Abstract:Understanding the training dynamics of deep neural networks remains a major open problem, with physics-inspired approaches offering promising insights. Building on this perspective, we develop a thermodynamic framework to describe the stationary distributions of stochastic gradient descent (SGD) with weight decay for scale-invariant neural networks, a setting that both reflects practical architectures with normalization layers and permits theoretical analysis. We establish analogies between training hyperparameters (e.g., learning rate, weight decay) and thermodynamic variables such as temperature, pressure, and volume. Starting with a simplified isotropic noise model, we uncover a close correspondence between SGD dynamics and ideal gas behavior, validated through theory and simulation. Extending to training of neural networks, we show that key predictions of the framework, including the behavior of stationary entropy, align closely with experimental observations. This framework provides a principled foundation for interpreting training dynamics and may guide future work on hyperparameter tuning and the design of learning rate schedulers.
| Comments: | Accepted at IJCAI-ECAI 2026 (the 35th International Joint Conference on Artificial Intelligence) |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2511.07308 [cs.LG] |
| (or arXiv:2511.07308v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2511.07308 arXiv-issued DOI via DataCite |
Submission history
From: Ildus Sadrtdinov [view email]
[v1]
Mon, 10 Nov 2025 17:10:01 UTC (20,385 KB)
[v2]
Wed, 13 May 2026 22:00:31 UTC (21,554 KB)