Beyond Uniform Approximation: Toward Efficient Computing in Logarithmic Number System for Large Language Models

Logarithmic Number System (LNS) offers a more efficient quantization format than integer system for data in Large Language Models (LLMs), with inherently simple implementation for multiplication operations. However, the addition operation in LNS is relatively complex and difficult to balance between precision and hardware overhead. In this work, we observe that the input patterns of the computing units in the inference phase of LLMs exhibit highly skewed frequencies, resulting in non-uniform contributions to the final error. Based on this observation, we propose the Beyond Uniform Approximation (BUA) to enhance the efficiency of LNS additions. From the perspective of Lookup Tables (LUTs), we propose LUT reshaping to efficiently reduce the approximation error, as well as LUT trimming, a heuristic method for simplifying LUTs within a given error budget. From the perspective of the entire LNS adder, we introduce auxiliary circuit approximation, which further reduces the hardware cost by simplifying computing data paths that are less frequently activated. Together, these techniques enable efficient approximate computing in LNS. Experimental results indicate that, compared to the Multiply-Accumulate (MAC) unit using the naive LNS adder, BUA significantly reduces perplexity across various LLMs while reducing Area-Delay Product (ADP) by 26.7% and Power-Delay Product (PDP) by 30.8%. Compared to the traditional integer MAC unit, LNS with BUA achieves higher zero-shot accuracy while reducing ADP by 35.1% and PDP by 44.0%. Moreover, we also offer additional approximation options that provide greater hardware savings with only minor degradation in accuracy.