Axis-aligned hyperrectangle subspace learner

Learns axis-aligned hyperrectangles (boxes with edges parallel to coordinate axes) that contain high-quality hyperparameter configurations.

Details

Geometry:

Fits a hyperrectangle defined by independent bounds per dimension: $$S = \{x \in \mathbb{R}^p : l_i \leq x_i \leq u_i\}$$

The transformation to unit hypercube is \(y = Ax + b\) where:

• \(A = \text{diag}(1/(u_i - l_i))\) is diagonal (independent scaling)

• \(b = -l/(u - l)\) is the translation vector

• \(x \in [l, u]\) are points in the fitted subspace

• \(y \in [0, 1]^p\) are points in the unit hypercube

Optimization (lambda specified): $$\min_{l,u,\xi} \frac{\lambda}{2}\|u-l\|_2^2 + \frac{1}{2n}\sum_t(\xi_t^- + \xi_t^+)$$ subject to: \(l - \xi_t^- \leq x^{(t)} \leq u + \xi_t^+\), \(\xi_t^-, \xi_t^+ \geq 0\)

Regularization parameter \(\lambda\) controls subspace size: larger values produce smaller subspaces with more outliers. Slack variables \(\xi_t^-, \xi_t^+\) allow configurations to violate bounds.

Uses ECOS solver. Special handling for univariate case (\(p=1\)).

Simple mode (lambda = NULL):

Uses coordinate-wise min/max: \(l_i = \min_t x_i^{(t)}\), \(u_i = \max_t x_i^{(t)}\)

No outliers are possible in this mode (hard constraints).

Key Properties:

• Most interpretable: bounds directly specify valid ranges

• Fastest: O(np) with diagonal matrices

• Best for independent hyperparameters

• Cannot capture correlations between hyperparameters

• Transformation matrix A is always diagonal

Comparison with other learners:

• Box: Axis-aligned only, fastest, most interpretable

• Polygon: Can rotate (L-infinity norm), maintains sharp corners

• Ellipsoid: Can rotate (L2 norm), smooth boundaries, most flexible

See also

LearnerSubspace for inherited methods and general workflow. LearnerSubspacePolygon for oriented hyperrectangles. LearnerSubspaceEllipsoid for ellipsoids. coef.LearnerSubspaceBox for extracting fitted parameters. augment.LearnerSubspaceBox for adding density parameters.

Super class

spacefinder::LearnerSubspace -> LearnerSubspaceBox

Methods

Public methods

• LearnerSubspaceBox$clone()

Inherited methods

• spacefinder::LearnerSubspace$initialize()
• spacefinder::LearnerSubspace$train()

---

Method clone()

The objects of this class are cloneable with this method.

Usage

LearnerSubspaceBox$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

Examples

if (FALSE) { # \dontrun{
# Create task and learner
task <- TaskSubspace$new(data, target_measure = "accuracy",
                         hps = c("learning_rate", "max_depth"))
learner <- LearnerSubspaceBox$new(task)

# Simple min/max bounds (no regularization)
learner$train(q_val = 0.9, lambda = NULL)
coef(learner)

# Regularized optimization
learner$train(q_val = 0.9, lambda = 0.1)
result <- learner$result
print(result$coefficients)  # Bounds per hyperparameter
print(result$n_violations)  # Number of outliers
print(result$outliers)      # Outlier indices

# With categorical hyperparameters
task <- TaskSubspace$new(data, target_measure = "accuracy",
                         hps = c("learning_rate", "max_depth"),
                         cat_hps = "optimizer")
learner <- LearnerSubspaceBox$new(task)
learner$train(q_val = 0.95, lambda = 0.05)
coef(learner)  # Returns separate bounds per optimizer

# Get transformation matrices (y = Ax + b)
coef(learner, vectorize = TRUE)  # Returns A (diagonal), b

# Check which configs are in the subspace
bounds <- coef(learner)
in_subspace <- learner$task$data[
  learning_rate >= bounds[hyperparameter == "learning_rate"]$min &
  learning_rate <= bounds[hyperparameter == "learning_rate"]$max &
  max_depth >= bounds[hyperparameter == "max_depth"]$min &
  max_depth <= bounds[hyperparameter == "max_depth"]$max
]
} # }