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Augment oriented hyperrectangle learner with beta densities

augment.LearnerSubspacePolygon.Rd

Transforms data from the fitted oriented (rotated) hyperrectangle subspace to the unit hypercube and fits univariate beta distributions to each hyperparameter dimension using weighted maximum likelihood estimation. The polygon learner uses the L-infinity norm, so the fitted subspace is \(\|Ax + b\|_\infty \leq 1\), which maps to \([-1, 1]^d\) before being rescaled to \([0, 1]^d\) for beta fitting.

Usage

# S3 method for class 'LearnerSubspacePolygon'
augment(x, regularize = TRUE, ...)

Arguments

x

A LearnerSubspacePolygon object with fitted subspace parameters

regularize

Logical indicating whether to enforce alpha >= 1 and beta >= 1 to avoid U-shaped densities. Default: TRUE

...

Additional arguments passed to fit_beta_mle_single (e.g., tol, max_iter, clip_eps)

Value

A data.table with columns:

  • parameter: Hyperparameter name

  • alpha: Fitted beta shape parameter (alpha > 0)

  • beta: Fitted beta shape parameter (beta > 0)

  • converged: Logical indicating whether MLE converged

  • iterations: Number of Newton-Raphson iterations used

  • cat_hp: Categorical level (only if task has categorical hyperparameters)

Details

Algorithm:

For each categorical level (or globally if no categorical hyperparameters):

  1. Transform data: \(z = A^{-1}(y - b)\) to map to \([-1, 1]^d\)

  2. Rescale to unit cube: \(x = (z + 1) / 2\) to map to \([0, 1]^d\)

  3. Filter points where all coordinates lie in \([0, 1]^d\)

  4. Fit \(Beta(\alpha, \beta)\) to each dimension via weighted MLE

  5. Apply regularization if requested: \(\alpha, \beta \geq 1\)

The key difference from the Box learner is that oriented hyperrectangles use the L-infinity norm \(\|z\|_\infty \leq 1\), which defines a hypercube in the transformed space \([-1, 1]^d\). We rescale this to \([0, 1]^d\) before fitting beta distributions.

Matrix Inversion:

Since \(A\) is a general positive definite matrix (not diagonal), uses Cholesky decomposition for efficient and stable inversion: \(A^{-1} = (R^T R)^{-1} = R^{-1}R^{-T}\) where \(R\) is the Cholesky factor.

Weights:

Data points are weighted by their performance (target measure values), normalized to sum to 1. This gives more influence to higher-performing configurations.

Regularization:

When regularize = TRUE (default), ensures \(\alpha \geq 1\) and \(\beta \geq 1\). This prevents U-shaped densities and ensures the mode exists in the interior of \([0,1]\).

Categorical Hyperparameters:

When the task includes categorical hyperparameters, separate beta distributions are fitted for each combination of hyperparameter and categorical level.

See also

LearnerSubspacePolygon for the learner class. fit_beta_mle_single for the univariate beta MLE algorithm. augment.LearnerSubspaceBox for the axis-aligned variant.

Examples

if (FALSE) { # \dontrun{
# Train learner
task <- TaskSubspace$new(data, target_measure = "auc",
                         hps = c("learning_rate", "max_depth"))
learner <- LearnerSubspacePolygon$new(task)
learner$train(q_val = 0.9, lambda = 0.1)

# Fit beta densities
densities <- augment(learner)
print(densities)

# Without regularization
densities_unreg <- augment(learner, regularize = FALSE)
} # }