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Sobolev rate for spline-series training risk

Statement

Assume AC. Let $a < b$ and let $X, Y$ be real random variables on a common probability space with $E ∣ Y ∣ < \infty$ . Choose a regular conditional law $Q (x, d y)$ of $Y$ given $X = x$ , and let $g_{0}$ be the selected Borel regression-function version from Regression target and chosen version. Suppose $P_{X}$ is supported on $[a, b]$ and has a Lebesgue density bounded almost everywhere by a finite constant $M_{X}$ , $g_{0} ∣_{(a, b)} \in W^{s, 2} (a, b)$ for an integer $s \geq 1$ , and $\int (y - g_{0} (x))^{2} Q (x, d y) \leq Δ$ for $P_{X}$ -almost every $x$ , where $0 \leq Δ < \infty$ . Fix a degree $p \in N_{0}$ with $p \geq s - 1$ . For each integer $J \geq 1$ , use the uniform partition into $J$ cells and the degree- $p$ B-spline space with each interior knot repeated $p + 1$ times, so its dimension is $K = (p + 1) J$ . For $n \geq 1$ iid observations from the joint law $P_{X} (d x) Q (x, d y)$ , let ${\hat{g}}_{K}$ be the fitted training vector and $g_{0} = (g_{0} (X_{1}), \dots, g_{0} (X_{n}))^{T}$ . Then $E [\frac{1}{n} ∥ {\hat{g}}_{K} - g_{0} ∥_{2}^{2}] \leq \frac{Δ K}{n} + C_{s, p, M_{X}, a, b} K^{- 2 s} ∥ D^{s} g_{0} ∥_{L^{2} (a, b)}^{2} .$ Choosing $J ≍ n^{1 / (2 s + 1)}$ (equivalently $K ≍ n^{1 / (2 s + 1)}$ for fixed $p$ ) gives expected empirical training risk $O (n^{- 2 s / (2 s + 1)})$ . This is not an independent-test-risk bound.

Facts & Assumptions

Given: AC; integer orders $s \geq 1$ and $p \geq s - 1$ ; bounded design density on $[a, b]$ ; $g_{0} \in W^{s, 2} (a, b)$ ; and the iid conditional-variance model of Expected empirical training risk for spline series least squares.

[A1]

Assume AC The Axiom of Choice, as required by the one-dimensional Sobolev representative supplier used in the approximation theorem.

[A2]

$E ∣ Y ∣ < \infty$ , $g_{0}$ is the selected regression-target version, $g_{0} ∣_{(a, b)} \in W^{s, 2} (a, b)$ , the design has a Lebesgue density bounded by $M_{X}$ , $0 \leq Δ < \infty$ , $s \in N$ , and $p \in N_{0}$ with $p \geq s - 1$ .

[F1]

The generic empirical training-risk bound is $Δ K / n$ plus the squared $L^{2} (P_{X})$ approximation error (Expected empirical training risk for spline series least squares).

[F2]

For a bounded local quasi-interpolant reproducing $P_{s - 1}$ , the $L^{2}$ approximation error is bounded by a constant times $h^{s} ∥ D^{s} g_{0} ∥_{2}$ ; the cellwise $L^{2}$ projector on the full-multiplicity uniform spline mesh is an admissible choice (Sobolev approximation by local spline quasi-interpolants).

[F3]

Under AC and $E ∣ Y ∣ < \infty$ , a supplied regular conditional law gives a Borel conditional-mean version $g_{0} (x) = \int y Q (x, d y)$ on integrable fibers; smoothness of this chosen version is an additional hypothesis (Regression target and chosen version).

[F4]

A clamped degree- $p$ knot vector with $J$ uniform cells, each of its $J - 1$ interior knots repeated $p + 1$ times, has $N = (J + 1) (p + 1)$ knots and therefore $K = N - p - 1 = (p + 1) J$ B-splines; the B-splines form a basis of the piecewise-polynomial space (B-spline spaces and knot conventions, B-spline basis properties and polynomial reproduction).

Proof

technique · direct

1.1A1A2F2F3F4givenalgebraconstruct

By [F3], the conditional mean used here is the selected Borel version required by the data-model interface. Let $f$ be its continuous Sobolev representative on $[a, b]$ from [F2]; since $P_{X}$ has a Lebesgue density, $f = g_{0}$ $P_{X}$ -almost surely. On each cell of length $h = (b - a) / J$ , use the cellwise $L^{2}$ projector $Q_{h}$ constructed in [F2] on $f$ . The resulting piecewise polynomial belongs to the spline space with full interior multiplicity $p + 1$ . Its dimension is $K = (p + 1) J$ by [F4], so the global basis theorem gives a deterministic coefficient vector ${\overset{ˉ}{β}}_{K}$ with $Q_{h} f = p_{K}^{T} {\overset{ˉ}{β}}_{K}$ . The projector family is local and reproduces $P_{p}$ , hence also $P_{s - 1}$ ; [F2] gives $∥ f - Q_{h} f ∥_{L^{2} (a, b)} \leq C_{s, p} h^{s} ∥ D^{s} f ∥_{L^{2} (a, b)}$ . The density bound implies $\int ∣ g_{0} - p_{K}^{T} {\overset{ˉ}{β}}_{K} ∣^{2} d P_{X} = ∥ f - Q_{h} f ∥_{L^{2} (P_{X})}^{2} \leq M_{X} ∥ f - Q_{h} f ∥_{L^{2} (a, b)}^{2}$ . Since $K = (p + 1) J$ and $p$ is fixed, $h ≍ K^{- 1}$ , so the squared approximation error is at most $C_{s, p, M_{X}, a, b} K^{- 2 s} ∥ D^{s} f ∥_{2}^{2}$ . This supplies the approximation hypothesis of [F1] with $γ = s$ and $C = C_{s, p, M_{X}, a, b}^{1 / 2} ∥ D^{s} f ∥_{2} \geq 0$ .

2.1A1A2F1step 1.1algebra∎

Insert this approximation bound into [F1]. The result is $Δ K / n + C_{s, p, M_{X}, a, b} K^{- 2 s} ∥ D^{s} g_{0} ∥_{2}^{2}$ . For $J ≍ n^{1 / (2 s + 1)}$ , both terms have order $n^{- 2 s / (2 s + 1)}$ because $K = (p + 1) J$ and $p$ is fixed. The criterion is evaluated at the training inputs in [F1], so no independent test-risk statement follows.

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Sources

Tom Lyche, Carla Manni, and Hendrik Speleers, B-Splines and Spline Approximation, Theorem 32, PDF pp. 28–29 (standard reference, not scraped)
Whitney K. Newey, Nonparametric and Semiparametric Estimation, MIT OCW 14.385, Fall 2007, Proposition 3 and proof, PDF pp. 10–11 (standard reference, not scraped)