Osgood-Calibrated Residual Step Size

Modify the residual scaling parameter, layer-drop controller, or continuous-depth solver rather than the optimizer update. Maintain a small calibration set of pairwise hidden states from each minibatch. Before applying block $\ell$, run a cheap trial with the current maximum step $\eta_{\max}$, compute $r_\ell$ and $r_\ell^{\mathrm{trial}}$ for each selected pair, and estimate $a_{ij,\ell}$ using one Hutchinson probe per endpoint. Use $\eta_\ell=\min(\eta_{\max},[G_\omega(r_\ell)+\delta-G_\omega(r_\ell^{\mathrm{trial}})]/(a_{ij,\ell}+\epsilon))$, clipped to $[\eta_{\min},\eta_{\max}]$. If the numerator is negative, halve $\eta$ and recompute the trial; if the budget is comfortably unused for several blocks, increase the next step by a factor such as $1.1$. Use a lookup table for $G_\omega$ and stop gradients through the controller initially, so it acts as a deterministic runtime policy. The mathematical component is the additive transformed-distance budget; empirical components are pair selection, probe-based Jacobian estimates, and safety margins. Start with a 32-layer residual MLP or small CIFAR ResNet, comparing fixed $\eta$, standard adaptive gradient clipping, and spectral normalization. Record accepted step sizes, number of trial evaluations, training loss at equal wall-clock time, gradient-norm spikes, and test accuracy under input perturbations. The hypothesis is that the controller permits larger average residual steps in smooth regions while preventing rare unstable expansions, producing faster loss descent and fewer divergence events.