# Exact information accounting for SGD methods

arXiv: https://arxiv.org/abs/2610.00446 · HTML: https://mlsanity.com/paper/2610.00446/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 105 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Cyclical Learning Rates for Training Neural Networks](https://mlsanity.com/paper/1506.01186/) (2015)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [The CMA Evolution Strategy: A Tutorial](https://mlsanity.com/paper/1604.00772/) (2016)
- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://mlsanity.com/paper/2305.14342/) (2023)
- [Super-Convergence: Very Fast Training of Neural Networks Using Large Learning Rates](https://mlsanity.com/paper/1708.07120/) (2017)
- [SOAP: Improving and Stabilizing Shampoo using Adam](https://mlsanity.com/paper/2409.11321/) (2024)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [Understanding Warmup-Stable-Decay Learning Rates: A River Valley Loss Landscape Perspective](https://mlsanity.com/paper/2410.05192/) (2024)
- [On the Heavy-Tailed Theory of Stochastic Gradient Descent for Deep Neural Networks](https://mlsanity.com/paper/1912.00018/) (2019)
- [Understanding and Improving Shampoo and SOAP via Kullback-Leibler Minimization](https://mlsanity.com/paper/2509.03378/) (2025)
- [Potential-Function Proofs for First-Order Methods](https://mlsanity.com/paper/1712.04581/) (2017)
- [Flat Minima and Generalization: Insights from Stochastic Convex Optimization](https://mlsanity.com/paper/2511.03548/) (2025)
- [Stopping Rules for Stochastic Gradient Descent via Anytime-Valid Confidence Sequences](https://mlsanity.com/paper/2512.13123/) (2025)
- [Adaptive Bayes exactly tracks information over intrinsic time](https://mlsanity.com/paper/2607.08789/) (2026)
- [Beyond Optimal Rates in Stochastic Optimization: Trajectory-Adaptive Stopping Rules](https://mlsanity.com/paper/2608.25551/) (2026)
- [The Exact Time-Uniform Rate Frontier for Stochastic Gradient Descent on Smooth Convex Objectives](https://mlsanity.com/paper/2609.08537/) (2026)

## Abstract (arXiv)

As an alternative to the standard geometric analyses, we give an exact, information-theoretic analysis of stochastic gradient descent (SGD) and its variants. We show that a preconditioned SGD step is the posterior-mean update of a Gaussian Bayes model, and that its one-step regret splits into an intrinsic-time cost and a change in comparator information. The split extends to an identity for the objective itself. Convex convergence, strict-saddle-point escape, the link between flatness and generalization, the standard learning-rate schedules, adaptive optimizers, and the noisy, momentum, heavy-tailed, and gradient-free variants of SGD each correspond to a term or a special case of this identity. We measure its terms on synthetic and real training runs. On real networks it attributes the slack of classical convergence bounds to the terms their derivations drop and separates optimizers that reach the same training loss. That separation follows the number and consistency of their steps. Its relation to which of them generalizes better differs between networks. For gradient-free SGD the identity determines how a curvature preconditioner should enter the update. The sharpness-based generalization certificate it yields, with a data-independent isotropic prior, is vacuous at network scale unless the curvature spectrum is nearly flat across all parameters.