Kalman Delta Networks: Uncertainty-aware Associative Memory
Kalman Delta Networks add uncertainty tracking to linear-attention associative memory, improving perplexity and downstream accuracy at 750M and 1.3B scales.
Kalman Delta Networks reformulate recurrent associative memory in linear-attention models as a linear-Gaussian state-space model, allowing the Kalman gain to weight each residual write by accumulated evidence and observation reliability; Delta-rule updates emerge as a special case lacking covariance tracking. Two scan-compatible approximations, Diagonal KDN (online mean-field variational inference) and Isotropic KDN (one uncertainty scalar per head), produce Mobius-map uncertainty recurrences enabling associative scans with logarithmic parallel depth. Controlled pretraining at 750M and 1.3B parameters consistently improves perplexity and mean downstream accuracy over state-of-the-art linear-attention models.
- Frames delta-rule associative memory as a linear-Gaussian state-space with Kalman-gain-weighted writes
- Diagonal and Isotropic KDN approximations are GPU-scan-compatible via Mobius map recurrences
- Consistent perplexity and downstream accuracy gains at 750M and 1.3B parameters
- Targets fixed-size recurrent memory write decisions for long-context linear attention
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Linear attention is increasingly used in frontier language models for efficient long-context inference and constant-memory decoding. Its fixed-size recurrent memory, however, requires an online decision at each token: what to write and how strongly to overwrite existing associations before knowing which information future queries will require. Delta-rule models learn this strength from the current token embedding but do not track confidence in the memory estimate, preventing each write from adapting to accumulated evidence. To represent this uncertainty explicitly, we reformulate recurrent associative memory as a linear--Gaussian state-space model, for which the Kalman filter is the optimal recursive estimator, and introduce a new family of models, Kalman Delta Networks (KDNs). Within KDNs, the transition propagates both the memory state and its uncertainty, allowing the Kalman gain to weight each residual write by accumulated evidence and observation reliability. Under this formulation, Delta-style updates emerge as a special case that substitutes a token-wise isotropic surrogate for predictive covariance and omits covariance tracking. Exact tracking, however, entails a dense, state-dependent Riccati recursion that is poorly suited to GPU-parallel linear-attention scans. To address this issue, we introduce two scan-compatible KDN approximations. Diagonal KDN projects each one-step posterior onto the diagonal Gaussian family through online mean-field variational inference, whereas Isotropic KDN uses an isotropic approximation with a single uncertainty scalar per head. Their uncertainty recurrences are Mobius maps, enabling associative scans with logarithmic parallel depth. Across controlled pretraining at 750M and 1.3B parameters, KDN variants consistently improve perplexity and mean downstream accuracy over state-of-the-art linear-attention models.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.07816