Lattice-based Deep Neural Networks: Regularity and Tailored Regularization

We give a nonasymptotic generalization theory for multilayer fully connected neural networks trained with path regularization, for general Lipschitz losses and without assuming the loss bounded, the architecture sparse, or any width or depth limit. Approximation error is modelled explicitly, and along the way we settle an open problem of E et al. on approximation rates in generalized Barron spaces. We treat the underparameterized and overparameterized regimes separately, which is what allows the resulting curve to be non-monotone in the width. We then prove lower bounds on the error itself, valid at every target and at every local minimizer, and combine them with the upper bounds into a theorem asserting that the error - not merely our bound on it - descends, rises and descends again as the width grows, with a peak exceeding the final plateau by a factor diverging in the sample size. Having both sides of the bound also identifies what drives the ascent, and it is not overfitting: once the regularization exceeds the dual norm of the noise, which the prescribed schedule does at every width, a slow-rate oracle inequality controls the error with no variance term at all. The rise is instead the shrinkage demanded by the regularization that the underparameterized regime requires, and we show it cannot be removed by any width-independent regularization under which the theory's guarantees hold. This mechanism is a property of the estimator analyzed here; we do not claim it accounts for the double descent reported for unregularized training.

We address the fundamental question of why deep neural networks generalize by establishing a pointwise generalization theory for fully connected networks. This framework resolves long-standing barriers to characterizing the rich nonlinear feature-learning regime and builds a new statistical foundation for representation learning. For each trained model, we characterize the hypothesis via a pointwise Riemannian Dimension, derived from the eigenvalues of the learned feature representations across layers. This establishes a principled framework for deriving hypothesis-dependent, representation-aware generalization bounds. These bounds offer a systematic upgrade over approaches based on model size, products of norms, and infinite-width linearizations, yielding guarantees that are orders of magnitude tighter in both theory and experiment. Analytically, we identify the structural properties and mathematical principles that explain the tractability of deep networks. Empirically, the pointwise Riemannian Dimension exhibits substantial feature compression, decreases with increased over-parameterization, and captures the implicit bias of optimizers. Taken together, our results indicate that deep networks are mathematically tractable in practical regimes and that their generalization is sharply explained by pointwise, feature-spectrum-aware complexity.

Production deep learning systems across enterprise domains operate under constraints that academic benchmarks routinely obscure: labeled data is expensive, inference budgets are tight, and models that cannot explain their behavior are difficult to trust and maintain. We present ChainzRule (CR), a neural architecture replacing typical activations with learnable polynomial layers governed by Differential Regularization (DREG), a layer-wise Jacobian penalty computed analytically during the forward pass at standard inference cost. The core claim is that bounding intermediate derivatives forces the network toward low-frequency, structurally stable representations, simultaneously reducing dependence on labeled data volume, improving robustness to distribution shift, and providing a measurable, gradient-based handle on model behavior. Evaluated across five domains, CR achieves $85.71\% \pm 2.01\%$ on Pima Diabetes (statistically superior to SVM and XGBoost), $46.20\% \pm 0.37\%$ on SST-5 sentiment classification with a frozen encoder (superior to RNTN using approximately 5% of its training data), $55.79\%$ on SST-5 with a fine-tuned BERT backbone (versus BERT-base linear head at $54.9\%$), $70.17\%$ on Yelp Full ordinal regression with 3.2M parameters versus a 10-model average of $66.35\%$, and $+2.32\%$ mean corruption accuracy on CIFAR-10-C. All results with reported $p$-values fall below the $α= 0.05$ threshold after Bonferroni correction. CR maintains a gradient tail ratio $τ$ (p99/mean) of $1.01$--$1.02$ against $1.07$--$1.09$ for all typical activation function baselines across every data fraction, a structural invariant we propose as the mechanistic driver of sample efficiency and a deployment-time proxy for model reliability.