# Bayesian filtering unifies adaptive and non-adaptive neural network optimization methods

arXiv: https://arxiv.org/abs/1807.07540 · HTML: https://mlsanity.com/paper/1807.07540/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jul 2018.

## Built on by (most cited first, 9 of 9)

- [The Bayesian Learning Rule](https://mlsanity.com/paper/2107.04562/) (2021)
- [Hebbian Deep Learning Without Feedback](https://mlsanity.com/paper/2209.11883/) (2022)
- [Gradient Descent on Neurons and its Link to Approximate Second-Order Optimization](https://mlsanity.com/paper/2201.12250/) (2022)
- [A Comprehensive Review of Digital Twin -- Part 1: Modeling and Twinning Enabling Technologies](https://mlsanity.com/paper/2208.14197/) (2022)
- [Clarifying Shampoo: Adapting Spectral Descent to Stochasticity and the Parameter Trajectory](https://mlsanity.com/paper/2602.09314/) (2026)
- [On Sequential Bayesian Inference for Continual Learning](https://mlsanity.com/paper/2301.01828/) (2023)
- [Robustness to corruption in pre-trained Bayesian neural networks](https://mlsanity.com/paper/2206.12361/) (2022)
- [Implicit Maximum a Posteriori Filtering via Adaptive Optimization](https://mlsanity.com/paper/2311.10580/) (2023)
- [Are vision language models robust to uncertain inputs?](https://mlsanity.com/paper/2505.11804/) (2025)

## Abstract (arXiv)

We formulate the problem of neural network optimization as Bayesian filtering, where the observations are the backpropagated gradients. While neural network optimization has previously been studied using natural gradient methods which are closely related to Bayesian inference, they were unable to recover standard optimizers such as Adam and RMSprop with a root-mean-square gradient normalizer, instead getting a mean-square normalizer. To recover the root-mean-square normalizer, we find it necessary to account for the temporal dynamics of all the other parameters as they are geing optimized. The resulting optimizer, AdaBayes, adaptively transitions between SGD-like and Adam-like behaviour, automatically recovers AdamW, a state of the art variant of Adam with decoupled weight decay, and has generalisation performance competitive with SGD.