# A Practical Sparse Approximation for Real Time Recurrent Learning

arXiv: https://arxiv.org/abs/2006.07232 · HTML: https://mlsanity.com/paper/2006.07232/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2020.

## Built on by (most cited first, 18 of 18)

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- [Hard ASH: Sparsity and the right optimizer make a continual learner](https://mlsanity.com/paper/2404.17651/) (2024)
- [A Truly Sparse and General Implementation of Gradient-Based Synaptic Plasticity](https://mlsanity.com/paper/2501.11407/) (2025)
- [Can Biologically Plausible Temporal Credit Assignment Rules Match BPTT for Neural Similarity? E-prop as an Example](https://mlsanity.com/paper/2506.06904/) (2025)
- [Investigating Action Encodings in Recurrent Neural Networks in Reinforcement Learning](https://mlsanity.com/paper/2605.16318/) (2026)
- [Frame forecasting in cine MRI using the PCA respiratory motion model: comparing recurrent neural networks trained online and transformers](https://mlsanity.com/paper/2410.05882/) (2024)
- [Generalising E-prop to Deep Networks](https://mlsanity.com/paper/2512.24506/) (2025)
- [Massive Redundancy in Gradient Transport Enables Sparse Online Learning](https://mlsanity.com/paper/2603.15195/) (2026)
- [Shared Phase and Retention Control for Efficient Adaptive Spectral Recurrence](https://mlsanity.com/paper/2609.39082/) (2026)

## Abstract (arXiv)

Current methods for training recurrent neural networks are based on backpropagation through time, which requires storing a complete history of network states, and prohibits updating the weights `online' (after every timestep). Real Time Recurrent Learning (RTRL) eliminates the need for history storage and allows for online weight updates, but does so at the expense of computational costs that are quartic in the state size. This renders RTRL training intractable for all but the smallest networks, even ones that are made highly sparse. We introduce the Sparse n-step Approximation (SnAp) to the RTRL influence matrix, which only keeps entries that are nonzero within n steps of the recurrent core. SnAp with n=1 is no more expensive than backpropagation, and we find that it substantially outperforms other RTRL approximations with comparable costs such as Unbiased Online Recurrent Optimization. For highly sparse networks, SnAp with n=2 remains tractable and can outperform backpropagation through time in terms of learning speed when updates are done online. SnAp becomes equivalent to RTRL when n is large.