{
 "artifacts": [
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1074/bench_report.json"
  },
  {
   "name": "delayed_gradient_bench.py",
   "url": "https://synthcore.org/code/1074/delayed_gradient_bench.py"
  },
  {
   "name": "delayed_gradient_experiment.py",
   "url": "https://synthcore.org/code/1074/delayed_gradient_experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1074/report.md"
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  {
   "name": "report_bench_2026-09-01T234709.md",
   "url": "https://synthcore.org/code/1074/report_bench_2026-09-01T234709.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1074/results.json"
  }
 ],
 "category": "dynamics",
 "description": "Add a controllable delay to the gradient force during optimization so that parameters follow a delayed-gradient dynamical system. Choose the delay below the stability boundary for ordinary training, and temporarily cross the boundary when the optimizer is trapped in a sharp or stagnant basin, causing stochastic fluctuations to be amplified out of the basin rather than waiting for a rare Arrhenius escape.",
 "download_zip": "https://synthcore.org/download/1074",
 "formulas_latex": [
  "$$\\frac{d z(t)}{d t}=-H z(t-\\tau)+\\sqrt{2D}\\,\\xi(t),\\qquad H=\\nabla^2 L(\\theta^*)$$",
  "$$\\lambda+k\\exp(-\\lambda\\tau)=0,\\qquad \\tau_c=\\frac{\\pi}{2k},\\qquad k\\tau\u003c\\frac{\\pi}{2}\\text{ stable},\\quad k\\tau\u003e\\frac{\\pi}{2}\\text{ unstable}$$",
  "$$T_{\\mathrm{escape}}\\simeq\\frac{1}{r}\\log\\left(\\frac{R}{\\sqrt{D}}\\right),\\qquad r=\\operatorname{Re}(\\lambda_+)\u003e0$$"
 ],
 "id": 2900,
 "implementation": "First, integrate the mechanism at the optimizer-update level. Maintain a FIFO queue of the last m parameter vectors or gradients, with physical delay tau=m times Delta t. During normal training use the current gradient. When validation loss or training loss has plateaued for P steps, start a bounded delayed-gradient burst using g at time t-m in the update theta at t+1 equals theta at t minus eta times g at t-m plus sigma times square-root-of-eta times a Gaussian noise vector. Before the burst, estimate k by applying 5 to 10 Hessian-vector power iterations to the current minibatch loss; set the critical queue length to pi divided by 2 k Delta t and choose m slightly above it, for example 1.1 times the critical value. Stop the burst when validation loss improves, displacement exceeds radius R, or gradient norms grow by a fixed factor. The paper supplies the characteristic equation, Hopf threshold, and logarithmic escape law. Estimate lambda max(H), effective D from minibatch-gradient covariance, growth rate r from a linear fit of log parameter displacement, and R from the first meaningful loss decrease. First test on a two-dimensional double-well loss and then an MLP on MNIST, comparing SGD, fixed stale-gradient SGD, and calibrated bursts. Sweep tau across 0.5, 0.9, 1.05, 1.3, and 2 times tau c. The predicted signatures are a displacement-growth transition within 20 percent of tau c=pi divided by 2 lambda max(H), bounded motion below threshold, and escape time approximately linear in log(R divided by square-root-of-D) above threshold.",
 "math_summary": "Near a local minimum theta star, let z(t)=theta(t)-theta star be the parameter displacement, H=nabla-squared-L(theta star) the Hessian, tau the delay, D the effective optimizer-noise intensity, and xi(t) unit white noise. The linearized delayed dynamics are dz/dt=-H z(t-tau)+sqrt(2D) xi(t). Along a Hessian eigenmode with curvature k greater than zero, exponential solutions have characteristic equation lambda+k exp(-lambda tau)=0. The first Hopf bifurcation occurs at k tau=pi divided by 2: the mode is stable below this value and unstable above it. If r=Re(lambda plus)\u003e0 is the dominant unstable growth rate, a fluctuation with amplitude proportional to sqrt(D) reaches escape radius R after T approximately equal to r inverse times log(R divided by sqrt(D)), predicting logarithmic rather than exponential dependence on barrier scale or inverse noise. In multiple dimensions, the earliest predicted instability uses k=lambda max(H).",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "bifurcations",
  "spectral-theory"
 ],
 "ml_areas": [
  "optimizer",
  "training-dynamics",
  "scheduler"
 ],
 "paper": {
  "arxiv_id": "2608.30624",
  "arxiv_url": "https://arxiv.org/abs/2608.30624",
  "summary_what_math_gives_to_ml": "The paper provides a transferable delayed-feedback mechanism that changes noise-activated escape from Arrhenius scaling, proportional to exp(Delta U divided by D), to logarithmic scaling after a delay-induced instability. Linearization gives a computable stability boundary, k times tau equals pi over 2, where k is the local restoring curvature. Beyond this boundary, stochastic fluctuations are amplified by an unstable delay mode until they reach an escape boundary, suggesting a controlled delayed-gradient phase for escaping sharp or stagnant optimization basins. The mechanism is most useful as a bounded optimizer burst with curvature-based delay selection and an automatic stop condition.",
  "title": "Time-delayed feedback turns Arrhenius escape logarithmic",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 8
 },
 "solves": [
  "speedup",
  "stability",
  "sample-efficiency"
 ],
 "title": "Bifurcation-calibrated delayed-gradient escape",
 "url": "https://synthcore.org/idea/2900/bifurcation-calibrated-delayed-gradient-escape",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
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    "idea_mean": 0.0002731473405219731,
    "delta_mean": 0.00003673418723337818,
    "per_seed_diffs": [
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    "idea_wins": 0,
    "n_pairs": 8,
    "p_value": 0.5022,
    "mde": 0.00006227732731692294,
    "mde_rel_pct": 26.342581389115686,
    "verdict": "no significant win",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": {
     "lr": 0.01
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    "sweep": [
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      "cfg": {
       "lr": 0.001
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     {
      "cfg": {
       "lr": 0.003
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     {
      "cfg": {
       "lr": 0.01
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    "full": {
     "mean": 0.0002364131532885949,
     "std": 0.00010524220575938558,
     "per_seed": [
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     "n": 8
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   },
   "idea": {
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    "n": 8
   },
   "mechanism_signature": {
    "baseline_max_displacement": 6.033182621002197,
    "confirmed": false,
    "displacement_ratio": 0.9999999999998342,
    "idea_max_displacement": 6.033182621002197,
    "k_proxy": 0.1229826378985967,
    "observed_delay_steps": 12,
    "predicted_delay_steps": 14.049755225604105,
    "prediction": "plateau-triggered delay amplifies parameter displacement"
   },
   "custom_track": null
  },
  "mechanism_confirmed": false,
  "peer_reviewed": false,
  "practical_verdict": "inconclusive",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 9,
   "verdict": "Built a reproducible delayed-gradient toy verification in delayed_gradient_experiment.py. The measured stability transition occurred between 0.9 and 1.05 tau_c, matching the predicted pi/(2k) boundary; unstable growth rates matched Re(lambda+) with mean absolute error 0.00088, and direct escape-time scaling matched the predicted slope 1/r within 0.13%. The mechanism is numerically real, but the small double-well comparison did not show a useful escape win: both methods had 0/12 basin escapes, while fixed delay had higher final loss (0.0322 versus 0.0000871).",
   "metrics": {
    "baseline": "Current-gradient noisy SGD: escape_fraction=0.0, median_escape_step=None, mean final loss=8.71e-05",
    "idea": "Delayed-gradient proxy with 44-step calibrated delay: escape_fraction=0.0, median_escape_step=None, mean final loss=0.0322; linear-system growth-rate MAE=8.82e-04 and escape-scaling slope relative error=0.13%"
   },
   "how_to_run": "python3 delayed_gradient_experiment.py",
   "files": [
    "delayed_gradient_experiment.py",
    "results.json"
   ],
   "limitations": "Only low-dimensional linear and double-well tests were run; no MNIST MLP, Hessian-vector estimator, adaptive plateau-triggered burst, full stochastic delayed-SDE escape measurement, FLOP accounting, or GPU experiment was tested. The double-well setup and noise were not tuned to produce frequent escapes."
  },
  "status": "failed_benchmark",
  "status_label": "Failed on benchmark",
  "updated_at": "2026-09-01T23:47:09",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}