Delay-aware event-triggered optimizer MVP

Run with:

/home/maxwelhelp/main/bin/python3 experiment.py

Outputs are written to results.json.

Mathematical checks

The envelope multiplier was evaluated as gamma = exp(2*mu*d)*(q+r*epsilon) with mu=0.08, q=0.72, and r=0.35. The predicted boundary is d*=log(1/(q+r*epsilon))/(2*mu). A sweep with delay resolution 0.005 gave:

| epsilon | predicted d* | observed d* | relative error | |---:|---:|---:|---:| | 0.00 | 2.0532 | 2.055 | 0.09% | | 0.20 | 1.4733 | 1.475 | 0.12% | | 0.50 | 0.6933 | 0.695 | 0.24% | | 0.75 | 0.1103 | 0.115 | 4.22% |

Thus increasing trigger tolerance shrinks the certified maximum delay, as predicted.

For constant drift magnitude B=0.37 and fixed V=2.25, the measured event time was swept over sigma={0.05,0.1,0.2,0.4,0.8}. Every observed gap equaled sigma*sqrt(V)/B (ratio 1.0), confirming linear scaling and a positive non-Zeno gap.

Optimizer toy experiment

On a 20-dimensional diagonal quadratic for 250 steps (eta=0.11, delay 3):

| method | final loss | events/steps | |---|---:|---:| | ordinary SGD, no queue | 2.91e-13 | 250/250 | | fixed-delay SGD | 2.70e-30 | 250/250 | | event-triggered delayed SGD | 4.61e-18 | 83/250 |

The event-triggered method therefore reduced transmissions by 66.8% while still reaching very small loss. In the trigger sweeps, increasing epsilon reduced events (for example 60 to 26 at delay 0) and increased final loss modestly but did not destabilize this conservative quadratic setup.

Limitations

The Lyapunov boundary test uses the scalar equality/worst-case recurrence, so it verifies the stated formula rather than estimating mu,q,r from an independent nonlinear trajectory. The optimizer implementation is a small deterministic quadratic, uses a practical gradient-based proxy for V, and does not test MNIST, stochastic gradients, wall-clock communication cost, or a large distributed system.