# Longhorn: State Space Models are Amortized Online Learners

arXiv: https://arxiv.org/abs/2407.14207 · HTML: https://mlsanity.com/paper/2407.14207/

## Key facts

- Cited by 59 papers in the map (data through 14 Oct 2026).
- Placed in Core SSM architectures for language, one of 10 directions in State-space sequence models (S4, Mamba).
- Builds on 47 of its 65 references that are in the map.
- First posted on arXiv 19 Jul 2024.

## Placed in

- State-space sequence models (S4, Mamba) / [Core SSM architectures for language](https://mlsanity.com/ssm/core-ssm-architectures-for-language/)

## Builds on (most cited first, 30 of 47)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [TriviaQA: A Large Scale Distantly Supervised Challenge Dataset for Reading Comprehension](https://mlsanity.com/paper/1705.03551/) (2017)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [BoolQ: Exploring the Surprising Difficulty of Natural Yes/No Questions](https://mlsanity.com/paper/1905.10044/) (2019)
- [Can a Suit of Armor Conduct Electricity? A New Dataset for Open Book Question Answering](https://mlsanity.com/paper/1809.02789/) (2018)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Know What You Don't Know: Unanswerable Questions for SQuAD](https://mlsanity.com/paper/1806.03822/) (2018)
- [Linformer: Self-Attention with Linear Complexity](https://mlsanity.com/paper/2006.04768/) (2020)
- [DeepSeek-V2: A Strong, Economical, and Efficient Mixture-of-Experts Language Model](https://mlsanity.com/paper/2405.04434/) (2024)
- [Vision Mamba: Efficient Visual Representation Learning with Bidirectional State Space Model](https://mlsanity.com/paper/2401.09417/) (2024)
- [Accelerating Large Language Model Decoding with Speculative Sampling](https://mlsanity.com/paper/2302.01318/) (2023)
- [DROP: A Reading Comprehension Benchmark Requiring Discrete Reasoning Over Paragraphs](https://mlsanity.com/paper/1903.00161/) (2019)
- [RWKV: Reinventing RNNs for the Transformer Era](https://mlsanity.com/paper/2305.13048/) (2023)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Language Modeling with Gated Convolutional Networks](https://mlsanity.com/paper/1612.08083/) (2016)
- [Retentive Network: A Successor to Transformer for Large Language Models](https://mlsanity.com/paper/2307.08621/) (2023)
- [Gated Linear Attention Transformers with Hardware-Efficient Training](https://mlsanity.com/paper/2312.06635/) (2023)
- [Linear Transformers Are Secretly Fast Weight Programmers](https://mlsanity.com/paper/2102.11174/) (2021)
- [Hungry Hungry Hippos: Towards Language Modeling with State Space Models](https://mlsanity.com/paper/2212.14052/) (2022)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Parallelizing Linear Transformers with the Delta Rule over Sequence Length](https://mlsanity.com/paper/2406.06484/) (2024)
- [Hyena Hierarchy: Towards Larger Convolutional Language Models](https://mlsanity.com/paper/2302.10866/) (2023)

## Built on by (most cited first, 30 of 59)

- [Titans: Learning to Memorize at Test Time](https://mlsanity.com/paper/2501.00663/) (2024)
- [Kimi Linear: An Expressive, Efficient Attention Architecture](https://mlsanity.com/paper/2510.26692/) (2025)
- [TTT3R: 3D Reconstruction as Test-Time Training](https://mlsanity.com/paper/2509.26645/) (2025)
- [Gated Slot Attention for Efficient Linear-Time Sequence Modeling](https://mlsanity.com/paper/2409.07146/) (2024)
- [DeltaProduct: Improving State-Tracking in Linear RNNs via Householder Products](https://mlsanity.com/paper/2502.10297/) (2025)
- [Test-time regression: a unifying framework for designing sequence models with associative memory](https://mlsanity.com/paper/2501.12352/) (2025)
- [Nested Learning: The Illusion of Deep Learning Architectures](https://mlsanity.com/paper/2512.24695/) (2025)
- [ATLAS: Learning to Optimally Memorize the Context at Test Time](https://mlsanity.com/paper/2505.23735/) (2025)
- [It's All Connected: A Journey Through Test-Time Memorization, Attentional Bias, Retention, and Online Optimization](https://mlsanity.com/paper/2504.13173/) (2025)
- [MesaNet: Sequence Modeling by Locally Optimal Test-Time Training](https://mlsanity.com/paper/2506.05233/) (2025)
- [Gated DeltaNet-2: Decoupling Erase and Write in Linear Attention](https://mlsanity.com/paper/2605.22791/) (2026)
- [Understanding and Improving Length Generalization in Recurrent Models](https://mlsanity.com/paper/2507.02782/) (2025)
- [Efficient Attention Mechanisms for Large Language Models: A Survey](https://mlsanity.com/paper/2507.19595/) (2025)
- [Lattice: Learning to Efficiently Compress the Memory](https://mlsanity.com/paper/2504.05646/) (2025)
- [Understanding and Mitigating Bottlenecks of State Space Models through the Lens of Recency and Over-smoothing](https://mlsanity.com/paper/2501.00658/) (2024)
- [Memory Caching: RNNs with Growing Memory](https://mlsanity.com/paper/2602.24281/) (2026)
- [Test-Time Training with KV Binding Is Secretly Linear Attention](https://mlsanity.com/paper/2602.21204/) (2026)
- [Gated KalmaNet: A Fading Memory Layer Through Test-Time Ridge Regression](https://mlsanity.com/paper/2511.21016/) (2025)
- [Mechanistic evaluation of Transformers and state space models](https://mlsanity.com/paper/2505.15105/) (2025)
- [RAD: Redundancy-Aware Distillation for Hybrid Models via Self-Speculative Decoding](https://mlsanity.com/paper/2505.22135/) (2025)
- [TNT: Improving Chunkwise Training for Test-Time Memorization](https://mlsanity.com/paper/2511.07343/) (2025)
- [Kalman Linear Attention: Parallel Bayesian Filtering For Efficient Language Modelling and State Tracking](https://mlsanity.com/paper/2602.10743/) (2026)
- [Preconditioned DeltaNet: Curvature-aware Sequence Modeling for Linear Recurrences](https://mlsanity.com/paper/2604.21100/) (2026)
- [Learning Memory Mechanisms for Decision Making through Demonstrations](https://mlsanity.com/paper/2411.07954/) (2024)
- [RoboSSM: Scalable In-context Imitation Learning via State-Space Models](https://mlsanity.com/paper/2509.19658/) (2025)
- [SAS: Simulated Attention Score](https://mlsanity.com/paper/2507.07694/) (2025)
- [Local Linear Attention: An Optimal Interpolation of Linear and Softmax Attention For Test-Time Regression](https://mlsanity.com/paper/2510.01450/) (2025)
- [MDN: Parallelizing Stepwise Momentum for Delta Linear Attention](https://mlsanity.com/paper/2605.05838/) (2026)
- [Learning in the Recurrent State: Gradient Descent with Linear Recurrent Networks](https://mlsanity.com/paper/2410.11687/) (2024)
- [Fast Spatial Memory with Elastic Test-Time Training](https://mlsanity.com/paper/2604.07350/) (2026)

## Abstract (arXiv)

Modern large language models are built on sequence modeling via next-token prediction. While the Transformer remains the dominant architecture for sequence modeling, its quadratic decoding complexity in sequence length poses a major limitation. State-space models (SSMs) present a competitive alternative, offering linear decoding efficiency while maintaining parallelism during training. However, most existing SSMs rely on linear recurrence designs that appear somewhat ad hoc. In this work, we explore SSM design through the lens of online learning, conceptualizing SSMs as meta-modules for specific online learning problems. This approach links SSM design to formulating precise online learning objectives, with state transition rules derived from solving these objectives. Based on this insight, we introduce a novel deep SSM architecture, Longhorn, whose update resembles the closed-form solution for solving the online associative recall problem. Our experimental results show that Longhorn outperforms state-of-the-art SSMs, including the Mamba model, on standard sequence modeling benchmarks, language modeling, and vision tasks. Specifically, Longhorn achieves a 1.8x improvement in sample efficiency compared to Mamba, and can extrapolate over contexts that are up to 16x longer during inference.