Voltic: Distinguishing Volatility from Stochasticity in Recurrent Memory

We introduce the design-model framework: a way to derive efficient recurrent sequence maps from explicit assumptions about memory. A design model writes evidence into memory by exact Bayesian filtering; a query-dependent readout produces a predictive distribution whose mean is the layer output. In our linear-Gaussian instantiation, the \emph{Bayesian Layer} propagates both a mean and a covariance: the covariance tracks uncertainty over stored associations, steering writes toward uncertain directions, attenuating gains as evidence accumulates, and preserving confident memories. The same framework unifies several sub-quadratic recurrences. Linear attention, GLA, and Mamba-2/SSD are exact filters under one design model, whereas DeltaNet and related Delta-rule models arise as covariance-reset reductions under another. Restoring the covariance yields closed-form predictions for retrieval dynamics, verified empirically, and improves robustness beyond the training regime across controlled collision studies, learned associative recall, and the Zoology MQAR benchmark; distilling Bayesian Layers into a pretrained 340M Gated DeltaNet improves RULER long-context retrieval at matched compute.

Linear attention enables efficient long-context inference by compressing token history into a fixed-size recurrent memory. This compression makes each update a trade-off between incorporating new information and preserving useful associations. Models such as DeltaNet, Gated DeltaNet, and KDA predict write strength from the current token representation, without explicitly tracking uncertainty in the stored memory. Yet this uncertainty matters: a new observation should have greater influence when the existing association is uncertain and less when it is already well supported. We introduce Kalman Delta Networks (KDNs), a family of linear-attention models that explicitly track memory uncertainty to guide each update. By formulating associative memory as a linear-Gaussian state-space model, KDNs propagate both the memory estimate and its uncertainty, using the Kalman gain to balance accumulated evidence against the reliability of new observations. This formulation also recovers standard delta-rule updates by replacing tracked covariance with a token-predicted isotropic surrogate. To support hardware-efficient training and inference, we derive Diagonal KDN and Isotropic KDN, which retain one uncertainty value per key channel and per head, respectively. Their uncertainty updates admit associative scans with logarithmic parallel depth, requiring only $O(d_k)$ and $O(1)$ auxiliary state per head. Across controlled pretraining at 750M and 1.3B parameters, both variants consistently improve perplexity and mean downstream accuracy over the evaluated state-of-the-art linear-attention baselines.

Delta-rule linear attention improves recurrent memory updates by correcting what is already stored at the current write address before writing new content. However, the active correction is still anchored to that same write address. As a result, stale information stored at a different address cannot be actively removed before new content is written elsewhere. We propose Erase-then-Delta Attention (EDA), a memory update rule that decouples where to erase from where to write. The key insight is that recurrent memory models should not only correct the current write, but also selectively suppress outdated memory at an independently chosen address. Concretely, our method first applies a targeted erase step along a learned erase direction, and then performs the standard delta-style corrective write along the current write direction. This preserves the corrective behavior of delta-rule updates while expanding their memory-management capacity. Language-model pretraining experiments across dense 2.5B and MoE 25B-A2.8B model families show that EDA performs best in both settings. The gain persists after 80B-token long-context midtraining of the MoE models, where EDA also performs best in long-context evaluations from 4k to 128k contexts. A compact update analysis and memory-state probes suggest why: EDA keeps the delta-rule corrective write intact while allocating an additional cleanup path most strongly when passive decay is weak. These results suggest that recurrent memory models should decide not only what to write, but also what stale information to erase and where.