schema_version: 1
title: Complex-Corrected Stochastic Gradient Descent for Noisy Observations
abstract: For Gaussian measurement error with known positive semidefinite covariance,
  an independent imaginary Gaussian perturbation yields exactly unbiased gradients
  for scores with an entire input extension. Under the manuscript's derivative envelope,
  gradient-dependent amplitude control and clean-score assumptions, it derives relative
  variance bounds, convergence and central limit laws. The 1/t averaging covariance
  differs from the interior Polyak–Ruppert covariance. Non-Gaussian extensions require
  separate conjugate-law and integrability conditions; neural-network experiments
  are empirical.
field: Optimization
license: CC-BY-4.0
type: paper
authors:
- eggtea
keywords:
- stochastic gradient descent
- measurement error
- complex corrected scores
- central limit theorem
main: main.pdf
takeaways: takeaways.md
verification:
- type: review
  path: verification/review.md
  status: materials-present-not-independently-verified
  model: Codex (GPT-6; current-session non-independent check)
  date: '2026-10-17'
  scope: Limited current-session claim/assumption and selected-proof consistency check
    of main.pdf SHA-256 3e222f197266ca0283b9618f4eb4d4cc1818c0e90a916d724c3981404da6eb02;
    includes executed formula checks; not an independent full mathematical review.
references:
- title: Corrected score estimation via complex variable simulation extrapolation
  authors: Steven J. Novick; Leonard A. Stefanski
  doi: 10.1198/016214502760047005
- title: Acceleration of stochastic approximation by averaging
  authors: B. T. Polyak; A. B. Juditsky
  doi: 10.1137/0330046
- title: A convergence theorem for non negative almost supermartingales and some applications
  authors: H. Robbins; D. Siegmund
  doi: 10.1016/B978-0-12-604550-5.50015-8