From Sublinear to Linear: Local Convergence in Finite-Width Networks via Locally Polyak-Lojasiewicz Regions
arXiv:2507.21429v3 Announce Type: replace-cross Abstract: We study local linear convergence of gradient descent for finite-width feedforward networks under the squared empirical loss. Prior work shows that GD can remain confined to a Locally Quasi-Convex Region (LQCR) around initialization, but only gives a sublinear rate. We show that if the empirical Neural Tangent Kernel is positive at initialization, Lipschitz stable on the LQCR, and compatible with the LQCR radius, then the squared loss satisfies a local Polyak-{\L}ojasiewicz inequality with constant $\mu = \lambda_0 - L_\Theta r(\Rcal) >
This academic paper contributes to the ongoing theoretical research into the convergence properties of neural networks and gradient descent optimization.
While highly technical, understanding the convergence dynamics of finite-width networks is foundational for developing more robust and efficient AI models in the future.
No immediate change, but it refines the theoretical understanding of local linear convergence in specific network architectures, potentially influencing future algorithm design.
Further theoretical understanding of neural network training dynamics is advanced, improving the academic foundation of AI.
Over the long term, these theoretical insights might contribute to the development of more stable and faster-training deep learning models.
Improved theoretical guarantees could eventually lead to more reliable and predictable AI systems for critical applications.
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