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This enables us to show that global minima with small weight norms exhibit strong generalization guarantees that are dimension-independent. In contrast, local minima that are \u201cflat\u201d can generalize poorly as the input dimension increases. We attribute this gap to a phenomenon we call neural shattering, where neurons specialize to extremely sparse input regions, resulting in activations that are nearly disjoint across data points. This forces the network to rely on large weight magnitudes, leading to poor generalization. Our analysis establishes an exponential separation between flat and low-norm minima. In particular, while flatness does imply some degree of generalization, we show that the corresponding convergence rates necessarily deteriorate exponentially with input dimension. 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