Condition-number-aware restarted PAGE

Integrate this into an optimizer for a model trained on a finite dataset, treating each example or minibatch shard as a component function. Maintain x_prev and v_prev. At the start of each restart epoch compute the exact full-data gradient v=∇F(x); inside the epoch, either refresh exactly with probability p or sample a minibatch B and set v_t=v_{t-1}+|B|^{-1}Σ_{i∈B}[∇f_i(x_t)-∇f_i(x_{t-1})]. Then update x_{t+1}=x_t−ηv_t. Every K steps estimate the local PL parameter from recent checkpoints: μ_hat=median(||g_j||²/(2 max(F_j−F_best,δ))). Estimate mean-squared smoothness using random probes u_j and minibatch gradients G: L_hat_ms²=median(||G(x+u_j)−G(x)||²/||u_j||²). Set κ_hat=L_hat_ms/max(μ_hat,δ). If κ_hat≤√n, select a short restart phase T from a grid {1,2,4,...,√n}; if κ_hat>√n, select T from {√n,2√n,...,κ_hat√n}. Adjust p approximately as one full refresh per phase, while clipping η to a conservative smoothness-based bound such as 0.1/L_hat_ms. Use stop-gradient around all estimates so they do not alter model gradients. First test on a strongly convex quadratic with controlled κ, then CIFAR-10 ResNet-18 and a 50M-parameter Transformer. Compare AdamW, fixed-schedule PAGE, and adaptive PAGE at equal optimizer steps, component-gradient FLOPs, parameters, and memory. Pre-register: for κ<√n, adaptive PAGE must reduce calls to a fixed objective gap by at least 20% versus fixed PAGE; for κ>√n, its cost must be within 10% of fixed PAGE; and removing exact refreshes must increase measured estimator variance by at least 2x. Record loss-versus-component-calls, gradient variance, estimated κ, and selected T. Falsify the transfer if the low-κ advantage is absent on controlled quadratics or if adaptive scheduling increases high-κ cost by more than 10%.