In this paper, we introduce Exact Flow Linear Attention~(EFLA), an exact-flow formulation of delta-rule linear attention. We show that the delta-rule update can be interpreted as an explicit Euler discretization of an underlying continuous-time system. EFLA replaces this first-order update with the exact closed-form flow. By exploiting the rank-1 structure of the dynamics matrix, both the matrix exponential and the input integral collapse to a simple update that preserves delta-rule linear attention's algebraic structure, parameter count, linear-time complexity, and chunkwise parallelism. This attention mechanism removes the Euler discretization error of the delta-rule dynamics without introducing additional parameters. Experiments on robustness tests, language modeling benchmarks, and the MAD synthetic benchmark show that EFLA improves stability under corrupted and high-energy inputs, reduces perplexity, and achieves stronger downstream performance compared to SSM and Euler-style baselines. These results establish exact-flow integration as a principled and scalable update mechanism for delta-rule linear attention.
Figures & tables
Perplexity ( ↓ )
Accuracy ( ↑ )
Model
Wiki.
LMB.
LMB.
PIQA
Hella.
Wino.
ARC-e
ARC-c
BoolQ
OBQA
SciQ
Avg.
ppl
ppl
acc
acc
acc_n
acc
acc
acc_n
acc
acc_n
acc
340M Parameters
Vanilla
Mamba-2
36.90
105.69
20.7
61.9
30.7
50.2
40.4
22.1
53.0
27.0
71.9
42.0
DeltaNet
38.09
96.26
22.5
60.7
30.1
51.9
41.7
21.6
60.1
29.0
70.4
42.2
Table 1 : Main language modeling results. The 340M models are trained for 8B tokens, whereas the 1.3B models are trained for 50B tokens on the same subset of the SlimPajama dataset ( Soboleva et al., 2023 ) with the Mistral ( Jiang et al., 2023 ) tokenizer. Perplexity: Lower ( ↓ ) is better. Accuracy: Higher ( ↑ ) is better. Best results are bolded.
Figure 1 : EFLA outperforms DeltaNet in both convergence speed and robustness on sMNIST. The plots illustrate training dynamics and robustness against dropout, intensity scaling, and additive Gaussian noise. EFLA maintains higher accuracy as perturbation intensity increases, especially under the larger learning-rate setting 3×10−3 .
Figure 2 : sMNIST results under different learning rates. EFLA maintains lower training loss and higher accuracy across the tested learning-rate range, indicating stronger robustness to learning-rate selection.
Linear Attention (LA) offers a promising paradigm for scaling large language models (LLMs) to long sequences by avoiding the quadratic complexity of self-attention. Recent LA models such as Mamba2 and GDN interpret linear recurrences as closed-form online stochastic gradient descent (SGD), but naive SGD updates suffer from rapid information decay and suboptimal convergence in optimization. While momentum-based optimizers provide a natural remedy, they pose challenges in simultaneously achieving training efficiency and effectiveness. To address this, we develop a chunkwise parallel algorithm for LA with a stepwise momentum rule by geometrically reordering the update coefficients. Further, from a dynamical systems perspective, we analyze the momentum-based recurrence as a second-order system that introduces complex conjugate eigenvalues. This analysis guides the design of stable gating constraints. The resulting model, Momentum DeltaNet (MDN), leverages Triton kernels to achieve comparable training throughput with competitive linear models such as Mamba2 and KDA. Extensive experiments on the 400M and 1.3B parameter models demonstrate consistent performance improvements over strong baselines, including Transformers, Mamba2 and GDN, across diverse downstream evaluation benchmarks. Code: https://github.com/HuuYuLong/MomentumDeltaNet .
Yulong Huang, Xiang Liu, Hongxiang Huang +5
The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China
Embedded Language Flows (ELF) rely primarily on full non-causal attention for iterative denoising, repeatedly incurring quadratic sequence-mixing cost at each sampling step. Gated Delta Networks (GDNs) provide an efficient recurrent alternative, but their standard causal formulation cannot directly capture the bidirectional context required by ELF. We introduce DeltaFlow, a noise-adaptive bidirectional GDN backbone for continuous language denoising. We study two variants: DeltaFlow-A, which alternates scan directions across layers, and DeltaFlow-P, which performs parallel forward and backward scans within each layer. We further introduce noise-adaptive memory control and scheduled Temporal State Consistency (TSC) to stabilize hidden representations across nearby noise levels. On OpenWebText, using a 32-step stochastic differential equation sampler, DeltaFlow-P reduces generated perplexity from 24.218 for the full-attention ELF baseline to 21.228 while maintaining comparable unigram entropy, with 36B training-token exposure compared with 45B for the baseline. In a denoiser-only benchmark, DeltaFlow-P achieves a 2.72x throughput speedup over the full-attention baseline at a sequence length of 16k. These results show that DeltaFlow is a promising alternative to dense attention for efficient continuous language denoising.
Guangfu Guo, Xiaoqian Lu, Linsey Pang +4
1Clemson University · University of Ottawa · 3PayPal +1
Linear attention replaces the unbounded cache of softmax attention with a fixed-size recurrent state, reducing sequence mixing to linear time and decoding to constant memory. The hard part is not just what to forget, but how to edit this compressed memory without scrambling existing associations. Delta-rule models subtract the current read before writing a new value, and Kimi Delta Attention (KDA) sharpens forgetting with channel-wise decay. But the active edit still uses a single scalar gate to control two different things: how much old content to erase on the key side and how much new content to commit on the value side. We introduce Gated DeltaNet-2, which generalizes both Gated DeltaNet and KDA by inheriting adaptive forgetting and channel-wise decay while addressing their shared limitation, the scalar tie between erasing and writing. Gated Delta Rule-2 separates these roles with a channel-wise erase gate b_t and a channel-wise write gate w_t, reducing to KDA when both gates collapse to the same scalar and to Gated DeltaNet when the decay also collapses. We derive a fast-weight update view, a chunkwise WY algorithm with channel-wise decay absorbed into asymmetric erase factors, and a gate-aware backward pass that preserves efficient parallel training. At 1.3B parameters trained on 100B FineWeb-Edu tokens, Gated DeltaNet-2 achieves the strongest overall results among Mamba-2, Gated DeltaNet, KDA, and Mamba-3 variants across language modeling, commonsense reasoning, and retrieval. Its advantage is most pronounced on long-context RULER needle-in-a-haystack benchmarks, where it improves the evaluated multi-key retrieval setting and remains strong in both recurrent and hybrid settings. Code is available at https://github.com/NVlabs/GatedDeltaNet-2.