# Multiplicative learning from observation-prediction ratios

arXiv: https://arxiv.org/abs/2503.10144 · HTML: https://mlsanity.com/paper/2503.10144/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 25 references that are in the map.
- First posted on arXiv 13 Mar 2025.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Difference Target Propagation](https://mlsanity.com/paper/1412.7525/) (2014)
- [How Auto-Encoders Could Provide Credit Assignment in Deep Networks via Target Propagation](https://mlsanity.com/paper/1407.7906/) (2014)
- [A Theoretical Framework for Target Propagation](https://mlsanity.com/paper/2006.14331/) (2020)
- [Mirror descent of Hopfield model](https://mlsanity.com/paper/2211.15880/) (2022)

## Abstract (arXiv)

Additive parameter updates, as used in gradient descent and its adaptive extensions, underpin most modern machine-learning optimization. Yet, such additive schemes often demand numerous iterations and intricate learning-rate schedules to cope with scale and curvature of loss functions. Here we introduce Expectation Reflection (ER), a multiplicative learning paradigm that updates parameters based on the ratio of observed to predicted outputs, rather than their differences. ER eliminates the need for ad hoc loss functions or learning-rate tuning while maintaining internal consistency. Extending ER to multilayer networks, we demonstrate its efficacy in image classification, achieving optimal weight determination in a single iteration. We further show that ER can be interpreted as a modified gradient descent incorporating an inverse target-propagation mapping. Together, these results position ER as a fast and scalable alternative to conventional optimization methods for neural-network training.