Anherutowa Calvo

Anherutowa Calvo

The curvature exponent $α$ in $h_k \propto σ_k^α$ -- governing how Hessian eigenvalues scale with gradient singular values -- varies systematically across layer types ($α\approx 2$ for convolutions, $\approx 1$ for transformer attention, $< 1$ for MLP up-projections). Why? We prove the Spectral Alignment Decomposition: $α= 2 + d\logΦ_k / d\logσ_k$, where $Φ_k$ measures alignment between Kronecker factor eigenbases and gradient singular directions. This reduces "why does $α$ vary?" to a geometric question we answer for LayerNorm, residual connections, and softmax heads. The decomposition implies a spectral transfer identity $s = αγ$ linking curvature exponent, effective gradient rank-decay $γ$, and Hessian decay exponent $s$. The identity is algebraic; its empirical content is that $α$ and $γ$, fit on independent data (HVPs vs. SVD), recover $s$ to ~2% median error across 93 layers, five architectures, and three datasets -- with no free parameters. A zeta-function bound on participation ratio shows curvature concentrates onto effectively one direction per layer. As a proof of concept, we derive the architecture-adaptive preconditioner $T(σ;α)$ and show that Spectral Newton -- implementing $T$ in the gradient singular basis -- outperforms AdamW on vision benchmarks where $α\approx 2$.

Anherutowa Calvo

Deep learning optimization exhibits structure that is not captured by worst-case gradient bounds. Empirically, gradients along training trajectories are often temporally predictable and evolve within a low-dimensional subspace. In this work we formalize this observation through a measurable framework for predictable gradient manifolds. We introduce two computable quantities: a prediction-based path length that measures how well gradients can be forecast from past information, and a predictable rank that quantifies the intrinsic temporal dimension of gradient increments. We show how classical online and nonconvex optimization guarantees can be restated so that convergence and regret depend explicitly on these quantities, rather than on worst-case variation. Across convolutional networks, vision transformers, language models, and synthetic control tasks, we find that gradient trajectories are locally predictable and exhibit strong low-rank structure over time. These properties are stable across architectures and optimizers, and can be diagnosed directly from logged gradients using lightweight random projections. Our results provide a unifying lens for understanding optimization dynamics in modern deep learning, reframing standard training as operating in a low-complexity temporal regime. This perspective suggests new directions for adaptive optimizers, rank-aware tracking, and prediction-based algorithm design grounded in measurable properties of real training runs.