When Do Early-Exit Networks Generalize? A PAC-Bayesian Theory of Adaptive Depth

Early-exit neural networks enable adaptive computation by allowing confident predictions to exit at intermediate layers, achieving 2-8$\times$ inference speedup. Despite widespread deployment, their generalization properties lack theoretical understanding -- a gap explicitly identified in recent surveys. This paper establishes a unified PAC-Bayesian framework for adaptive-depth networks. (1) Novel Entropy-Based Bounds: We prove the first generalization bounds depending on exit-depth entropy $H(D)$ and expected depth $\mathbb{E}[D]$ rather than maximum depth $K$, with sample complexity $\mathcal{O}((\mathbb{E}[D] \cdot d + H(D))/\epsilon^2)$. (2) Explicit Constructive Constants: Our analysis yields the leading coefficient $\sqrt{2\ln 2} \approx 1.177$ with complete derivation. (3) Provable Early-Exit Advantages: We establish sufficient conditions under which adaptive-depth networks strictly outperform fixed-depth counterparts. (4) Extension to Approximate Label Independence: We relax the label-independence assumption to $\epsilon$-approximate policies, broadening applicability to learned routing. (5) Comprehensive Validation: Experiments across 6 architectures on 7 benchmarks demonstrate tightness ratios of 1.52-3.87$\times$ (all $p < 0.001$) versus $>$100$\times$ for classical bounds. Bound-guided threshold selection matches validation-tuned performance within 0.1-0.3%.