Asymptotic Training-Horizon Controller

Integrate this at the training loop's checkpoint and evaluation path, not inside the forward or backward pass. Every K optimizer steps, record a smoothed validation loss y_i, the checkpoint index t_i, and optionally training loss and gradient norm as separate series. Choose a small candidate basis, for example phi(t)=[1,t^{-1/2},t^{-1},t^{-3/2}], after rescaling t by the first checkpoint to avoid numerical magnitude problems. For each new checkpoint and each window size w in {8,16,32}, solve the ridge system (X^T X+lambda D^T D)theta=X^T y, with D=diag(0,1,1,1), lambda between 1e-4 and 1e-2, and compute y_infinity=theta[0]. Estimate expected improvement over H future checkpoints with theta^T(phi(t)-phi(t+H)). Compute a stability score from the standard deviation of y_infinity across window sizes and from the fit residual; do not stop unless both are below thresholds for three consecutive evaluations. Pseudocode: record y; for each w construct X and solve the ridge system; obtain y_infinity,w and delta_w; if median(delta_w)<epsilon and std(y_infinity,w)<tau, terminate training, otherwise continue. The mathematics supplies the estimator and extrapolated limit; epsilon, tau, basis exponents, and H must be calibrated empirically. First test on CIFAR-10 with a small ResNet-18 and on a 100M-parameter language-model fine-tuning run, comparing fixed-budget training and ordinary patience-based early stopping. Measure compute saved at equal final validation loss, premature-stop rate, and correlation between predicted and realized future improvement. A useful signal is at least 15% fewer optimizer steps with no statistically significant degradation in final validation loss; instability or large disagreement across windows should correctly disable stopping.