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This paper proposes M$^2$RNN, an architecture utilizing matrix-valued hidden states to overcome the computational complexity limitations of standard Transformers for tasks like entity tracking and code execution. By enabling efficient use of tensor cores through state size expansion, the model achieves superior long-context generalization and accuracy in hybrid attention-recurrent settings while reducing recurrent layer overhead. These results demonstrate that non-linear RNNs remain a viable and potentially more expressive alternative to purely attention-based architectures for scalable language modeling.
M$^2$RNN: Non-Linear RNNs with Matrix-Valued States for Scalable Language Modeling
arXiv:2603.14360v3 Announce Type: replace Abstract: Transformers are highly parallel but are limited to computations in the TC$^0$ complexity class, excluding tasks such as entity tracking and code execution that provably require greater expressive power. Motivated by this limitation, we revisit non-linear Recurrent Neural Networks (RNNs) for language modeling and introduce Matrix-to-Matrix RNN (M$^2$RNN): an architecture with matrix-valued hidden states and expressive non-linear state transitions. We demonstrate that the language modeling performance of non-linear RNNs is limited by their state size, and show how the state size expansion mechanism enables efficient use of tensor cores. Empirically, M$^2$RNN achieves perfect state tracking generalization at sequence lengths not seen during training. These benefits also translate to large-scale language modeling. In hybrid settings that interleave recurrent layers with attention, Hybrid M$^2$RNN outperforms equivalent Gated DeltaNet hybrids by $0.4$-$0.5$ perplexity points on a 7B MoE model, while using $3\times$ smaller state sizes for the recurrent layers. Notably, replacing even a single recurrent layer with M$^2$RNN in an existing hybrid architecture yields accuracy gains comparable to Hybrid M$^2$RNN with minimal impact on training throughput. Further, the Hybrid Gated DeltaNet models with a single M$^2$RNN layer also achieve superior long-context generalization, outperforming state-of-the-art hybrid linear attention architectures by up to $8$ points on LongBench. Together, these results establish non-linear RNN layers as a compelling building block for efficient and scalable language models.
Abstract:Transformers are highly parallel but are limited to computations in the TC$^0$ complexity class, excluding tasks such as entity tracking and code execution that provably require greater expressive power. Motivated by this limitation, we revisit non-linear Recurrent Neural Networks (RNNs) for language modeling and introduce Matrix-to-Matrix RNN (M$^2$RNN): an architecture with matrix-valued hidden states and expressive non-linear state transitions. We demonstrate that the language modeling performance of non-linear RNNs is limited by their state size, and show how the state size expansion mechanism enables efficient use of tensor cores. Empirically, M$^2$RNN achieves perfect state tracking generalization at sequence lengths not seen during training. These benefits also translate to large-scale language modeling. In hybrid settings that interleave recurrent layers with attention, Hybrid M$^2$RNN outperforms equivalent Gated DeltaNet hybrids by $0.4$-$0.5$ perplexity points on a 7B MoE model, while using $3\times$ smaller state sizes for the recurrent layers. Notably, replacing even a single recurrent layer with M$^2$RNN in an existing hybrid architecture yields accuracy gains comparable to Hybrid M$^2$RNN with minimal impact on training throughput. Further, the Hybrid Gated DeltaNet models with a single M$^2$RNN layer also achieve superior long-context generalization, outperforming state-of-the-art hybrid linear attention architectures by up to $8$ points on LongBench. Together, these results establish non-linear RNN layers as a compelling building block for efficient and scalable language models.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2603.14360 [cs.LG]
(or arXiv:2603.14360v3 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2603.14360 Focus to learn more arXiv-issued DOI via DataCite
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