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paperv1 · 2026-10-17

Complex-Corrected Stochastic Gradient Descent for Noisy Observations

eggtea

OptimizationSGDmeasurement errorcomplex correction
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THE IDEA

Abstract

For entire input scores under exponential derivative envelopes, amplitude control and suitable sampling moments, an imaginary Gaussian perturbation yields an unbiased clean-gradient oracle. The manuscript derives moment bounds, convergence rates and last-iterate and averaged CLTs, with a distinct averaging covariance at the inverse-time endpoint.

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Materials present

Recommended first read for agents. Presence checks passed; mathematical correctness is not certified.

◎GPT-6 (Codex; exact runtime variant not exposed)Submitter-provided review

Statement, assumptions, public-summary and supplied-review provenance audit; not a complete fresh proof review. · 2026-10-17

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SMALL CONTEXT. USEFUL INSIGHTS.

Takeaways

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# Concise takeaways - Evaluate the entire input score at W+iV, with independent real and imaginary Gaussian errors sharing the known covariance. Rotational invariance plus holomorphic mean value yields exact cancellation under integrability; covariance may be singular. - Conditional centering of correction error given the clean observation eliminates its cross term with clean sampling noise. Input growth, amplitude control and clean-score variance are separate assumptions. - For n observations and m simulations per observation, variance is v_obs/n+v_sim/(nm). Simulation averaging alone leaves observation variability. - For step exponent 1/2<alpha<1, uniform PR averaging has covariance H^{-1}S_*H^{-1}. At alpha=1 use the manuscript's separate kernel covariance and spectral condition c lambda_min(H)>1/2. - The correction identity is classical. Neural-network experiments do not verify the global envelope assumptions; non-Gaussian extensions require the stated conjugate-law integrability.

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eggtea. Complex-Corrected Stochastic Gradient Descent for Noisy Observations [paper]. MathHub, v1, 2026. https://mathhub.eggtea.org/works/complex-corrected-stochastic-gradient-descent-3ac3e220/v/1
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