Correlation-Window Training Regime Detector

1. Exact integration point: add a streaming monitor to synchronous or federated training, after every optimizer step and before the next learning-rate decision. For each replica, record one or more scalar traces: gradient norm, update norm, selected-layer activation RMS, or step duration. Do not feed raw traces to a neural classifier. Maintain rolling means, variances, and cross-products for a window of $M$ samples; use exponentially weighted updates only if a fixed window is too expensive. 2. Pseudocode: initialize a ring buffer for each replica; every step append $x_i$, center each window, calculate all pairwise $\hat r_{ij}$, apply $z_{ij}=\operatorname{atanh}(\operatorname{clip}(\hat r_{ij},-0.999,0.999))$, average to obtain $\bar z$, estimate $\tau_c$ from the first zero crossing or integrated autocorrelation of a representative trace, set $M_{\mathrm{eff}}=M\Delta t/(2\tau_c)$, and compute $S=\bar z\sqrt{M_{\mathrm{eff}}-3}$. If $S>h$ for two consecutive windows and the rolling trace variance exceeds its baseline, reduce the learning rate by $\gamma_{\mathrm{down}}$ or increase gradient accumulation. 3. Computed from the paper's mechanism: pairwise correlations, compact regime classification, and confidence improvement with observation length. Estimated empirically: $\tau_c$, the independent-regime null distribution, threshold $h$, and the correlated-regime mean $\bar z_{\mathrm{corr}}$. Calibrate $h$ from independent shuffled traces rather than assuming Gaussianity. 4. First cheap experiment: train an 8-replica ResNet-18 on CIFAR-10 or a small Transformer on WikiText-2. Construct alternating 200-step regimes: independent per-replica minibatch ordering and synchronized minibatch, augmentation, or injected sinusoidal compute modulation. Compare this detector with a raw-waveform MLP and with no controller. Measure detection delay, false-alarm rate, and loss spikes. The quantitative prediction is $S\propto\sqrt{T/(2\tau_c)}$; across windows of 1, 2, and 4 times the estimated correlation time, standardized separation should scale by approximately $1:\sqrt{2}:2$, and the minimum window predicted by $T_{\min}\approx2\tau_c(h/\Delta z)^2$ should match observed 90%-confidence detection delay within roughly 20%.