Or Ordentlich

Or Ordentlich, Yury Polyanskiy

This work investigates the problem of quantized matrix multiplication (MatMul), which has become crucial for the efficient deployment of large language models (LLMs). We consider two settings: 1) Generic MatMul, where both matrices must be quantized (weight+activation quantization); and 2) weight-only quantization, where the second matrix is only known through covariance matrix $Σ_X$ of its columns. For each setting, we first review the fundamental information-theoretic tradeoff between quantization rate and distortion (high-rate theory), and then analyze the performance of several popular quantization schemes, comparing them to these fundamental limits. Specifically, we discuss rate loss (compared to information theoretic optima) of absmax INT and floating-point (FP) quantization, for which we also derive remarkably accurate heuristic approximations. Weight-only quantization is related to the problem of weighted mean squared error (WMSE) source coding, whose classical (reverse) waterfilling solution dictates how one should distribute rate between coordinates of the vector. We show how waterfilling can be used to improve practical LLM quantization algorithms (GPTQ), which at present allocate rate equally. This new scheme (termed ``WaterSIC'') only uses scalar INT quantizers, but its high-rate performance is basis free (it depends only on the determinant of $Σ_X$ and, thus, unlike existing schemes, is immune to applying random rotations) and is within a multiplicative factor of $\frac{2πe}{12}$ (or 0.25 bit/entry) of the information-theoretic distortion limit (!). GPTQ's performance is affected by the choice of basis, but for a random rotation and actual $Σ_X$ from Llama-3-8B we find GPTQ to be within 0.1 bit (depending on the layer type) of WaterSIC, suggesting that GPTQ with random rotation is also near optimal (for high-rate quantization).

Alina Harbuzova, Or Ordentlich, Yury Polyanskiy

Fast computation of a matrix product $W^\top X$ is a workhorse of modern LLMs. To make their deployment more efficient, a popular approach is that of using a low-precision approximation $\widehat W$ in place of true $W$ ("weight-only quantization''). Information theory demonstrates that an optimal algorithm for reducing precision of $W$ depends on the (second order) statistics of $X$ and requires a careful alignment of vector quantization codebook with PCA directions of $X$ (a process known as "waterfilling allocation''). Dependence of the codebook on statistics of $X$, however, is highly impractical. This paper proves that there exist a universal codebook that is simultaneously near-optimal for all possible statistics of $X$, in the sense of being at least as good as an $X$-adapted waterfilling codebook with rate reduced by 0.11 bit per dimension. Such universal codebook would be an ideal candidate for the low-precision storage format, a topic of active modern research, but alas the existence proof is non-constructive. Equivalently, our result shows existence of a net in $\mathbb{R}^n$ that is a nearly-optimal covering of a sphere simultaneously with respect to all Hilbert norms.

Egor Lifar, Semyon Savkin, Or Ordentlich et al.

This paper considers the problem of converting a given dense linear layer to low precision. The tradeoff between compressed length and output discrepancy is analyzed information theoretically (IT). It is shown that a popular GPTQ algorithm may have an arbitrarily large gap to the IT limit. To alleviate this problem, a novel algorithm, termed ''WaterSIC'', is proposed and is shown to be within a rate gap of 0.255 bits to the IT limit, uniformly over all possible covariance matrices of input activations. The key innovation of WaterSIC's is to allocate different quantization rates to different columns (in-features) of the weight matrix, mimicking the classical IT solution known as ''waterfilling''. Applying WaterSIC to the Llama and Qwen family of LLMs establishes new state-of-the-art performance for all quantization rates from 1 to 4 bits.