Partial Training

By default, fit() trains all control points of the Bézier simplex simultaneously. The freeze argument lets you hold a subset of control points constant during training, so that only the remaining (free) control points are updated.

This is useful when:

Boundary constraints are known — If single-objective optimizations have already determined the vertex solutions, you can freeze those vertices and train only the interior and edge control points.
Incremental refinement — Fit a low-degree model first, reuse its control points as the initialization for a higher-degree model, and freeze the already-accurate points to stabilize training.
Encoding prior knowledge — Pin control points whose values are theoretically or physically determined.

The example below demonstrates how to freeze the two vertices of a Bézier curve while training its interior control points.

The freeze argument takes a list of multi-index lists (e.g., [[3, 0], [0, 3]]). Any control point whose multi-index appears in freeze is excluded from gradient updates.

import torch
import torch_bsf

ts = torch.tensor(  # parameters on a simplex
   [
      [8/8, 0/8],
      [7/8, 1/8],
      [6/8, 2/8],
      [5/8, 3/8],
      [4/8, 4/8],
      [3/8, 5/8],
      [2/8, 6/8],
      [1/8, 7/8],
      [0/8, 8/8],
   ]
)
xs = 1 - ts * ts  # values corresponding to the parameters

# Initialize 2D control points of a Bézier curve of degree 3
init = {
   # index: value
   (3, 0): [0.0, 0.1],
   (2, 1): [1.0, 1.1],
   (1, 2): [2.0, 2.1],
   (0, 3): [3.0, 3.1],
}

# Or, generate random control points in [0, 1)
init = torch_bsf.bezier_simplex.rand(n_params=2, n_values=2, degree=3)

# Or, load control points from a file
init = torch_bsf.bezier_simplex.load("control_points.yml")

# Train the edge of a Bézier curve while its vertices are frozen
bs = torch_bsf.fit(
   params=ts,  # input observations (training data)
   values=xs,  # output observations (training data)
   init=init,  # initial values of control points
   freeze=[[3, 0], [0, 3]],  # freeze vertices of the Bézier curve
)

# Predict with the trained model
t = [
   [0.2, 0.8],
   [0.7, 0.3],
]
x = bs(t)
print(x)