Grouped Sequency-arranged Rotation: Optimizing Rotation Transformation for Quantization for Free

Abstract: LLMs face deployment challenges due to high computational costs, and while Post-Training Quantization (PTQ) offers a solution, existing rotation-based methods struggle at very low bit-widths like 2-bit. We introduce a novel, training-free approach to construct an improved rotation matrix, addressing the limitations of current methods. The key contributions include leveraging the Walsh-Hadamard transform with sequency ordering, which clusters similar frequency components to reduce quantization error compared to standard Hadamard matrices, significantly improving performance. Furthermore, we propose a Grouped Sequency-arranged Rotation (GSR) using block-diagonal matrices with smaller Walsh blocks, effectively isolating outlier impacts and achieving performance comparable to optimization-based methods without requiring any training. Our method demonstrates robust performance on reasoning tasks and Perplexity (PPL) score on WikiText-2. Our method also enhances results even when applied over existing learned rotation techniques.

• The paper introduces GSR, a novel rotation technique using grouped Walsh matrices to efficiently reduce quantization errors in low-bit quantization scenarios.
• The paper details a sequency reordering strategy that isolates outlier influences within quantization blocks, achieving a perplexity drop from 20.29 to 11.59 on WikiText-2.
• The paper demonstrates that GSR attains performance comparable to optimization-based methods without retraining, offering a practical enhancement for LLM deployments.

Grouped Sequency-arranged Rotation: Optimizing Rotation Transformation for Quantization for Free

Introduction to LLM Quantization Challenges

The deployment of LLMs often encounters high computational and memory demands, particularly in constrained environments. Post-Training Quantization (PTQ) emerges as a solution by minimizing model size while maintaining performance. Within this paradigm, rotation-based transformations are a key technique to manage quantization issues, but such methods, including QuaRot, exhibit substantial performance drops at very low bit-widths like 2-bit quantization.

Proposed Methodology: Grouped Sequency-arranged Rotation (GSR)

The paper introduces a novel rotation technique termed Grouped Sequency-arranged Rotation (GSR), intended to be a zero-training enhancement to the rotation matrix construction process for PTQ. Leveraging the Walsh-Hadamard transform, GSR arranges frequency components to attenuate quantization errors more efficiently than previous Hadamard-based approaches. Specifically, GSR deploys block-diagonal matrices with grouped Walsh matrices, isolating outliers’ influence within quantization blocks. This innovation achieves comparable performance to existing optimization-based methods, eliminating the need for training phases.

Figure 1: Overall diagram of rotation scheme. We applied Grouped Sequency-arranged Rotation (GSR) on $R_1$.

Theoretical Framework

Walsh-Hadamard Transform and Sequency

The basis for GSR is the transformation of Hadamard matrices into Walsh matrices via sequency ordering, where rows are reordered to manage bit flips strategically—this process clusters elements with similar frequency characteristics. The implication is a reduction in intra-group variance, beneficial for mitigating quantization errors.

Rotation for LLM Quantization

Within the context of LLM activation paths, different rotation matrices ($R_1$, $R_2$, $R_3$, $R_4$) perform unique roles. The sequencing optimization of GSR efficiently handles outliers by distributing their effect across smaller partitions, as opposed to global rotations that spread the influence extensively (Figure 2).

Figure 2: Global rotation applies a full-matrix transformation across all dimensions and spreads outlier effects widely.

Empirical Validation

Experimental Setup

GSR was validated across several LLM benchmarks including WikiText-2 and zero-shot reasoning tasks with Llama-2 models. Comparison with incumbent techniques—QuaRot, SpinQuant, and OSTQuant—demonstrated GSR’s superiority particularly in low-bit quantization scenarios.

Results and Analysis

Across tested configurations, GSR consistently reduced perplexity (PPL) scores: for instance, from QuaRot’s GH at 20.29 on WikiText-2 to GSR’s adjusted 11.59. Zero-shot task performance also benefitted significantly from GSR’s efficient transformation, affirming its efficacy without supplementary model training.

Future Implications and Extensions

The potential of GSR extends beyond current benchmarks, suggesting applicability in various model architectures requiring PTQ without intensive retraining. However, as quantization precision increases, gains from GSR diminish, directing future research towards broadening the method’s scalability and efficiency improvements for higher bit-width scenarios.

Conclusion

GSR presents an advance in PTQ for LLMs through a signal processing-inspired strategy, enhancing model adaptability without training overhead. Its methodical application of sequency principles fosters robust performance across quantization levels, establishing a foundation for future explorations into efficient model deployment strategies within resource-limited settings.