{
 "artifacts": null,
 "category": "dynamics",
 "description": "Treat the optimization error as a Lyapunov-like state and adapt the learning rate so that its measured decrease follows a chosen stability degree. Instead of requiring exponential decrease, the controller targets dE/dt approximately equal to -c E^(1+m), which is appropriate near flat minima or marginally stable training regimes where exponential contraction may be impossible.",
 "formulas_latex": [
  "$$dE/dt \u003c= -c E^(1+m),   c \u003e 0,   m \u003e= 0$$",
  "$$E(t) \u003c= E(0) exp(-ct)  if m = 0;   E(t) \u003c= [E(0)^(-m) + mct]^(-1/m)  if m \u003e 0$$",
  "$$c_hat_k = (E_k - E_(k+1))/(delta_t E_k^(1+m))$$",
  "$$eta_k \u003c- eta_k times clip((c_hat_k/c_star)^alpha, eta_min/eta_k, eta_max/eta_k)$$"
 ],
 "id": 3126,
 "implementation": "Integrate this controller into SGD or Adam after computing a minibatch gradient but before permanently applying the update. Maintain a nonnegative monitored error E_k: on supervised learning use the current loss minus a running lower bound, clipped below by epsilon_E. Pseudocode: compute gradient g_k; propose theta_trial = theta_k - eta_k g_k; evaluate a small fixed monitoring minibatch and obtain E_trial; compute c_hat = (E_k - E_trial)/(E_k^(1+m) delta_t); reject the proposal and halve eta if E_trial exceeds E_k; otherwise accept it and update eta using the displayed multiplicative controller toward c_star. Smooth c_hat with an exponential moving average to reduce minibatch noise. The Lyapunov comparison law and degree interpretation come from the paper; c_star, the loss floor, smoothing coefficient, and rejection thresholds are empirical. First run a two-layer MLP on MNIST and a small ResNet on CIFAR-10, comparing SGD, Adam, and controllers with m equal to 0, 1, and 2. Plot loss on log-linear and log-log axes. The falsifiable prediction is exponential decay for m equal to zero and a final log-log slope near -1/m for m greater than zero; estimate the slope over the final third of training and require agreement within 20 percent. Also test learning rates around the acceptance boundary: the controller should reject steps when c_hat becomes nonpositive, and the measured transition from stable to divergent updates should be reproducible across random seeds.",
 "math_summary": "Let E(t) be a nonnegative training error, c greater than zero a dissipation constant, and m greater than or equal to zero the stability degree. If dE/dt is at most -c E^(1+m), comparison theory gives exponential decay E(t) at most E(0) exp(-ct) for m equal to zero, and algebraic decay E(t) at most [E(0)^(-m) + mct]^(-1/m) for m greater than zero. In a discrete optimizer, estimate c at step k by c_hat_k = (E_k - E_(k+1)) divided by delta_t E_k^(1+m). Adjust the step size toward a target c_star while rejecting steps that violate monotonic decrease. The measurable asymptotic signature is a log-log slope of -1/m for m greater than zero.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "optimization"
 ],
 "ml_areas": [
  "optimizer",
  "training-dynamics",
  "scheduler"
 ],
 "paper": {
  "arxiv_id": "2609.03120",
  "arxiv_url": "https://arxiv.org/abs/2609.03120",
  "summary_what_math_gives_to_ml": "The paper introduces a quantitative stability degree that distinguishes exponential convergence from algebraic convergence, with degree m greater than zero corresponding to decay proportional to t^(-1/m). Its transferable mechanism is a Lyapunov dissipation inequality whose exponent determines the long-time convergence law, together with direct and converse tests for certifying that exponent. In neural-network training, this can become an adaptive learning-rate controller and an online stability monitor that targets a prescribed decay exponent rather than merely checking whether the loss decreases.",
  "title": "On a Gradation for Asymptotic Stability",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 8
 },
 "solves": [
  "stability",
  "speedup"
 ],
 "title": "Polynomial-Lyapunov Training Controller",
 "url": "https://synthcore.org/idea/3126/polynomial-lyapunov-training-controller",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}