New Upper Bounds Derived for Learning Coefficients in Three-Layer Neural Networks
A new paper on arXiv presents a formula for computing upper bounds on local learning coefficients at singular realization parameters in three-layer neural networks. The learning coefficient, also known as the real log canonical threshold, governs Bayesian asymptotic behavior in singular learning models, but general methods for evaluating it in neural networks have remained limited. The result extends prior work to a broader class of activation functions and offers a systematic framework for understanding how weight parameters influence learning behavior.
Researchers have derived a formula providing upper bounds on local learning coefficients for a class of singular realization parameters in three-layer neural networks, addressing a gap in existing methods. Three-layer neural networks are singular learning models, meaning their Bayesian generalization behavior is governed by the real log canonical threshold, a quantity that has been difficult to evaluate broadly. Prior approaches using semiregular model formulas were restricted to nonsingular points and produced bounds that diverged substantially from known values at singular points. The new formula can be interpreted as a counting rule subject to budget, demand, and supply constraints, and applies in general non-polynomial real-analytic settings, with an additional restriction to networks with no hidden units in the polynomial case. It covers a wide range of activation functions, including the swish function and polynomial activations under the stated restriction. The authors also demonstrate that, when the input dimension is one, their upper-bound formula matches previously known exact learning coefficient values, providing a useful validation. The work offers a more systematic perspective on how the weight parameters of three-layer networks shape their learning coefficients.
What's missing
The paper is a preprint and has not yet undergone formal peer review. The upper-bound formula's tightness at general singular points beyond the one-dimensional input case remains an open question, as exact learning coefficient values for higher-dimensional inputs are not yet established for comparison.
What different sources said
- arXiv cs.LGCenter
Upper Bounds for Local Learning Coefficients of Three-Layer Neural Networks
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