# Multiplicative learning from observation-prediction ratios

Source: [Multiplicative learning from observation-prediction ratios](<https://www.sota2.com/research/paper/68fa77621e209131cea72e43>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2026-03-26T05:00:04.121Z

Authors: Han Kim, Hyungjoon Soh, Vipul Periwal, Junghyo Jo

Published: 2025-03-13T08:14:00.000Z

## Abstract

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.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2503.10144>)

[PDF](<https://arxiv.org/pdf/2503.10144.pdf>)