Emergent Low-Dimensional Topological Dynamics Dictate Generalization and Escape Mechanics in Billion-Parameter Neural Networks
The remarkable generalization of modern over-parameterized neural networks defies classical statistical learning bounds. Here we establish that optimization trajectories in billion-parameter models do not sample the ambient dimension $\mathbb{R}^N$ homogeneously; instead, they are dynamically constrained to a low-dimensional execution submanifold $\mathcal{M}_{\text{exec}}$ (with empirically observed effective dimension $d_{\text{eff}} \le 12$). By equipping the parameter space with a regularized Hessian metric $g_{ij} = \nabla_i \nabla_j L + εδ_{ij}$ and analyzing stochastic Langevin diffusion on loss sublevel sets, we derive a rigorous Topological Generalization Bound grounded in projected subspace persistent homology. Furthermore, by linking Morse index transitions to Betti number annihilation ($β_1 \to 0$), we formulate an exact mathematical mechanism for exponential escape from strict saddles under anisotropic noise. We validate our framework across 80+ architectures (ResNets, Vision Transformers, and 7B LLMs). To bridge theoretical topology with computational feasibility, we resolve the curse of dimensionality by computing projected persistent homology proxies via randomized Lanczos iterations restricted strictly within active Hessian subspaces. Building upon this mathematical foundation, we introduce Topological-Adaptive Optimization (Top-SGD and Top-AdamW)---a family of wrapper algorithms that dynamically modulate learning rates using active curvature signatures. Extensive benchmarks demonstrate a 14.1\% reduction in total wall-clock training time on ImageNet with all computational and memory overheads fully accounted for. Code is publicly available at https://github.com/imslab-research/Topological-SGD.