From Sublinear to Linear: Fast Convergence in Deep Networks via Locally Polyak-Lojasiewicz Regions

Gradient descent (GD) on deep neural network loss landscapes is non-convex, yet often converges far faster in practice than classical guarantees suggest. Prior work shows that within locally quasi-convex regions (LQCRs), GD converges to stationary points at sublinear rates, leaving the commonly observed near-exponential training dynamics unexplained. We show that, under a mild local Neural Tangent Kernel (NTK) stability assumption, the loss satisfies a PL-type error bound within these regions, yielding a Locally Polyak-Lojasiewicz Region (LPLR) in which the squared gradient norm controls the suboptimality gap. For properly initialized finite-width networks, we show that under local NTK stability this PL-type mechanism holds around initialization and establish linear convergence of GD as long as the iterates remain within the resulting LPLR. Empirically, we observe PL-like scaling and linear-rate loss decay in controlled full-batch training and in a ResNet-style CNN trained with mini-batch SGD on a CIFAR-10 subset, indicating that LPLR signatures can persist under modern architectures and stochastic optimization. Overall, the results connect local geometric structure, local NTK stability, and fast optimization rates in a finite-width setting.

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