Paper Detail
Tommaso Cerruti, Tim Rieder, George Rowlands, Lingfeng Jin, Imanol Schlag
Self-attention lets each token retrieve information from the full context, but its quadratic cost in sequence length limits training and inference at long context. This paper presents a comparative study of softmax attention and four recent recurrent linear-attention architectures: DeltaNet, Gated DeltaNet, Kimi Delta Attention, and Gated DeltaNet-2. We express these mechanisms in a common recurrent-memory notation, making explicit how they differ in expressivity, memory decay, erase and write control, training throughput, and implementation complexity. Our experiments center on 350M-parameter models trained for 15B tokens, and include optimizer and learning-rate comparisons, hybrid-versus-pure stack comparisons, sequence-length runtime measurements, larger DeltaNet runs at 1.3B and 3B parameters, and a small set of downstream evaluations. The reported speed results measure training throughput and iteration time; we do not provide an empirical inference-speed benchmark. Within the reported 350M-parameter, 15B-token sweep, Kimi Delta Attention with Muon reaches the lowest final validation loss, a pure Gated DeltaNet stack trained with AdamW has the highest normalized training throughput, hybrid stacks generally improve loss at a throughput cost, and Muon consistently lowers final validation loss relative to AdamW in the matched architecture settings we evaluate. We introduce and evaluate lightweight cross-layer routing mechanisms for DeltaNet-style memories. The most natural DeltaNet-inspired formulation, forwarding a lower layer's delta-rule write error into the next layer's value target, does not improve over matched baselines. Routing into the aligned hidden stream and forwarding the write value instead yields a modest improvement in the matched runs we report: Cross-Layer Value Routing (CLVR) lowers final validation loss for both DeltaNet and Gated DeltaNet.
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@misc{cerruti2026linear,
title = {Linear Attention Architectures: Mechanisms, Trade-offs, and Cross-Layer Routing},
author = {Tommaso Cerruti and Tim Rieder and George Rowlands and Lingfeng Jin and Imanol Schlag},
year = {2026},
abstract = {Self-attention lets each token retrieve information from the full context, but its quadratic cost in sequence length limits training and inference at long context. This paper presents a comparative study of softmax attention and four recent recurrent linear-attention architectures: DeltaNet, Gated DeltaNet, Kimi Delta Attention, and Gated DeltaNet-2. We express these mechanisms in a common recurrent-memory notation, making explicit how they differ in expressivity, memory decay, erase and write c},
url = {https://huggingface.co/papers/2607.07953},
keywords = {self-attention, softmax attention, recurrent linear-attention, DeltaNet, Gated DeltaNet, Kimi Delta Attention, Gated DeltaNet-2, recurrent-memory notation, memory decay, erase and write control, training throughput, implementation complexity, attention mechanism, sequence length, validation loss, optimizer, learning rate, hybrid stacks, pure stack, cross-layer routing, delta-rule write error, hidden stream, Cross-Layer Value Routing, CLVR, code available, huggingface daily},
eprint = {2607.07953},
archiveprefix = {arXiv},
}
{}