# Inverse-Square Adaptive Parameter Reset

- ID: 2755
- Canonical URL: https://synthcore.org/idea/2755/inverse-square-adaptive-parameter-reset
- API JSON: https://synthcore.org/api/idea/2755.json
- API Markdown: https://synthcore.org/api/idea/2755.md
- Verification status: mechanism_failed
- Source: [arXiv:2608.27090](https://arxiv.org/abs/2608.27090)
- Category: dynamics
- Solves: stability, generalization
- ML areas: optimizer, training-dynamics, regularization, rnn
- Math tags: statistical-mechanics, stochastic-processes, dynamical-systems, bifurcations
- Ratings: usefulness 7/10; difficulty 5/10; novelty 8/10

## Idea 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.

## Mathematical statement

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.

## Key formulas

- 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 > 1, equivalently r0/D > 2.
- r(z) proportional to |z|^lambda: lambda > -2 gives stretched-exponential tails, lambda = -2 gives power-law tails, and lambda < -2 gives no asymptotic localization.

## Implementation notes

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 < 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 < 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.

## Verification

- Status: mechanism_failed
- Mechanism evidence: yes
- Mechanism confirmed: no
- 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.

### Mechanism check

- 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.
- Confidence: 9/10
- 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.

## Artifacts

- [experiment.py](https://synthcore.org/code/1088/experiment.py)
- [report.md](https://synthcore.org/code/1088/report.md)
- [results.json](https://synthcore.org/code/1088/results.json)
- [Download all files as ZIP](https://synthcore.org/download/1088)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.