we choose parameter \(c=0\).

The acquisition function can be specified using the `acquisition_function` parameter.

## Exploring challenger sets

After selecting challenger hyperparameter sets, the optimisation score of models trained using these hyperparameter sets are compared against the best-known score in a run-off intensify step. During this step, models will be trained on new bootstrap data, compared, and then trained on another set of bootstraps, and so on. Pruning the set of challenger hyperparameters after each iteration reduces computational load. Familiar implements the following methods:

- `single_shot` (default): The set of alternative parameter sets is not pruned, and each intensify iteration contains only a single (initial) intensification step that only uses a single bootstrap. This is the exploration method that under default settings does the least in-depth exploration, with the advantage of being fast. Extensive internal tests show that this exploration method yields similarly performant models as more extensive exploration methods.

- `successive_halving`: The set of alternative parameter sets is pruned by removing the worst performing half of the sets after each step (Jamieson and Talwalkar 2016). The set of investigated parameter sets gets progressively smaller.

- `stochastic_reject`: The set of alternative parameter sets is pruned by comparing the performance of each parameter set with that of the incumbent best parameter set using a paired Wilcoxon test.

- `none`: The set of alternative parameter sets is not pruned. Compared to `single_shot`, hyperparameter sets are assessed using multiple bootstraps (`smbo_step_bootstraps` * `smbo_intensify_steps`).

The method used to steer exploration can be set using the `exploration_method` parameter.

## Providing hyperparameters manually

It is possible to set hyperparameters manually. This can be used to change hyperparameters that are fixed by default, to set a fixed value for randomised hyperparameters, or to provide a different search range for randomised hyperparameters.

Hyperparameters can be provided using the `hyperparameter` tag. For the `glm_logistic` learner an example tag may for example look as follows:

    <hyperparameter>
      <glm_logistic>
        <sign_size>5</sign_size>
      </glm_logistic>
    </hyperparameter>

Or as a nested list passed as the `hyperparameter` argument to `summon_familiar`:

    hyperparameter = list("glm_logistic"=list("sign_size"=5))

More than one value can be provided. The behaviour changes depending on whether the hyperparameter is categorical or numeric variable. In case of categorical hyperparameters, the provided values define the search range. For numerical hyperparameters, providing two values sets the bounds of the search range, whereas providing more than two values will define the search range itself.

## Configuration options for hyperparameter optimisation

Hyperparameter optimization may be configured using the tags/arguments in the table below.

Configuration options for hyperparameter optimisation.
tag / argument | description | default
--- | --- | ---
`optimisation_bootstraps` | Maximum number of bootstraps created for optimisation. | `20`
`optimisation_determine_vimp` | If `TRUE`, compute variable importance for each bootstrap. If `FALSE` use variable importance computed during the feature selection step. | `TRUE`
`smbo_random_initialisation` | If `random` initial parameter sets are generated randomly from default ranges. If `fixed` or `fixed_subsample`, the initial parameter sets are based on a grid in parameter space. | `fixed_subsample`
`smbo_n_random_sets` | Sets the number of hyperparameter sets drawn for initialisation. Ignored if `smbo_random_initialisation` is `fixed`. If `smbo_random_initialisation` is `fixed_subsample`, the number of selected hyperparameters may be lower. | `100`
`max_smbo_iterations` | Maximum number of intensify iterations of the SMBO algorithm. | `20`
`smbo_stop_convergent_iterations` | Number of subsequent convergent SMBO iterations required to stop hyperparameter optimisation early. | `3`
`smbo_stop_tolerance` | Tolerance for recent optimisation scores to determine convergence. | `0.1 * 1 / sqrt(n_samples)`
`smbo_time_limit` | Time limit (in minutes) for the optimisation process. | `NULL`
`smbo_initial_bootstraps` | Number of bootstraps assessed initially. | `1`
`smbo_step_bootstraps` | Number of bootstraps used within each step of an intensify iteration. | `3`
`smbo_intensify_steps` | Number of intensify steps within each intensify iteration. | `5`
`optimisation_metric` | The metric used for optimisation, e.g. `auc`. See the vignette on performance metrics for available options. More than one metric may be specified. | `auc_roc` (`binomial`, `multinomial`); `mse` (`continuous`); and `concordance_index` (`survival`)
`optimisation_function` | The optimisation function used (see Assessing goodness of hyperparameter sets). | `validation`
`hyperparameter_learner` | Learner used to predict optimisation scores for new hyperparameter sets (see Predicting optimisation score for new hyperparameter sets). | `gaussian_process`
`acquisition_function` | The function used to quantify utility of hyperparameter sets (see Acquisition functions for utility of hyperparameter sets). | `expected_improvement`
`exploration_method` | Method used to explore challenger hyperparameter sets (see Exploring challenger sets). | `single_shot`
`smbo_stochastic_reject_p_value` | The p-value level for stochastic pruning. | `0.05`
`parallel_hyperparameter_optimisation` | Enables parallel processing for hyperparameter optimisation. Ignored if `parallel = FALSE`. Can be `outer` to process multiple optimisation processes simultaneously. Overrides `parallel = TRUE`. | `TRUE`

# Model recalibration

Even though learners may be good at discrimination, model calibration can be lacklustre. Some learners are therefore recalibrated as follows:

1. 3-fold cross-validation is performed. Responses are predicted using a new model trained on each training fold.

2. Three recalibration models are trained using the responses for the training folds as input (Niculescu-Mizil and Caruana 2005).

3. The three recalibration models are then applied to (new) responses of the full model, and the resulting values averaged for each sample.

The following learners currently undergo recalibration:

learner | outcome | recalibration model
--- | --- | ---
`xgboost_lm`, `xgboost_lm_logistic` | binomial, multinomial | `glm_logistic`
`xgboost_tree`, `xgboost_tree_logistic` | binomial, multinomial | `glm_logistic`
`xgboost_lm_cox` | survival | `glm_cox`
`xgboost_tree_cox` | survival | `glm_cox`
`boosted_glm_cindex`, `boosted_glm_gehan` | survival | `glm_cox`
`boosted_tree_cindex`, `boosted_tree_gehan` | survival | `glm_cox`

Familiar currently does not recalibrate models by inverting the model calibration curves, but may do so in the future.

# References

Boser, Bernhard E, Isabelle M Guyon, and Vladimir N Vapnik. 1992. “A Training Algorithm for Optimal Margin Classifiers.” Proceedings of the Fifth Annual Workshop on Computational Learning Theory, July, 144–52.
Breiman, Leo. 2001. “Random Forests.” Mach. Learn. 45 (1): 5–32.
Chang, Chih-Chung, and Chih-Jen Lin. 2011. “LIBSVM: A Library for Support Vector Machines.” ACM Trans. Intell. Syst. Technol. (New York, NY, USA) 2 (3): 27:1–27.
Chen, Tianqi, and Carlos Guestrin. 2016. “XGBoost: A Scalable Tree Boosting System.” Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, August, 785–94.
Cox, D R. 1972. “Regression Models and Life-Tables.” J. R. Stat. Soc. Series B Stat. Methodol. 34 (2): 187–202.
Cui, Yin, Menglin Jia, Tsung-Yi Lin, Yang Song, and Serge Belongie. 2019. “Class-Balanced Loss Based on Effective Number of Samples.” 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (Long Beach, CA, USA), June, 9268–77.
Efron, Bradley. 1977. “The Efficiency of Cox’s Likelihood Function for Censored Data.” J. Am. Stat. Assoc. 72 (359): 557–65.
Gramacy, Robert B. 2016. “laGP: Large-Scale Spatial Modeling via Local Approximate Gaussian Processes in R.” Journal of Statistical Software 72 (1): 1–46.
Hastie, Trevor, Robert Tibshirani, and Jerome Friedman. 2009. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Second Edition. Springer Series in Statistics. Springer Science+Business Media, LLC.
Hutter, Frank, Holger H Hoos, and Kevin Leyton-Brown. 2011. “Sequential Model-Based Optimization for General Algorithm Configuration.” In Learning and Intelligent Optimization, edited by Carlos A Coello Coello, vol. 6683. Lecture Notes in Computer Science. Springer Berlin Heidelberg.
Ishwaran, Hemant, Udaya B Kogalur, Eugene H Blackstone, and Michael S Lauer. 2008. “Random Survival Forests.” Ann. Appl. Stat. 2 (3): 841–60.
Ishwaran, Hemant, Udaya B Kogalur, Xi Chen, and Andy J Minn. 2011. “Random Survival Forests for High-Dimensional Data.” Stat. Anal. Data Min. 4 (1): 115–32.
Jamieson, Kevin, and Ameet Talwalkar. 2016. “Non-Stochastic Best Arm Identification and Hyperparameter Optimization.” In Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, edited by Arthur Gretton and Christian C Robert, vol. 51. Proceedings of Machine Learning Research. PMLR.
Kaufmann, Emilie, Olivier Cappé, and Aurélien Garivier. 2012. “On Bayesian Upper Confidence Bounds for Bandit Problems.” Artificial Intelligence and Statistics, 592–600.
Lausen, Berthold, and Martin Schumacher. 1992. “Maximally Selected Rank Statistics.” Biometrics 48 (1): 73.
Meyer, David, Evgenia Dimitriadou, Kurt Hornik, Andreas Weingessel, and Friedrich Leisch. 2021. E1071: Misc Functions of the Department of Statistics, Probability Theory Group (Formerly: E1071), TU Wien. https://CRAN.R-project.org/package=e1071.
Nelder, J A, and R W M Wedderburn. 1972. “Generalized Linear Models.” J. R. Stat. Soc. Ser. A 135 (3): 370–84.
Niculescu-Mizil, Alexandru, and Rich Caruana. 2005. “Predicting Good Probabilities with Supervised Learning.” Proceedings of the 22nd International Conference on Machine Learning, August, 625–32.
Oshiro, Thais Mayumi, Pedro Santoro Perez, and José Augusto Baranauskas. 2012. “How Many Trees in a Random Forest?” Machine Learning and Data Mining in Pattern Recognition, 154–68.
R Core Team. 2024. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. https://www.R-project.org/.
Schapire, Robert E. 1990. “The Strength of Weak Learnability.” Mach. Learn. 5 (2): 197–227.
Schmid, Matthias, Marvin N Wright, and Andreas Ziegler. 2016. “On the Use of Harrell’s C for Clinical Risk Prediction via Random Survival Forests.” Expert Syst. Appl. 63 (November): 450–59.
Shahriari, B, K Swersky, Z Wang, R P Adams, and N de Freitas. 2016. “Taking the Human Out of the Loop: A Review of Bayesian Optimization.” Proc. IEEE 104 (1): 148–75.
Simon, Noah, Jerome Friedman, Trevor Hastie, and Rob Tibshirani. 2011. “Regularization Paths for Cox’s Proportional Hazards Model via Coordinate Descent.” J. Stat. Softw. 39 (5): 1–13.
Sparapani, Rodney, Charles Spanbauer, and Robert McCulloch. 2021. “Nonparametric Machine Learning and Efficient Computation with Bayesian Additive Regression Trees: The BART R Package.” Journal of Statistical Software 97 (1): 1–66.
Srinivas, N, A Krause, S M Kakade, and M W Seeger. 2012. “Information-Theoretic Regret Bounds for Gaussian Process Optimization in the Bandit Setting.” IEEE Trans. Inf. Theory 58 (5): 3250–65.
Therneau, Terry M, and Patricia M Grambsch. 2000. Modeling Survival Data: Extending the Cox Model. Statistics for Biology and Health. Springer Science & Business Media.
Venables, W. N., and B. D. Ripley. 2002. Modern Applied Statistics with s. Fourth. Springer. https://www.stats.ox.ac.uk/pub/MASS4/.
Wright, Marvin N, Theresa Dankowski, and Andreas Ziegler. 2017. “Unbiased Split Variable Selection for Random Survival Forests Using Maximally Selected Rank Statistics.” Stat. Med. 36 (8): 1272–84.
Wright, Marvin N, and Andreas Ziegler. 2017. “Ranger : A Fast Implementation of Random Forests for High Dimensional Data in c++ and R.” J. Stat. Softw. 77 (1).