# Noise-Debiased Thermodynamic Variance for Local Learning Coefficient Probes

arXiv: https://arxiv.org/abs/2606.22389 · HTML: https://mlsanity.com/paper/2606.22389/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 15 references that are in the map.
- First posted on arXiv 21 Jun 2026.

## Builds on (most cited first, 9 of 9)

- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks](https://mlsanity.com/paper/1710.11029/) (2017)
- [The Local Learning Coefficient: A Singularity-Aware Complexity Measure](https://mlsanity.com/paper/2308.12108/) (2023)
- [Deep Learning is Singular, and That's Good](https://mlsanity.com/paper/2010.11560/) (2020)
- [Loss Landscape Degeneracy and Stagewise Development in Transformers](https://mlsanity.com/paper/2402.02364/) (2024)
- [Differentiation and Specialization of Attention Heads via the Refined Local Learning Coefficient](https://mlsanity.com/paper/2410.02984/) (2024)
- [Estimating Real Log Canonical Thresholds](https://mlsanity.com/paper/1906.01341/) (2019)

## Abstract (arXiv)

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 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.