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An NTK eigenmode decays exponentially under least squares

Example

Assume AC. Fix $N, n_{0} \geq 1$ , a depth $L \geq 2$ , a bias scale $β > 0$ , and a globally Lipschitz nonpolynomial activation $σ : R \to R$ . Use the source-normalized network with scalar output and independent standard-normal initialization, and let $x_{1}, \dots, x_{N}$ be pairwise distinct points of $S^{n_{0} - 1} = {x \in R^{n_{0}} : ∥ x ∥_{2} = 1}$ . Let $K_{\infty}$ be the limiting neural tangent Gram matrix on these inputs given by the sequential infinite-width theorem. Choose a unit eigenvector $v \in R^{N}$ of $K_{\infty}$ with eigenvalue $λ > 0$ . For labels $y \in R^{N}$ and $a \neq 0$ , initialize the limiting empirical least-squares kernel flow by $f_{0} = y + a v$ and solve

$\dot{f} (t) = - \frac{1}{N} K_{\infty} (f (t) - y) .$

Then $f (t) = y + a e^{- t λ / N} v$ . The error stays entirely in this one kernel mode, and its Euclidean norm decays by the factor $e^{- t λ / N}$ .

Example references

Jacot, Gabriel, and Hongler, §5, least-squares kernel eigenmodes

Distinct sphere inputs give strictly positive Gram eigenvalues

The deterministic limiting Gram matrix from the sequential width limit

Least-squares NTK flow decays mode by mode and fits distinct sphere data: Each residual eigenmode evolves by $e^{- t λ / N}$

Facts & Assumptions

Given: AC; the source-normalized scalar-output model; $N, n_{0} \geq 1$ , $L \geq 2$ , $β > 0$ , a globally Lipschitz nonpolynomial activation; pairwise distinct unit-sphere inputs; a unit eigenvector $v$ of their sequential limiting Gram matrix with eigenvalue $λ > 0$ ; target vector $y$ ; and initial prediction $f_{0} = y + a v$ with $a \neq 0$ .

[F1]

Distinct sphere inputs under the proposition's assumptions have a positive-definite limiting NTK Gram matrix (The limiting neural tangent kernel is strictly positive definite on every unit sphere).

[F2]

Least-squares kernel flow multiplies an eigenmode residual by $e^{- t λ / N}$ (Least-squares NTK flow decays mode by mode and fits distinct sphere data).

Verification

technique · substitute the single-mode initial residual into the exact eigenmode formula

1.1F1F2givenalgebra

The initial residual is $f_{0} - y = a v$ . Since $v$ is a unit eigenvector with eigenvalue $λ > 0$ , [F2] gives $f (t) - y = a e^{- t λ / N} v$ .

2.1F2step 1.1algebra∎

Taking norms and using $∥ v ∥_{2} = 1$ yields $∥ f (t) - y ∥_{2} = ∣ a ∣ e^{- t λ / N}$ , which tends to zero because $λ / N > 0$ . This is the single positive mode asserted in the Example.

Depends on

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Sources

Arthur Jacot, Franck Gabriel, and Clément Hongler, Neural Tangent Kernel: Convergence and Generalization in Neural Networks (standard reference, not scraped)