Yifan Zhang, Steve Ta, Jasper Zhang +8cs.LG cs.CL stat.ML
Recurrent fast-weight memories and selective state-space models compress an expanding context into a fixed-size recurrent state, making the state transition an online learning rule. We study this rule under read-after-write autoregressive semantics. For the prefix-prediction objective considered here, the local fast-memory example revealed at step $t$ is the prefix-aligned pair $(\mathbf{x}_t,\mathbf{y}_t)=(φ(\mathbf{k}_{t-1}),\mathbf{v}_t)$. The common same-step association $(φ(\mathbf{k}_t),\mathbf{v}_t)$ remains causal, but optimizes a different internal objective. We derive normalized first-order updates for squared-error regression and negative inner-product objectives. The regression family comprises Falcon-1 (a scalar NLMS update), Falcon-2 (its per-column extension), and Falcon-3 (a sliding-window mini-batch update); Falcon-1A/Falcon-2A/Falcon-3A are the corresponding inner-product variants. We provide recurrent, masked-parallel, and chunk-parallel forms, together with numerically stable positive-decay renormalization. Representative variants remain competitive in language modeling and improve length extrapolation on variable-digit addition. This framework separates temporal alignment, plasticity, forgetting, and bounded rehearsal in recurrent sequence models.
Delta-Rule recurrent models maintain a fixed-size state, enabling $O(1)$ inference memory but potentially becoming unstable under extreme-context extrapolation. By tracking RWKV-7 over sequences of up to 100M tokens, we empirically identify a distinct failure pattern: \textbf{localized norm explosion atop a relatively sparse substrate}, rather than global state saturation. Analysis of the recurrent update suggests that persistent decay keeps weakly updated entries small, whereas uneven injections allow a few channels to accumulate extreme values. Motivated by this diagnosis, we propose \textbf{State Anomaly Neutralization (SANE)}, which applies adaptive $\tanh$ compression at chunk boundaries while preserving the intra-chunk parallel structure. Within a safe threshold range ($3 \le α\le 5$), SANE matches the baseline on 11 short-context reasoning benchmarks with no statistically significant degradation. After a 100M-token prefix, which exceeds the training length by over $24{,}000\times$, SANE retains functional reasoning ($33.46$--$35.56$) while the baseline encounters numerical overflow. In contrast, overly permissive thresholds ($α\ge 8$) remain numerically stable but lose reasoning capability entirely, showing that numerical stabilization alone does not guarantee functional reasoning and revealing a capacity--stability trade-off in state compression.
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.
Recurrent models must forget in order to remember, yet the state of the art decides what to erase without consulting what is stored -- the gate sees only the arriving token, not the memory it is about to modify. This memory-blind gating is one of three coupled defects in the leading delta-rule architecture (GDN-2): the value-axis erase mask wastes parameters at the scale of the value projection, and -- as we prove -- mathematically prevents the WY-form triangular chunk solver that makes recurrent training competitive with Transformers. We introduce CARVE (Content-Aware Recurrent with Value Efficiency), which resolves all three problems through one principle: erase only on the key axis. This is provably necessary and sufficient for the WY-form solver to remain valid. Within it, CARVE reuses the recurrent output tensor -- already written to GPU memory -- as a free content signal for the erase gate, and replaces the per-value write-gate projection with a single scalar per head. At initialisation CARVE is bit-identical to GDN-2; any quality difference emerges from what the content gate learns. At 1.3B parameters trained on 100B tokens, CARVE achieves WikiText perplexity 15.72 (minus 0.18 vs. GDN-2, a 4.5-sigma effect), leads every recurrent baseline on nine common-sense reasoning benchmarks, and sets state of the art on every RULER retrieval probe -- at 0.4% throughput overhead, 13% lower peak memory, and 19% fewer parameters. Six formal theorems cover memory capacity, Lyapunov stability, gradient flow, expressivity separation, Pareto-optimal chunk size, and hybrid optimality.