{
 "artifacts": [
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/1088/experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1088/report.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1088/results.json"
  }
 ],
 "category": "dynamics",
 "description": "Add a state-dependent stochastic reset to a neural-network parameter vector, optimizer state, or recurrent hidden state. The reset hazard is weak at large displacement but has the marginal inverse-square scaling that produces a predicted power-law excursion distribution and a sharp transition between localized training and runaway parameter drift.",
 "download_zip": "https://synthcore.org/download/1088",
 "formulas_latex": [
  "dz_t = sqrt(2D) dW_t, and z is reset to 0 at rate r(z) = r0/(z^2 + epsilon^2).",
  "D p''(z) - r(z) p(z) = 0, with p(z) proportional to |z|^(-beta), and beta(beta + 1) = r0/D.",
  "beta = (sqrt(1 + 4r0/D) - 1)/2, and localization occurs if and only if beta \u003e 1, equivalently r0/D \u003e 2.",
  "r(z) proportional to |z|^lambda: lambda \u003e -2 gives stretched-exponential tails, lambda = -2 gives power-law tails, and lambda \u003c -2 gives no asymptotic localization."
 ],
 "id": 2755,
 "implementation": "First, integrate the mechanism as an outer wrapper around SGD or Adam on a small model, resetting the full parameter vector to the best validation checkpoint. A safer first variant applies it only to optimizer momentum or to an RNN/SSM hidden state. Define the scalar excursion z = norm(theta - theta_ref), where theta_ref is the current reference checkpoint. After every optimizer update, compute the reset probability q = 1 - exp(-delta_t times r0 divided by (z squared plus epsilon squared)). Draw a uniform random number U; if U \u003c q, restore theta to theta_ref and optionally set Adam moments to zero. Pseudocode is: compute minibatch gradient; perform baseline optimizer update; compute z; compute q; sample U; restore the checkpoint if U \u003c q; otherwise continue. Keep theta_ref fixed initially, then test a version updated only after validation improvement. The paper supplies the inverse-square hazard, the power-law exponent, and the critical ratio r0/D = 2. Estimate D empirically from the covariance of parameter increments after subtracting their local mean drift: D approximately equals trace of the increment covariance divided by twice the step duration. The first cheap experiment should use a two-layer MLP on MNIST, comparing ordinary SGD, constant-rate checkpoint resets, and inverse-square resets across a grid of r0 values. Record excursion histograms, reset rates, validation loss, maximum excursion, and fitted tail exponent. The quantitative prediction is a transition near r0/D = 2: above it, excursions remain localized; below it, the complementary cumulative excursion distribution follows a power law with exponent approximately beta minus 1. The measured boundary should be within 20 percent of 2. As controls, a hazard proportional to z raised to minus 3 should fail to localize, while a hazard proportional to z raised to minus 1 should produce stretched-exponential rather than power-law tails.",
 "math_summary": "Approximate one scalar training excursion coordinate z by noisy diffusion dz = square root of 2D times dW, where D is effective minibatch-noise diffusivity and W is standard Brownian motion. Reset z to zero with hazard r(z) = r0 divided by z squared plus epsilon squared, where r0 is reset strength and epsilon prevents a singularity at the origin. Away from the reset point, the stationary Fokker-Planck equation is D times p double-prime minus r times p equals zero. For large absolute z, substituting p(z) proportional to |z| raised to minus beta gives beta times beta plus one equal to r0 divided by D. Therefore beta equals (square root of (1 + 4 r0/D) minus 1) divided by 2. The stationary distribution is normalizable only for beta greater than 1, equivalently r0/D greater than 2. The predicted delocalization boundary is therefore r0 = 2D. More generally, reset rates scaling as |z| raised to lambda produce stretched-exponential tails for lambda greater than -2, power-law tails at lambda equal to -2, and asymptotic nonlocalization for lambda less than -2.",
 "math_tags": [
  "statistical-mechanics",
  "stochastic-processes",
  "dynamical-systems",
  "bifurcations"
 ],
 "ml_areas": [
  "optimizer",
  "training-dynamics",
  "regularization",
  "rnn"
 ],
 "paper": {
  "arxiv_id": "2608.27090",
  "arxiv_url": "https://arxiv.org/abs/2608.27090",
  "summary_what_math_gives_to_ml": "The paper identifies a sharp localization threshold for diffusion with position-dependent stochastic resetting: if the reset rate scales as r(x) proportional to |x|^lambda, localization occurs for lambda greater than -2, while lambda less than -2 cannot confine the diffusive particle. The marginal inverse-square case, r(x) = r0 divided by x squared, produces power-law stationary tails and a finite-noise delocalization transition because normalizability depends on the ratio r0/D. A transferable neural-network mechanism is an adaptive stochastic reset of parameters, optimizer states, or recurrent hidden states toward a checkpoint, with reset hazard proportional to inverse squared distance; this gives a measurable stability boundary rather than an arbitrary reset schedule.",
  "title": "Localization Delocalization Transition in Diffusion with Adaptive Resetting",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 8,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "generalization"
 ],
 "title": "Inverse-Square Adaptive Parameter Reset",
 "url": "https://synthcore.org/idea/2755/inverse-square-adaptive-parameter-reset",
 "verification": {
  "peer_reviewed": false,
  "stage1_mechanism_check": {
   "worked": false,
   "confidence": 9,
   "verdict": "Built an inverse-square stochastic reset diffusion simulator, parameter sweeps, lambda controls, and a small SGD checkpoint-reset wrapper in experiment.py. The algebraic beta formula and critical ratio r0/D=2 are internally consistent, but measured tail exponents stayed near 4 across the sweep and did not track predicted beta values; the lambda classification was also not observed. The optimizer appeared better in one confounded sine-regression test, but this is not evidence for the proposed mechanism because it used online best-validation checkpoint restoration and an arbitrary hazard scale.",
   "metrics": {
    "baseline": "SGD validation MSE 6.2002 on the small noisy sine-regression stress test",
    "idea": "Inverse-square checkpoint-reset wrapper validation MSE 0.5158 with 353 resets; toy fitted density exponents approximately 3.90-4.70 for predicted beta values 0.366-2.372, with no confirmed transition near r0/D=2"
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "results.json",
    "REPORT.txt"
   ],
   "limitations": "The diffusion experiment used shortened trajectories and a finite reflecting box after the initial timeout, so asymptotic tail estimation was noisy and finite-time effects may matter. No MNIST experiment, rigorous stationary-distribution solver, fair multi-seed optimizer benchmark, or empirical estimation of D from minibatch parameter increments was performed."
  },
  "status": "mechanism_failed",
  "status_label": "Mechanism failed",
  "updated_at": "2026-09-02T00:07:39",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}