On the Expressive Power of Weight Quantization in Large Language Models

Post-training quantization reduces the deployment cost of large language models, yet how severely a quantized model degrades is not determined by bit-width alone. We systematically study weight-only post-training quantization across bit-widths, quantization methods, model scales and downstream tasks on multiple model families. We observe that such degradation varies substantially across these factors: 4-bit quantization usually preserves performance, 2-bit often causes broad degradation, and at 3-bit, degradation becomes apparent but varies markedly with task type, quantization method and model scale. To explain this variability, we use the signal-to-noise ratio (SNR) to measure how strongly quantization perturbs full-precision representations. We trace degradation back to two linked processes: how quantization errors arise within individual modules, and how they accumulate across layers. First, a source SNR decomposition shows that newly introduced errors depend on three factors: the magnitude of the weight error, the strength of the task-specific signal, and how strongly the quantization error aligns with task-specific activations. Different factors affect these components in distinct ways. Second, a cross-layer propagation analysis shows that these errors can be attenuated, preserved, or amplified as they pass across layers, and that larger models benefit from weaker error amplification. Together, these results establish that quantization degradation is governed by how errors are introduced at the source and how they accumulate across the network.

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.

Large language models (LLMs) have driven major progress in NLP, yet their substantial memory and compute demands still hinder practical deployment. Binarization can compress weights to 1 bit, fundamentally lowering compute and bandwidth cost. However, existing methods cannot address activation heavy tails and thus must keep activations in high precision, preventing true end-to-end acceleration. To overcome this limitation, we propose BWLA (Binarized Weights and Low-bit Activations), the first post-training quantization framework that preserves high accuracy while achieving 1-bit weight quantization together with low-bit activations (e.g., 6 bits). The Orthogonal-Kronecker Transformation (OKT) learns an orthogonal mapping via EM minimization, converting unimodal weights into symmetric bimodal forms while suppressing activation tails and incoherence. The Proximal SVD Projection (PSP) then performs lightweight low-rank refinement through proximal SVD projection, further enhancing quantizability with minimal overhead. On Qwen3-32B, BWLA reaches a Wikitext2 perplexity of 11.92 under 6-bit activations (vs. 38 from SOTA), improves five zero-shot tasks by more than 70%, and delivers 3.26 times inference speedup, demonstrating strong potential for real-world LLM compression and acceleration.