Cross-validation and degree search

When degree selection matters

Polynomial degree controls the flexibility of a Bézier model. A degree-one curve is a straight segment; degree two can represent a quadratic bend; higher degrees permit more complicated variation and introduce more unknown controls. A larger degree often lowers training error even when its predictions between observations get worse. Select degree using observations that were not used to fit the corresponding model.

This is particularly useful when a surrogate will interpolate costly simulator outputs, when the data contain noise, or when one region has much weaker coverage than another. It also gives evidence for retaining a low degree: if higher degrees do not reduce held-out error, additional coefficients and compute cost have little practical justification.

cross_validate evaluates one space. select_degree evaluates an explicit list of candidate degrees on identical folds. Both retain results rather than silently converting validation into a fitted ensemble or full-data refit. Inspect the evidence, choose a degree, and then explicitly train the model that the application will use.

A reproducible search

This example adds fixed-seed noise to a curved scalar response. The barycentric inputs [1-x, x] represent a segment rather than two independent features. The same observations and fold partition are used for all degrees.

import torch
from torch_bsf import BezierSpace, Samples, fit
from torch_bsf.model_selection import KFold, cross_validate, select_degree

x = torch.linspace(0, 1, 18, dtype=torch.float64)
t = torch.stack([1 - x, x], dim=1)
generator = torch.Generator().manual_seed(17)
y = torch.sin(torch.pi * x) + .2 * x
y = y + .1 * torch.randn(18, generator=generator, dtype=torch.float64)
samples = Samples(t, y[:, None])
template = BezierSpace.simplex(1, degree=1)
cv = KFold(n_splits=3, shuffle=True, seed=7)
search = select_degree(template, samples, degrees=[1, 2, 3, 5, 9], cv=cv)
assert search.success
for candidate in search.candidates:
    assert candidate.validation.success
    assert len(candidate.validation.folds) == 3
chosen = fit(search.best_space, samples).require_success()
assert chosen.model.space.compatible(search.best_space)

Use candidate.degree, candidate.validation.metrics and individual fold results to produce a report. search.best_degree is the candidate minimizing held-out pooled_mse among successful candidates. best_space is its compiled space, not a trained model. If all candidates fail, success is false and both properties are None. Check success before passing the selected space to fit.

No early stopping occurs: every requested candidate is evaluated. This makes it possible to discover a later improvement after a mediocre intermediate degree. It also means that a long candidate list has a real cost. Start with a small range and expand only when the best candidate is at its upper boundary and there are enough observations to identify the larger space.

Read the validation evidence

Each FoldResult keeps train_indices, test_indices, the complete FitResult, held-out mse and mae, and total held-out observation weight. Every observation appears in one test fold. The test targets do not participate in that fold’s numerical solve. A failed solve remains visible and makes that candidate ineligible; it is never omitted from an apparently successful mean.

The reported metrics answer different questions:

Table 6 Validation metrics

Metric	Interpretation
mean_mse	Arithmetic mean of fold MSEs. Every fold contributes equally.
std_mse	Population standard deviation of fold MSEs. Large values flag unstable partitions.
pooled_mse	Fold MSEs weighted by their held-out weight sums. This is the selection criterion.
Fold mae	Weighted absolute error, averaged over output coordinates; useful beside squared error.

With unequal fold sizes or unequal observation weights, mean_mse and pooled_mse need not match. Output columns are averaged equally in their current units. A large-scale output may dominate both; inspect per-output errors separately if the application has different tolerances per response. A small average can conceal unacceptable error near an endpoint or boundary. Plot residuals against parameter position as well as comparing summary errors.

Fig. 12 Before selection: degree one underfits the curved target, while degree nine introduces unnecessary flexibility. After selection: refit the degree with the smallest pooled three-fold error. The target and data are identical across curves. Right: candidate errors on a logarithmic scale.

The figure uses the example’s 18 observations, noise seed 17 and fold seed 7. Reproduce it with python examples/generate_guide_figures.py. The generated metrics JSON retains every candidate error. This synthetic comparison illustrates the workflow; its chosen degree is not a universal default for other responses or sample counts.

Tune candidates and folds

Increasing degree increases controls combinatorially with domain dimension. A degree \(d\) simplex of dimension \(a\) has \(\binom{a+d}{d}\) controls. A product has the product of these counts over factors. Check space.n_controls before requesting a large search. A cubic triangle has ten controls; a cubic segment has four. The same number of observations therefore supports very different searches on those domains.

Increasing KFold.n_splits fits more models per candidate. Each training set contains a larger fraction of observations, while each test set becomes smaller and may provide a noisier local estimate. Three to five folds are often a reasonable first experiment, provided each training fold has positive weight and sufficient coverage. The requested split count must be at least two and cannot exceed the number of observations. A fold with zero total train or test weight is invalid; it does not become a zero-error fold.

shuffle=True randomizes indices with a local generator. seed fixes that partition for reproducibility; it does not change the regression solver’s precision. shuffle=False makes contiguous test blocks in input row order. This can test interpolation gaps when rows are spatially sorted, but is not an automatic time-series or grouped validation scheme. For correlated repeated measurements, independent random folds can be optimistic. Construct a separate evaluation protocol reflecting the future prediction task instead of reporting random-fold error as evidence for extrapolation or unseen experimental groups.

For a product space, provide degree tuples such as [(1, 1), (2, 1), (2, 2)]. Increasing one factor degree adds flexibility along that factor. Keep factor order consistent with concatenated barycentric blocks. A tuple must match the number of factors. Degree search calls space.with_degree; use manually constructed compatible candidate spaces and explicit validation when changing topology or specifying different cellwise degree assignments.

Penalties and final deployment

Use the same penalty and solver in candidate evaluation and the final refit. A degree chosen with strong smoothness is not evidence for an unregularized fit of that degree. To compare several penalty weights, run separate searches with the same KFold settings and record all trials. Avoid repeatedly tuning against the final test set; retain untouched data for an honest final evaluation if the choice matters operationally.

cross_validate also accepts fixed for evaluating one space with exact controls. select_degree does not take fixed controls, because a global row number may mean a different coefficient at a different degree. For a constrained degree search, build each space, resolve its endpoint or face indices anew, and call cross_validate explicitly with the corresponding constraints.

The final full-data model generally differs from every fold model because it uses more observations. Recheck numerical status and inspect errors after refitting. If you want an ensemble instead, explicitly construct Ensemble(validation.models) from a single compatible space; see Explicit ensembles and active learning. Save the chosen model with io.save_model and record the candidate list, weights, fold settings and errors alongside it. Persistence of model geometry does not by itself preserve the reasoning behind selection.