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Beyond 4 bits: Data-Centric Post-training Quantization and Recovery
arxiv.orgpaper

Beyond 4 bits: Data-Centric Post-training Quantization and Recovery

This paper studies 2-bit post-training quantization (PTQ) of six dense and mixture-of-experts models from 3.8B to 32B parameters. It finds the quantization representation matters most: vector codebooks give usable models where scalar PTQ falls to chance. Calibration and recovery data must be chosen jointly. The recipe uses about 2.3 effective bits per weight, keeps 89 to 97% of full-precision QA accuracy, and needs 1.4 to 30x less compute than the strongest QAT baseline.

TL;DR Shows that vector codebooks, task-aligned calibration data and codebook-only recovery make 2-bit PTQ viable where scalar PTQ collapses.
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What it claims

At 2 bits, the quantization representation decides whether a model works at all. On Phi-4-mini, scalar PTQ (SpinQuant) falls from a 7-task QA average of 69.2 at 4 bits to 29.3 at 2 bits, which is the random floor. 2-bit YAQA, a vector quantizer, holds 61.8 against a BF16 ceiling of 71.0. Every tested scalar pipeline scores 0.0 on generation at 2 bits on all three main models.

Calibration data is the second-largest factor, and it interacts with the recovery data. Generation losses depend on the model. HumanEval retention ranges from 45% (Phi-4-mini, Qwen3-8B) to 81% (Qwen3-30B-A3B).

How it works

Attention and MLP projections are quantized, including every MoE expert. Routers, embeddings, norms and the LM head stay in half precision. A router ablation suggests the router is robust to quantization. All methods land at 2.09 to 2.33 effective bits per weight once indices, codebooks and scales are counted.

The vector baselines are VPTQ (a 2^16-entry codebook over 8-dimensional vectors) and YAQA (Fisher-weighted trellis rounding). Both run in an incoherence-rotated frame. Learned rotations did not help at 2 bits.

Calibration mixtures are cheap to sweep. The objective depends on the data only through the input Hessian. Hessians are precomputed per domain (web, instruction, OpenR1-Math, code) and reweighted as H_α = Σ α_j H_j. Each mixture still needs its own quantization pass. For YAQA, the Kronecker input and output factors are mixed separately.

Recovery freezes the backbone and trains only the codebook, with assignments fixed. It distills from the full-precision model using α·CE + (1−α)·τ²·KL, with α = 0.5. Memory is about 2P + bP/8 bytes, against about 14P for QAT.

Evidence

The recovery budget is 3,200 examples (3.3M tokens), with three seeds per configuration. Results below are AvgGen, the mean of GSM8K 5-shot and HumanEval pass@1.

  • Phi-4-mini (BF16 AvgGen 72.3). Web calibration without recovery scores 11.1. Math calibration plus web recovery scores 45.7, the best cell (GSM8K 56.9, HumanEval 34.6).
  • Llama-3.1-8B (BF16 AvgGen 44.0). The best cell is math calibration with no recovery, at 27.4. Every recovery lowered it.
  • Calibration effect. Switching from web to math calibration adds +22.5, +12.2 and +16.5 AvgGen on Phi-4-mini, Llama-3.1-8B and Qwen3-8B.
  • Interaction. Math recovery helps a web-calibrated base but not a math-calibrated one. The interaction contrast is −8.5 [−12.6, −4.3] on Phi-4-mini and −2.7 [−5.0, −0.7] on Llama-3.1-8B.
  • Pre-registered test. For Qwen3-8B, the design and three predictions were registered before any data was collected. All three held, with an interaction of −9.4 [−13.3, −5.3].
  • Robustness. The interaction stays negative on held-out tasks (MBPP, HumanEval+, MATH-500), on re-quantized bases and on fresh recovery samples.
  • Compute. Vector PTQ without training uses 39 to 304x less compute than trained scalar methods (GSQ, BitDistiller). With recovery it uses 1.4 to 30x less than the strongest QAT baseline.

Math calibration has a cost on QA. On Phi-4-mini, AvgQA drops from 63.1 with web calibration to 58.3 with math calibration.

How to use it

The stated recipe has three parts:

  1. Use a vector codebook quantizer (YAQA or VPTQ).
  2. Calibrate on a small task-aligned mixture.
  3. Run short codebook-only recovery, choosing the recovery data together with the calibration data.

The paper describes a cheap per-model check for deciding whether to recover at all. Code, Hessian mixtures and quantized models are promised "upon acceptance" and are not yet released.

Limits

The interaction result covers one domain pair (web and math) at one short budget. On prompt-independent tasks alone, the interaction is clear only on Phi-4-mini.

Recovery is not universally useful. It hurt the base Llama-3.1-8B in every tested cell.

Evaluation uses dequantized weights, and storage figures are packed-size estimates. No inference kernels or latency are reported. The paper is a preprint under review dated October 2026, and the fetched text stops at Table 3, so the MoE and Qwen3-32B results were not visible.

Organisations named here

  • arXiv

Technologies named here

  • LLM
  • Mixture of Experts

Techniques named here

  • Post-training quantization
  • Vector codebook quantization representation
  • Task-aligned calibration mixture
  • Codebook-only recovery stage
  • Quantization-aware training

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