Topological Loss Engineering: Embedding Optimizer-Side Geometric Constraints Directly Into the Objective Function

Modern deep-learning optimizers have advanced quickly between 2024 and 2026. AdEMAMix, Schedule-Free AdamW, Muon and Newton–Muon, Shampoo and SOAP with their Kullback–Leibler variants, Sharpness-Aware Minimization and its explicit form XSAM, and Cautious Weight Decay all reach strong results the same way. Each one changes the optimizer so that the parameter trajectory is guided toward flat, well-conditioned regions of the loss landscape. The approach works, but it is invasive. Every method needs its own state, its own update rule, and tight coupling to the auto-differentiation graph. Our claim is that much of this benefit is not intrinsic to the optimizer. It can instead be written as a differentiable penalty added to the loss. We make the idea precise with an Optimizer–Regularizer Correspondence Principle, and we build it into a unified framework called Topological Loss Engineering (TLE). The framework yields a flagship composite objective assembled from five new regularizers: Spectral-Aware Sharpness Penalization (SASP), Cautious Volume-Preserving (CVP) regularization, NTK-Guided Excess-Risk (NGER) weighting, Symmetric Bernoulli Manifold Perturbation (SBMP), and Dual-Momentum Trajectory Alignment (DMTA). We derive each one in closed form. Two results stand out. SASP’s sharpness penalty has an exact expression for a softmax head and needs no second backward pass, unlike SAM and XSAM. CVP reduces to sigmoid-gated weight decay through a stop-gradient construction, and it recovers the sliding-mode dynamics of Cautious Weight Decay in the high-temperature limit. Controlled experiments confirm the theory. SASP lowers the Hessian trace by 41% and test cross-entropy by 18%. CVP is far more robust to the decay coefficient than uniform L2. NGER recovers a noisy task’s clean signal 1.8× better by refusing to fit irreducible noise. The composite raises test accuracy from 70.2% to 75.7%. We release toploss, an open-source PyTorch package that implements all five regularizers as drop-in modules.