Regularized Calibration with Successive Rounding for Post-Training Quantization

TL;DR

This work tackles the memory and latency challenges of deploying large language models by enhancing post-training quantization. It introduces regularized asymmetric calibration via an α-weighted interpolation between full-precision and quantized activations, decomposing the objective into a symmetric reconstruction term plus an asymmetric regularizer, and provides closed-form and stochastic strategies to select α. Building on this, the authors derive a shifted-Hessian, triangular-discrete least-squares formulation and develop two rounding algorithms: a fast greedy SNRQ and a higher-quality, bounded-search K-SNRQ. Empirical results across multiple model families, bit-widths, and benchmarks show consistent perplexity and accuracy gains with modest computational overhead, confirming the practical utility of regularized calibration with successive rounding for PTQ.

Abstract

Large language models (LLMs) deliver robust performance across diverse applications, yet their deployment often faces challenges due to the memory and latency costs of storing and accessing billions of parameters. Post-training quantization (PTQ) enables efficient inference by mapping pretrained weights to low-bit formats without retraining, but its effectiveness depends critically on both the quantization objective and the rounding procedure used to obtain low-bit weight representations. In this work, we show that interpolating between symmetric and asymmetric calibration acts as a form of regularization that preserves the standard quadratic structure used in PTQ while providing robustness to activation mismatch. Building on this perspective, we derive a simple successive rounding procedure that naturally incorporates asymmetric calibration, as well as a bounded-search extension that allows for an explicit trade-off between quantization quality and the compute cost. Experiments across multiple LLM families, quantization bit-widths, and benchmarks demonstrate that the proposed bounded search based on a regularized asymmetric calibration objective consistently improves perplexity and accuracy over PTQ baselines, while incurring only modest and controllable additional computational cost.

Figures (12)

• Figure 1: Mean activation error (MAE) $\lvert X_f - X_q\rvert$ across transformer layers of L2-7B on C4, evaluated on the calibration set (left) and a validation set (right) under greedy layer-wise rounding. Although $\alpha = 1$ achieves the lowest calibration error, intermediate values of $\alpha$ reduce error on the validation set.
• Figure 2: Regularized layer-wise calibration using interpolated activations $X_\alpha^l = \alpha X_f^l + (1-\alpha) X_q^l$ that blend full precision and quantized features during weight optimization at layer $l$.
• Figure 3: Mean Wiki2 perplexity ($\pm$ std. over 5 seeds) for a fixed $\alpha=\mathbb{E}[\alpha]$ vs. sampled $\alpha$. Sampling reduces variability across runs and achieves equal or lower mean perplexity across all models.
• Figure 4: Overview of $K$-SNRQ beam search. Columns of the weight matrix are quantized sequentially from $j\!=\!n$ to $1$, maintaining a beam of size $K$. At each step, candidate extensions are scored using the incremental loss and pruned to the top-$K$ beams.
• Figure 5: Performance-efficiency trade-offs of $K$-SNRQ for 3-bit L2-7B. Wiki2 and C4 perplexity versus quantization time (left) and peak GPU memory (right) as beam width $K$ increases.

Theorems & Definitions (14)

• Proposition 3.1: Decomposition of the interpolated proxy
• Proposition 3.2: Optimal $\alpha$
• Theorem 4.1: Regularized calibration as least-squares problem
• proof
• Proposition 1.1: Columnwise decomposition of the triangular proxy
• proof
• Proposition 2.1: Decomposition of the regularized calibration proxy
• proof
• Proposition 2.2: Optimal $\alpha$
• proof