analiza aiqbench
This paper establishes a theoretical lower bound demonstrating that the minimax risk in Gaussian regression exhibits an intrinsic quadratic dependence on network depth for deep ReLU architectures. By constructing explicit local packings, the authors prove this scaling is fundamental rather than an artifact of specific model choices or insufficient sample sizes. The findings clarify how representation-limited behavior transitions at smaller radii, offering critical insights into the capacity limits and optimization landscapes of very deep neural networks.
Depth Enables Local Entropy: Quadratic Depth Dependence in Deep Variation-Norm ReLU Regression
arXiv:2608.17434v1 Announce Type: new Abstract: We study Gaussian regression over the explicit vector-valued Parhi--Nowak deep-RBV^2 architecture with depth L, width w, layer-sum variation budget A, and output bound B. For this O(L w^2)-parameterized architecture, the known lower and upper bounds differ by one factor of depth. We construct a local packing showing that the quadratic depth dependence is intrinsic under an explicit sample-size-dependent radius condition. The packing has log-cardinality Omega(L^2 w^2 log w); its codewords lie in an O(lambda) L^2 ball and are pairwise Omega(lambda)-separated. The main ingredients are a bias-corrected bounded-coefficient approximation theorem and balanced amplification: multiplying a depth-D ReLU network by q can be implemented using one constant channel so that every coefficient grows by only q^(1/D). Translation to vector-valued RBV^2 blocks then has layer-sum cost O(D w^2 q^(1/D)). Gaussian Fano yields a radius-explicit lower bound governed by the output, testing, and representation scales. Under A=B=R, sigma proportional to R, and the stated radius condition, this gives minimax risk at least of order L^2 w^2 log(w) R^2/n. A pseudodimension-based finite-net upper bound gives O-tilde(L^2 w^2 R^2/n) for unbounded Gaussian responses. Thus the minimax risk has quadratic polynomial dependence on depth, up to logarithmic factors, and exhibits a transition to representation-limited behavior at smaller radius.
Abstract:We study Gaussian regression over the explicit vector-valued Parhi--Nowak deep-RBV^2 architecture with depth L, width w, layer-sum variation budget A, and output bound B. For this O(L w^2)-parameterized architecture, the known lower and upper bounds differ by one factor of depth. We construct a local packing showing that the quadratic depth dependence is intrinsic under an explicit sample-size-dependent radius condition. The packing has log-cardinality Omega(L^2 w^2 log w); its codewords lie in an O(lambda) L^2 ball and are pairwise Omega(lambda)-separated. The main ingredients are a bias-corrected bounded-coefficient approximation theorem and balanced amplification: multiplying a depth-D ReLU network by q can be implemented using one constant channel so that every coefficient grows by only q^(1/D). Translation to vector-valued RBV^2 blocks then has layer-sum cost O(D w^2 q^(1/D)). Gaussian Fano yields a radius-explicit lower bound governed by the output, testing, and representation scales. Under A=B=R, sigma proportional to R, and the stated radius condition, this gives minimax risk at least of order L^2 w^2 log(w) R^2/n. A pseudodimension-based finite-net upper bound gives O-tilde(L^2 w^2 R^2/n) for unbounded Gaussian responses. Thus the minimax risk has quadratic polynomial dependence on depth, up to logarithmic factors, and exhibits a transition to representation-limited behavior at smaller radius.
Artificial Intelligence (cs.AI)
arXiv:2608.17434 [cs.AI]
(or arXiv:2608.17434v1 [cs.AI] for this version)
https://doi.org/10.48550/arXiv.2608.17434 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Text integral sub licența deschisă a publicației · 295 cuvinte
Subiecte conexe
Latent reasoning architectures would likely undermine CoT, our strongest oversight tool
AI Alignment Forum · 23 sept. 2026
J++ Lens: Jacobian Filtering Enables More Faithful Workspace Lenses
LessWrong · 8 oct. 2026
PIRAMID: Progress and Plans
LessWrong · 27 iul. 2026
Expenditure Horizon: Measuring Optimization Ability, with an Application to NanoGPT
METR · 21 iul. 2026