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From Approximation Rates to Loss-Landscape Barrier Decay in Shallow ReLU Networks

We study pathwise connectivity of sublevel sets for one-hidden-layer ReLU networks with constrained first-layer weights and an $\ell_1$ penalty on the output layer. The data term is assumed convex and globally Lipschitz in the scalar logit. We first give a finite-width construction that connects any two points of a common sublevel through a path controlled by a loss-consistent compression functional and a first-order perturbation term. The proof replaces the quadratic perturbation estimate in the Freeman--Bruna mechanism by a direct Lipschitz bound. Positive homogeneity is then used in a direction that is compatible with the penalty: every active atom is moved monotonically from the unit ball to the unit sphere while its output coefficient is reduced. Sphere covering and cluster merging consequently give $O(m^{-1/(n-1)})$ fixed-level thickening for $n\ge2$, while the one-dimensional two-ray dictionary gives exact connectivity for every $m\ge4$. We also prove internally that the regularized approximation values satisfy $e(l)-e_\infty=O(l^{-1/2})$. More generally, a rate $O(l^{-s})$ transfers to a near-optimal barrier rate $O(m^{-s/((n-1)s+1)})$; under the standing assumptions, this yields the explicit rate $O(m^{-1/(n+1)})$. A theorem-aligned finite-distribution experiment complements the analysis. The primary Huber run yields a maximal best certified upper gap $1.66\times10^{-5}$ over 720 recorded pairs at widths $m\ge16$; a matched binary-cross-entropy rerun and a 720-endpoint dense-representation stress test probe loss robustness and the active cluster-merging mechanism.

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