Deadline-Adaptive Gradient Flow

1. Integration point: implement this as an optimizer wrapper around SGD, initially without momentum. At the beginning of a training phase, record a smoothed initial loss estimate $L_0$, choose a deadline $T$ in optimizer steps, target $\varepsilon$, exponent $q\in\{0.25,0.5,0.75\}$, and numerical floor $\delta=10^{-12}$. Estimate the local PL constant using $\widehat\mu=\operatorname{clip}(\|g\|^2/(2L),\mu_{\min},\mu_{\max})$ with exponential smoothing. 2. Pseudocode: compute minibatch loss $L$ and gradient $g$; update $\widehat\mu$; compute $k_{\mathrm{target}}$ from the displayed formula; optionally use $r=\operatorname{clip}((L/L_0)^{-a},r_{\min},r_{\max})$ with $a=0.25$; form $v=(\|g\|^2+\delta)^{(q-1)/2}g$; update $\theta\leftarrow\theta-hk_{\mathrm{target}}rv$, where $h$ is the integration step. If the observed loss is above the predicted comparison trajectory, increase $r$ modestly; if it is below, decrease it. Clip updates only for numerical safety and log clipping frequency. 3. Computed versus estimated: the differential inequality and gain formula are analytical; $L_0$, $\widehat\mu$, minibatch gradients, and the actual settling time are empirical. The PL assumption should be checked by plotting $\|g\|^2/(2L)$ and its lower quantiles. 4. First cheap experiment: train a two-layer MLP on MNIST and a small ResNet on CIFAR-10 using vanilla SGD, cosine SGD, Adam, and this method at equal compute. Test deadlines equal to 25%, 50%, and 100% of baseline training length. The quantitative prediction is that, during a PL-like late-training segment, $L(t)^{(1-q)/2}$ decreases approximately linearly with slope $-k(1-q)(2\widehat\mu)^{(q+1)/2}/2$, and the time to reach $\varepsilon$ scales approximately as $1/k$ and matches the requested deadline within 20%. Multiply the predicted gain by 2, 4, and 8 to identify a reproducible divergence or oscillation boundary.