Noise-Debiased Thermodynamic Variance for Local Learning Coefficient Probes

Abstract:Local learning coefficient (LLC) probes offer a singularity-aware view of neural-network training, but mean-energy methods require a local loss baseline that is ambiguous at transient checkpoints. Thermodynamic variance avoids this input; under mini-batch evaluation, however, direct variance mixes cross-state loss fluctuations with same-state noise. We operationalize this route with the \emph{Shift-Invariant Variance Estimator} (SIVE), which estimates and subtracts the latter component using repeated evaluations. Conditional on any fixed retained path, unclipped SIVE is unbiased for noiseless path variance without requiring MCMC stationarity. The finite-scale diagnostic remains indexed by localization scale $h$---even a locally linear loss has tether-dependent variance---while interpretation as a Real Log Canonical Threshold (RLCT) requires additional stationary low-temperature conditions. Toy experiments recover calibrated finite-scale targets. At the primary localization scale, all five MNIST MLP trajectories exhibit a mid-training trough followed by a rebound in SIVE, while Raw Variance decreases from Epoch 40 to 100 in every trajectory. Across four localization scales, the joint early-drop/late-rise criterion is met in 19 of 20 trajectory--scale pairs. At Epoch 40, the estimated observation-noise correction accounts for $77.5\%$ of Raw Variance. Same-state debiasing thus reveals a reproducible turning structure masked by time-varying observation noise.

27 pages, 1 figure, 9 tables. Code is available at this https URL

Machine Learning (cs.LG)

arXiv:2606.22389 [cs.LG]

(or arXiv:2606.22389v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2606.22389 Focus to learn more arXiv-issued DOI via DataCite

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