{
 "artifacts": [
  {
   "name": "REPORT.md",
   "url": "https://synthcore.org/code/1073/REPORT.md"
  },
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1073/bench_report.json"
  },
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/1073/experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1073/report.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1073/results.json"
  },
  {
   "name": "stage2_bench.py",
   "url": "https://synthcore.org/code/1073/stage2_bench.py"
  }
 ],
 "category": "dynamics",
 "description": "Replace every-step parameter communication or correction by an impulsive update emitted only when the local optimization state has drifted sufficiently from its last transmitted value. The correction is executed after a known or measured delay, and the trigger threshold is selected so that stale updates remain inside a Lyapunov-certified stability region while reducing communication and redundant optimizer work.",
 "download_zip": "https://synthcore.org/download/1073",
 "formulas_latex": [
  "$$\\dot z=F(z),\\qquad z((t_k+d_k)^+)=z((t_k+d_k)^-)+K e_k,\\qquad e_k=z(t_k)-\\hat z(t_k),\\quad 0\\le d_k\\le d_{\\max}.$$",
  "$$\\dot V\\le 2\\mu V+\\alpha\\|e_k\\|^2,\\qquad V((t_k+d_k)^+)\\le qV((t_k+d_k)^-)+r\\|e_k\\|^2.$$",
  "$$\\exp(2\\mu d_{\\max})(q+r\\varepsilon)\u003c1,\\qquad \\|e_k\\|^2\\le\\varepsilon V(t_k)\\ \\Longrightarrow\\ V_{k+1}\\le\\rho V_k,\\quad \\rho\u003c1.$$",
  "$$\\tau_{k+1}-\\tau_k\\gtrsim\\frac{\\sigma\\sqrt{V(t_k)}}{B},\\qquad \\|e(t)\\|\\ge\\sigma\\sqrt{V(t)}\\ \\text{is the event condition},\\quad B\\ge\\|\\dot e(t)\\|.$$"
 ],
 "id": 2890,
 "implementation": "Integrate the mechanism at the optimizer communication or update layer, initially using SGD on a small quadratic problem. Maintain parameters theta, a last-transmitted reference theta_hat, and a queue of corrections with future execution times. At optimizer step k, compute gradient g_k and the proposed correction u_k=-eta g_k. Trigger an event only when the observable drift satisfies ||theta_k-theta_hat||^2 \u003e epsilon V_k, using V_k=||g_k||^2+lambda||theta_k-theta_hat||^2 for a practical proxy. When triggered, enqueue u_k with execution time k+d and set theta_hat=theta_k. At each step, apply all queued corrections whose execution time has arrived; this explicitly creates stale delayed impulses. Pseudocode is: compute g; perform the local optimizer step; evaluate drift and V; if drift^2 exceeds epsilon V, enqueue the correction and refresh theta_hat; apply due corrections; record V and event times. Estimate mu from the maximum observed local secant growth rate, and estimate q and r by fitting the ratio of post-event to pre-event quadratic energies. The theoretical mechanism is the multiplier exp(2 mu d)(q+r epsilon); these quantities are empirically estimated rather than assumed exact. First test one-dimensional and 20-dimensional quadratic regression, then MNIST logistic regression, comparing ordinary SGD, fixed-delay SGD, and the event-triggered variant. Sweep delay d and threshold epsilon. The quantitative prediction is a divergence boundary near exp(2 mu d)(q+r epsilon)=1: below the boundary, the energy envelope decays geometrically; above it, delayed corrections cause oscillation or divergence. The measured boundary should agree with the estimated condition within 20 percent, and event times should have a positive minimum gap rather than accumulating.",
 "math_summary": "Let z(t) denote an optimization state, F(z) its continuous or interpolated training dynamics, e_k=z(t_k)-z_hat(t_k) the drift from the last transmitted state, K the correction matrix, and d_k the actuation delay with 0 \u003c= d_k \u003c= d_max. Use the quadratic Lyapunov function V(z)=z^T P z, where P is positive definite. Assume the dynamics satisfy the one-sided Lipschitz bound (z-y)^T P(F(z)-F(y)) \u003c= mu times ||z-y||_P^2. During the delay, suppose the Lyapunov function obeys V_dot \u003c= 2 mu V + alpha ||e_k||^2. At the delayed impulse, suppose V_plus \u003c= q V_minus + r ||e_k||^2. If the trigger enforces ||e_k||^2 \u003c= epsilon V(t_k), a sufficient delayed contraction condition is exp(2 mu d_max)(q+r epsilon)\u003c1. If the drift satisfies ||e_dot|| \u003c= B and an event is emitted when ||e|| \u003e= sigma sqrt(V), the local inter-event time is bounded below by approximately sigma sqrt(V)/B, preventing Zeno accumulation.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "optimization",
  "stability"
 ],
 "ml_areas": [
  "optimizer",
  "training-dynamics",
  "scheduler"
 ],
 "paper": {
  "arxiv_id": "2608.24074",
  "arxiv_url": "https://arxiv.org/abs/2608.24074",
  "summary_what_math_gives_to_ml": "The paper provides a transferable mechanism for stabilizing coupled nonlinear systems when control events are detected at one time but executed after an actuation delay. Its main assets are a delay-dependent Lyapunov contraction condition, a positive lower bound on inter-event times that excludes Zeno behavior, and a Laplacian spectral rule for selecting pinned nodes. A neural-network analogue is an event-triggered optimizer that applies stale queued updates only when parameter or representation drift exceeds a threshold, together with a spectral strategy for anchoring a small subset of communicating modules or workers.",
  "title": "Event-Triggered Pinning Impulsive Control of Complex Networks with Actuation Delays: Stability Analysis and Zeno-Free Conditions",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 8
 },
 "solves": [
  "stability",
  "speedup"
 ],
 "title": "Delay-aware event-triggered optimizer",
 "url": "https://synthcore.org/idea/2890/delay-aware-event-triggered-optimizer",
 "verification": {
  "peer_reviewed": false,
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 9,
   "verdict": "Built a delayed event-triggered SGD MVP with queued stale corrections, practical Lyapunov-proxy triggering, and deterministic toy sweeps. The predicted delayed-contraction boundary matched the observed sweep within 0.09–4.22%, and inter-event gaps matched the predicted linear sigma*sqrt(V)/B scaling exactly. On a 20-dimensional quadratic, event triggering reduced transmissions by 66.8% while reaching final loss 4.61e-18, although this is only a small deterministic toy result.",
   "metrics": {
    "baseline": "Ordinary SGD: final loss 2.91e-13 after 250 steps; fixed-delay SGD: 2.70e-30 with 250/250 events.",
    "idea": "Event-triggered delayed SGD: final loss 4.61e-18 after 250 steps, 83/250 events, 66.8% communication reduction, minimum event gap 3 steps. Contraction-boundary relative errors were 0.09%, 0.12%, 0.24%, and 4.22% for epsilon values 0, 0.2, 0.5, and 0.75."
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "results.json",
    "REPORT.md"
   ],
   "limitations": "The Lyapunov boundary uses an exact scalar worst-case recurrence rather than independently estimated dynamics. The optimizer test is a small diagonal quadratic with deterministic gradients; MNIST, stochastic gradients, wall-clock communication cost, learned or measured Lyapunov constants, and distributed hardware were not tested."
  },
  "status": "unverified",
  "status_label": "Unverified",
  "updated_at": "2026-09-01T23:48:59",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}