Calibrated Stable-Window Early Stopping

Integrate the method into the training controller using validation loss sampled once per epoch or every fixed number of optimizer updates as $X_i$. Do not use raw minibatch losses directly; aggregate validation observations or block-average noisy measurements first. Maintain a rolling horizon of the most recent $n$ observations, normalize indices as $t_i=i/(n-1)$, choose a stable duration $\delta$ such as 0.1--0.2 of the horizon, and set $\Delta$ in validation-loss units per epoch. At each controller update: (1) fit a local linear regression around every interior index using weights $w_{ij}=K((t_j-t_i)/h)$, obtaining slope $\hat d_i$; (2) let $m=\lceil\delta(n-1)\rceil$ and compute each discrete window score $q_r=\max_{i=r,\ldots,r+m-1}|\hat d_i|$; (3) set $\hat d_{\infty}=\min_r q_r$ and record the minimizing window; (4) fit residuals from the local regressions and generate moving-block bootstrap sequences by resampling contiguous blocks of length $\ell\approx n^{1/3}$; (5) refit and recompute $\hat d_{\infty}^{*(b)}$ for $B=200$ replicates; (6) estimate a one-sided lower confidence bound from the empirical bootstrap quantile. Reduce the learning rate only when the bound supports $d_{\infty}<\Delta$ and the selected window reaches the current time. Stop only if this condition persists for two checks and the best checkpoint lies in or immediately after the detected window. The paper supplies the sliding-window statistic and localization principle; bandwidth $h$, block length $\ell$, and tolerance $\Delta$ are hyperparameters estimated or selected empirically. First compare against fixed patience and ReduceLROnPlateau using a small ResNet-18 on CIFAR-10 or a 2-layer Transformer on a toy language corpus. Track final test accuracy, optimizer updates, false plateau detections, and controller delay. A successful result is fewer updates at matched accuracy with fewer premature decisions under minibatch-size and noise changes.