Generalization Bounds for Nonlinear Least Squares via Learned Feature Geometry
Researchers derived new generalization error bounds for ridge-regularized nonlinear least-squares models by analyzing the geometry of learned features through the empirical Jacobian Gram matrix. The bounds depend on data-dependent effective dimension and learned geometry rather than parameter count, with explicit mechanisms for networks like one-hidden-layer ReLU. This work advances theoretical understanding of how neural networks generalize by connecting algorithmic stability to the actual geometry of trained models.
A new theoretical analysis of nonlinear least-squares models provides generalization error bounds based on the geometry of learned features at trained parameters, rather than at initialization. The approach uses on-average algorithmic stability and the empirical Jacobian Gram matrix to derive bounds that scale with intrinsic data dimension for manifold-supported data and piecewise Lipschitz Jacobians. For linear models, the analysis recovers classical effective dimension results, while for nonlinear cases it introduces a residual-curvature term reflecting model nonlinearity. The bounds are derived using the Brascamp-Lieb inequality under strongly log-concave noise assumptions. Experiments on synthetic manifolds, clustered distributions, and benchmark datasets validate the tightness of the bounds and show agreement with observed generalization gaps, while also demonstrating compression of trained Jacobian features.
What's missing
The paper does not discuss computational complexity of calculating the proposed bounds in practice, nor does it compare empirical tightness against other recent generalization bound approaches (e.g., PAC-Bayes, margin-based bounds). The analysis assumes strongly log-concave noise, which may not hold for all real-world datasets.
What different sources said
- arXiv cs.LGCenter
Generalization Error Curves for Analytic Spectral Algorithms under Power-law Decay
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