Hybrid Associative Memories

Recurrent-attention hybrid language models (LMs), which interleave attention and recurrent layers, are increasingly used to combine the efficiency of the recurrent layers with the strong performance of attention layers. Prior work suggests that attention and recurrent layers offer complementary pathways to use past information: attention supports precise memory recall from earlier tokens, while recurrent layers support consolidation of disparate information over long contexts. However, we observe that simply having access to both pathways does not mean that hybrid LMs are effectively using them. We find that they rely substantially more on attention than on the recurrent state. Standard supervised fine-tuning improves overall performance but does not improve how the two memory pathways are coordinated: the model becomes more reliant on information propagated by attention layers, while its use of information propagated by recurrent layers remains limited. To encourage better coordination between the two memory pathways, we add an auxiliary loss that limits attention's access to earlier context while the recurrent state propagates through the full sequence. This objective encourages the model to retain and use information through the recurrent pathway alongside attention. It improves overall performance, with particularly strong gains on tasks involving longer contexts or requiring information aggregation, consistent with the strengths of recurrent layers observed in analysis. Crucially, this imbalance and the benefit of our auxiliary loss generalize: they apply to multiple recurrent-attention LMs in question-answering and agentic tasks, as well as to attention-based LMs that combine different forms of memory. Together, our findings show that simply providing multiple memory pathways does not ensure their effective use, and that targeted supervision is needed to better coordinate them.

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.

Linear attention reduces the quadratic cost of softmax attention to $\mathcal{O}(T)$, but its memory state grows as $\mathcal{O}(T)$ in Frobenius norm, causing progressive interference between stored associations. We introduce Variational Linear Attention (VLA), which reframes the memory update as an online regularised least-squares problem with an adaptive penalty matrix maintained via the Sherman-Morrison rank-1 formula. We prove that normalising the write direction to unit length gives the recurrence Jacobian spectral norm exactly $1$ for all sequence lengths and head dimensions (Proposition 2), and that the state norm is self-limiting under bounded inputs (Proposition 1). Empirically, VLA reduces $\|S_t\|_F$ by $109\times$ relative to standard linear attention at $T{=}1{,}000$, achieves near-perfect exact-match accuracy on multi-query associative recall within the effective per-head memory regime ($n_\text{pairs} < d_h$), maintaining substantially higher retrieval performance than DeltaNet and standard linear attention under increasing memory load, and maintains 62% accuracy at the per-head capacity boundary. A Triton-fused kernel achieves $14\times$ speedup over sequential Python and $\mathcal{O}(T)$ scaling, crossing below softmax attention latency at approximately 43,000 tokens.