Low-Bit Recurrent States in Hybrid Language Models

Low-bit quantization shrinks language models but treats precision as a single global hyper-parameter: every weight uses the same bit-width. We introduce Variable Bit-width Quantization (VBQ), a training-time method in which each contiguous group of 64 weights learns its own resolution from {1,2,4,8} bits via a Gumbel-Softmax relaxation, trained jointly by an alternating optimization that gives the precision logits a clean, task-aligned signal. VBQ discovers a consistent, strongly heterogeneous allocation within individual projection types, not merely across layers, impossible to express with per-layer methods: 69% of groups collapse to 1 bit, the LM head averages 1.09 bits, while the first MLP block keeps ~2.5 bits. This pattern is stable enough to freeze into a fixed recipe and reuse without further search. The recipe yields a "bigger-but-smaller" regime: a 131M model at 1.82 mean bits reaches perplexity 4.2 on TinyStories, beating a 55M FP16 model (PPL 4.4) at 3.8x less storage, and lets a 1.46B model on FineWeb-Edu match a 593M FP16 control at ~3.7x less storage with 2.5x more parameters. As quality-per-byte, VBQ is 3.9-8.4x more efficient than FP16. The recipe maps directly to packed low-bit storage, so it also accelerates inference: with custom fused dequantize-and-multiply kernels, memory-bandwidth-bound autoregressive decode is faster at equal output, and the speedup grows with scale (parity at 131M, 1.9x at 1.0B, 4.7x at 9B on Apple silicon). A distributional analysis (KL divergence and argmax-flip rate) reveals a striking mechanism: deeper layers progressively self-heal the quantization error injected by early layers. The win is a from-scratch, train-time phenomenon; scaling the search economically beyond 1.5B parameters remains open. VBQ reframes precision as a learnable, non-uniform resource and shows that spending a fixed bit budget unevenly beats spending it uniformly.

Linear attention and recurrent models compress past tokens into a fixed-size state that is updated at every step. However, storing these states in high precision incurs substantial memory and bandwidth costs at large batch sizes, motivating low-bit state quantization. Unlike weights and key--value caches, a recurrent state is requantized after every update, so its quantization errors accumulate across steps. Our analysis shows that these errors are highly coherent across steps, so reducing per-step errors alone is insufficient. This coherence arises mainly from small updates that are lost in rounding. To preserve these updates, we intervene both before and at rounding. Before rounding, we propose Low-Rank Bypass (LRB), which accumulates small updates in a compact low-rank buffer and periodically merges its sufficiently large elements into the state. At rounding, we observe that a state element can even move opposite to its update, and such reversals prove particularly harmful. Therefore, we propose Bounded Directional Rounding (BDR), which rounds reversed elements toward their updates without additional storage. Although BDR enlarges the rounding error, it improves the model, suggesting that the direction of an update matters more than its rounding error at a single step. Across Gated DeltaNet, GLA, Mamba2, and RetNet, our training-free method substantially outperforms INT4, while reducing state memory by up to 5.3$\times$ and speeding up decoding by up to 1.9$\times$ relative to FP32 states.

Mamba-style and hybrid language models compress their past into a fixed-size recurrent state that is rewritten at every generated token. Storing this state in low precision saves memory bandwidth, but every rounding error is fed back into the next update and can accumulate over long generations. Production systems round the state stochastically; we ask which rounding rule such caches should use. We find that a deterministic golden-ratio Weyl dither, which needs no random numbers, consistently brings the quantized model closer to the full-precision one than stochastic rounding, across pure and hybrid models, storage formats, and long decoding horizons, at no extra cost. Round-to-nearest behaves differently: because it discards small updates, its error keeps growing, so it can look best in short evaluations yet falls far behind over long generations. A discrepancy analysis explains this ordering, and we document implementation pitfalls that silently remove the benefit.