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arXiv · 2505.15064

Why and When Deep is Better than Shallow: Implementation-Agnostic State-Transition Model of Deep Learning

Sho Sonoda·Yuka Hashimoto·Isao Ishikawa·Masahiro Ikeda
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Abstract

Why and when does depth improve generalization? We study this question in an implementation-agnostic state-transition model, where a depth-$k$ predictor is a readout class $H$ composed with the word ball $B(k,F)$ generated by hidden state transitions. Generalization bounds separate implementation error, approximation error, and statistical complexity, and upper bound the depth-dependent variance term by a Dudley entropy integral over $B(k,F)$, with a conditional lower-bound diagnostic under readout separation. We identify geometric and semigroup mechanisms that keep this entropy contribution saturated or polynomial, and contrast them with separation mechanisms that recover the classical exponential-growth obstruction. Coupling these variance upper bounds with approximation rates gives typical depth trade-off patterns, clarifying that depth is statistically favorable when approximation improves rapidly while the transition semigroup remains geometrically tame.

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Sho Sonoda, Yuka Hashimoto, Isao Ishikawa, Masahiro Ikeda. 2025-05-21. Why and When Deep is Better than Shallow: Implementation-Agnostic State-Transition Model of Deep Learning. https://arxiv.org/abs/2505.15064

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