Low-Bit N-Gram Embeddings for Language Models

Conditional memory expands language-model capacity through selective lookup, but the full embedding table must remain available for every possible input. We introduce Conditional Program (CP), a compressed representation that reconstructs only the rows an input requests. CP represents each row by a short code and a local scale, with a reconstruction program shared across the table. We instantiate this representation with a Gaussian trellis, norm restoration and independently decodable row records. CP's two-bit code lowers output KL by 41--63% relative to 2-bit round-to-nearest. Analysis across three trained n-gram memory families shows that Gaussian-like coordinate statistics coexist with learned relationships among rows; at matched storage budgets, the shared trellis outperforms the fitted codebooks and low-rank residual variants we test. For DeepSeek, it reduces table storage from 203GB to 51GB, leaving room for larger workloads when the table is kept on GPU.

In a small language model the vocabulary is the single largest part. In Gemma 3 270M the embedding table holds 168M of 268M parameters, and because quantisation methods leave the embedding in 16-bit, it is also most of the file once the rest of the model is compressed. We ask what that table actually buys. We replace it with one frozen table of 512-bit codes, 16 MB, with no trainable parameters, derived once from a pretrained embedding by principal-component projection and sign binarisation, and we train the same Gemma-shaped body from scratch on the same 2.05B tokens with and without it. The learned table buys 0.20 nats of validation loss for 2.6 times the trainable parameters, but on LAMBADA, HellaSwag and PIQA the frozen-table model is better or tied; only ARC-Easy favours the learned table. With a ternary BitNet b1.58 body the whole model is a 46 MB file, still ties the float model on LAMBADA and PIQA, and on a laptop CPU generates 615 to 740 tokens per second in 62 MB of memory, five to six times the speed of the float model's 8-bit GGUF in llama.cpp at a seventh of the memory. The frozen table wins by 1.7 to 2.3 nats on rare words, accepts 10,000 new words after training with a 3% cost where the learned table collapses by 43%, and all of it is released with the weights, the data and every evaluation answer.

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.