Oriented hyperrectangle subspace learner

Learns oriented (rotated) hyperrectangles that contain high-quality hyperparameter configurations. Allows arbitrary rotation while maintaining rectangular shape.

Details

Geometry:

Fits a rotated hyperrectangle defined by: $$\{x \in \mathbb{R}^p : \|Ax + b\|_\infty \leq 1\}$$

The transformation matrix \(A \in \mathbb{R}^{p \times p}\) is positive definite but not restricted to diagonal, allowing rotation in hyperparameter space. The L-infinity norm constraint maintains rectangular shape (sharp corners).

Optimization (lambda specified): $$\min_{A \succeq 0, b, s} \lambda \cdot (-\log\det(A)) + \frac{1}{n}\sum_{t=1}^n s_t$$ subject to: \(\|Ax^{(t)} + b\|_\infty \leq 1 + s_t\), \(s_t \geq 0\)

Uses SCS solver for semidefinite programming. Volume minimization via \(-\log\det(A)\).

Simple mode (lambda = NULL):

Minimizes volume without slack variables. All points must satisfy \(\|Ax^{(t)} + b\|_\infty \leq 1\).

Key Properties:

• Intermediate flexibility: captures correlations between hyperparameters

• Maintains rectangular shape with sharp corners (unlike ellipsoids)

• More flexible than Box (can rotate), less flexible than Ellipsoid

• Computational cost between Box and Ellipsoid

• Good balance of interpretability and expressiveness

Comparison with other learners:

• Box: Axis-aligned, diagonal A, fastest, most interpretable

• Polygon: Can rotate, general A, rectangular shape, intermediate cost

• Ellipsoid: Can rotate, general A, smooth boundaries, slowest

See also

LearnerSubspace for inherited methods and general workflow. LearnerSubspaceBox for axis-aligned hyperrectangles (faster). LearnerSubspaceEllipsoid for ellipsoids (more flexible). coef.LearnerSubspacePolygon for extracting fitted parameters. augment.LearnerSubspacePolygon for adding density parameters.

Super class

spacefinder::LearnerSubspace -> LearnerSubspacePolygon

Methods

Public methods

• LearnerSubspacePolygon$clone()

Inherited methods

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

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Method clone()

The objects of this class are cloneable with this method.

Usage

LearnerSubspacePolygon$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

Examples

if (FALSE) { # \dontrun{
# Create task and learner
task <- SubspaceTask$new(data, target_measure = "accuracy")
learner <- LearnerSubspacePolygon$new(task)

# Minimum-volume oriented box (hard constraints)
learner$train(lambda = NULL)
result <- learner$result
print(result$A)  # Shape matrix (not diagonal)
print(result$b)  # Translation vector

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

# Check orientation
eigendecomp <- eigen(result$A)
print(eigendecomp$vectors)  # Principal directions
print(1 / eigendecomp$values)  # Box widths along principal axes

# With categorical hyperparameters
task <- SubspaceTask$new(data, target_measure = "accuracy",
                         cat_hps = "optimizer")
learner <- LearnerSubspacePolygon$new(task)
learner$train(q_val = 0.95, lambda = 0.05)
coef(learner, vectorize = TRUE)  # Separate A, b per optimizer
} # }