# Bounded Telegraph Exploration for Optimizers

- ID: 2817
- Canonical URL: https://synthcore.org/idea/2817/bounded-telegraph-exploration-for-optimizers
- API JSON: https://synthcore.org/api/idea/2817.json
- API Markdown: https://synthcore.org/api/idea/2817.md
- Verification status: unverified
- Source: [arXiv:2608.29226](https://arxiv.org/abs/2608.29226)
- Category: dynamics
- Solves: stability, generalization, sample-efficiency
- ML areas: optimizer, training-dynamics, regularization
- Math tags: stochastic-processes, dynamical-systems, probability
- Ratings: usefulness 7/10; difficulty 4/10; novelty 7/10

## Idea description

Add a bounded colored exploration force to an optimizer by filtering a sum of independent two-state telegraph signals through a stable linear relaxation equation. Unlike Gaussian momentum noise, the perturbation has a strict amplitude bound and a tunable finite correlation time, reducing rare destructive parameter excursions while retaining structured exploration.

## Mathematical statement

For each parameter coordinate or parameter block, define u(t) by du/dt = -gamma u(t) + sum_k v_k s_k(t), where gamma > 0 is the relaxation rate, v_k > 0 are amplitudes, and s_k(t) in {-1,+1} are independent symmetric telegraph processes switching at rate lambda_k. The stationary process obeys |u(t)| <= V/gamma, where V = sum_k v_k. For symmetric switching, E[u] = 0, Var(u) = sum_k v_k^2/[gamma(gamma+2 lambda_k)], and the autocovariance is C_u(tau) = sum_k v_k^2[exp(-2 lambda_k |tau|) - exp(-gamma |tau|)]/[gamma^2 - 4 lambda_k^2] when gamma is not 2 lambda_k. These relations give exact amplitude, variance, and correlation-time targets for optimizer noise.

## Key formulas

- du/dt = -gamma u(t) + sum_k v_k s_k(t), with s_k(t) in {-1,+1} and flip probability lambda_k dt over a small interval dt.
- |u(t)| <= (sum_k v_k)/gamma = u_max.
- Var(u) = sum_k v_k^2/[gamma(gamma + 2 lambda_k)].
- C_u(tau) = sum_k v_k^2 [exp(-2 lambda_k |tau|) - exp(-gamma |tau|)]/[gamma^2 - 4 lambda_k^2], for gamma != 2 lambda_k.

## Implementation notes

Integrate the mechanism into a momentum-like optimizer state rather than directly perturbing weights. For parameter vector theta, maintain telegraph force u and update theta <- theta - eta [g(theta) + alpha u], where g is the minibatch gradient, eta is the learning rate, and alpha scales exploration. Use one scalar telegraph process per parameter block or layer to avoid storing a sign for every parameter. With step size dt, flip each sign s_k independently with probability 1 - exp(-lambda_k dt), then apply the exact linear relaxation update u <- exp(-gamma dt) u + [1 - exp(-gamma dt)] sum_k v_k s_k/gamma. Pseudocode: initialize u = 0 and random signs; for each minibatch compute g; flip signs; update u; optionally clamp only roundoff excursions to [-u_max,u_max]; apply theta <- theta - eta(g + alpha u). The paper supplies the bounded-support and stationary-moment mechanism; estimate empirical variance and autocorrelation from the run. First test a small MLP and ResNet-18 on CIFAR-10 against SGD and SGD with Gaussian OU noise at matched variance. Sweep gamma, K, and lambda_k. The quantitative prediction is |u| <= u_max at every step, with violations below numerical precision. Measured stationary variance should match the closed form within 10%, and the autocorrelation should fit the predicted sum of exponentials. As K increases with fixed total variance, standardized excess kurtosis should decrease toward zero.

## Disclaimer

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