Smooth Barrier Tube Controller

Use this as a training-time controller for either parameters or selected hidden states. For an MLP, define margins $h_j$ for every layer as $H-\|z_j\|_2$ and optionally define a representation-separation margin $h_{\mathrm{sep}}=\|\mu_a-\mu_b\|_2-c$ for class prototypes. For a transformer, add attention-entropy and logit-range margins, for example $h=H_{\max}-\|qk^T/\sqrt d\|_\infty$. Compute the ordinary gradient $u_0=-\nabla\mathcal L$, compute barrier gradients by automatic differentiation, and use $u=u_0-\kappa\nabla B$. Before applying the update, estimate the right side of the step condition with current margins and gradient norms; reduce $\eta$ by backtracking until $\eta\|u\|\leq\alpha\min_j h_j/\|\nabla h_j\|$. If a margin is already nonpositive, restore the last safe checkpoint and use a squared hinge recovery term instead of the logarithm. The paper-derived mechanism is continuous barrier-based obstacle avoidance combined with a nominal objective; the exact neural safety margins and their thresholds must be selected empirically. The first cheap test should use a small transformer on CIFAR-10 or a 2-layer MLP on MNIST, comparing ordinary training, gradient clipping, and the barrier controller under a deliberately high learning rate and activation perturbations. Sweep $\eta$ and record the first step at which any $h_j\leq0$. The quantitative prediction is that the barrier method maintains $h_j>0$ whenever the adaptive step inequality holds, while unconstrained training will cross the boundary; near the boundary, the barrier gradient magnitude should scale as $1/h_{\min}$ and the accepted step size should scale linearly with $h_{\min}$.