SGD as Free Energy Minimization: A Thermodynamic View on Neural Network Training
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Bibliographic record
Abstract
We present a thermodynamic interpretation of the stationary behavior of stochastic gradient descent (SGD) under fixed learning rates (LRs) in neural network training. We show that SGD implicitly minimizes a free energy function $F=U-TS$, balancing training loss $U$ and the entropy of the weights distribution $S$, with temperature $T$ determined by the LR. This perspective offers a new lens on why high LRs prevent training from converging to the loss minima and how different LRs lead to stabilization at different loss levels. We empirically validate the free energy framework on both underparameterized (UP) and overparameterized (OP) models. UP models consistently follow free energy minimization, with temperature increasing monotonically with LR, while for OP models, the temperature effectively drops to zero at low LRs, causing SGD to minimize the loss directly and converge to an optimum. We attribute this mismatch to differences in the signal-to-noise ratio of stochastic gradients near optima, supported by both a toy example and neural network experiments.
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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