Convolution-Calibrated Persistent-Noise Optimizer

Integrate the mechanism into SGD, Adam, or Langevin SGD inside a trust region, initially on one parameter block or a low-dimensional projection. Estimate a local curvature lambda using a Hessian-vector product, diagonal Gauss-Newton estimate, or finite-difference curvature. Maintain one persistent sign sigma per block. At step k with duration h, flip sigma with probability 1 - exp(-alpha h/2), compute the minibatch gradient g, and update theta by theta <- theta + h(-mu g + v sigma) + sqrt(2 D_T h) epsilon, where epsilon is standard normal noise. Use gradient clipping or a trust-region projection to prevent the persistent force from leaving the local basin. The paper provides the stationary run-and-tumble density and its Ornstein-Uhlenbeck convolution; curvature, effective thermal noise D_T, and the validity of the quadratic approximation must be estimated from training trajectories. The first cheap experiment should use a one-dimensional and two-dimensional quadratic loss, followed by a small MLP on MNIST. Compare SGD, Gaussian Langevin SGD, and persistent-noise SGD at matched total variance. Record parameter histograms after equilibration and fit both the predicted convolution and a Gaussian. The quantitative prediction is that the convolution fit wins at small r or chi, while the histogram becomes Gaussian-like when r or chi is large. The measured variance should match a^2/(2r+1) + D_T/(mu lambda) within 20 percent. For the discrete deterministic part, instability should begin when |1-h mu lambda| reaches one, with the upper boundary h mu lambda = 2.