Time-Reversed Terminal-Gradient Optimizer

Integrate this at the optimizer-update level, not inside the neural network. Maintain two optimizer variants: a conventional forward schedule and a time-reversed schedule with the same number of updates per training block. Parameterize a horizon-N momentum optimizer by positive learning rates alpha_k and bounded momentum coefficients beta_k, initialized from a cosine-decayed SGD schedule or standard momentum. Choose a proxy curvature set Lambda consisting of {mu, sqrt(mu L), L}, or use a logarithmic grid over [mu,L]. For every lambda, initialize x_{-1}=x_0=1 and run: g_k=lambda*x_k; x_next=x_k-alpha_k*g_k+beta_k*(x_k-x_{k-1}); then set x_{k-1}=x_k and x_k=x_next. Optimize R=max_lambda |lambda*x_N(lambda)| with an added penalty for intermediate amplification, such as gamma times max over k and lambda of |x_k(lambda)| squared. Reverse the learned coefficient arrays to create the second optimizer; do not reverse minibatch order or data augmentation. Apply the selected coefficients to every parameter tensor in each N-update block. Estimate mu and L either from a broad normalized proxy range or from minibatch gradient directional ratios; the proxy recurrence is exact, while the useful curvature range is empirical. Start with N=8 or 16 and compare SGD with momentum, cosine-decayed SGD, the learned forward schedule, and its reversed schedule on an MLP or ResNet-18 trained on MNIST or CIFAR-10. Use equal update counts and equal optimizer FLOPs. Record loss, global gradient norm at block endpoints, validation accuracy, peak gradient norm, and the fraction of unstable blocks. Success means lower terminal gradient norm and faster loss descent at equal compute, with no increase in gradient explosions; failure is no improvement over cosine SGD at matched update budget.