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Kernel regression estimates a regression function using localized kernel weights rather than a fixed finite-dimensional functional form. The Nadaraya–Watson estimator is the local-constant version; local polynomial regression improves its local approximation by fitting a polynomial.

This local-constant estimator is one particular kernel regression method.

The Nadaraya–Watson estimator is the degree-zero local polynomial estimateAt a support boundary it generally has first-order bias because it reproduces constants but not linear functions.

For a differentiable neural network $f(\theta ,x)$ and initialization law $P$, the expected neural tangent kernel is $k_{\mathrm{NTK}}(x,x^{\prime })={\mathbb{E}}_{\theta \sim P}⟨{\nabla }_{\theta }f(\theta ,x),{\nabla }_{\theta }f(\theta ,x^{\prime })⟩$, assuming finite diagonal expectations. At a fixed parameter value the corresponding inner product defines the empirical neural tangent kernel. Both are positive-semidefinite kernels, since a finite quadratic form is respectively an expected squared norm or a squared norm. In suitable infinite-width scalings, the empirical kernel matrix converges to a deterministic limit and remains constant during training. Squared-error gradient flow then has the same training-prediction dynamics as kernel regression with this limit, provided the initial predictions match.

Evaluating the negative functional gradient at $x_j$ yieldsThus the evaluation vector $v(t)=(g_t(x_1),\dots ,g_t(x_n){)}^T$ followsThis is exactly the limiting prediction dynamics in part (ii). If $v(0)=u(0)$, subtracting the two systems gives $\frac{d}{dt}(v-u)=-H^{\ast }(v-u)$ with zero initial value, so $v(t)=u(t)$ for every $t$.

An infinitely wide neural network in this regime follows the same training-prediction dynamics as kernel regression with its neural tangent kernel and matching initial predictions. For this squared-error loss function, the relevant algorithm is gradient flow in the Reproducing kernel Hilbert space. Matching predictions at arbitrary additional inputs also requires the corresponding cross-input neural tangent kernel to freeze and the initial predictions there to match. A finite-width neural network can change its feature map during training, so the equivalence depends on the assumptions rather than holding for every architecture and parameterization.