Optimal Quantization for Matrix Multiplication
Recent work in machine learning community proposed multiple methods for performing lossy compression (quantization) of large matrices. This quantization is important for accelerating matrix multiplication (main component of large language models), which is often bottlenecked by the speed of loading these matrices from memory. Unlike classical vector quantization and rate-distortion theory, the goal of these new compression algorithms is to be able to approximate not the matrices themselves, but their matrix product. Specifically, given a pair of real matrices $A,B$ an encoder (compressor) is applied to each of them independently producing descriptions with $R$ bits per entry. These representations subsequently are used by the decoder to estimate matrix product $A^\top B$. In this work, we provide a non-asymptotic lower bound on the mean squared error of this approximation (as a function of rate $R$) for the case of matrices $A,B$ with iid Gaussian entries. Algorithmically, we construct a universal quantizer based on nested lattices with an explicit guarantee of approximation error for any (non-random) pair of matrices $A$, $B$ in terms of only Frobenius norms $\|\bar{A}\|_F, \|\bar{B}\|_F$ and $\|\bar{A}^\top \bar{B}\|_F$, where $\bar{A},\bar{B}$ are versions of $A,B$ with zero-centered columns, respectively. For iid Gaussian matrices our quantizer achieves the lower bound and is, thus, asymptotically optimal. A practical low-complexity version of our quantizer achieves performance quite close to optimal. In addition, we derive rate-distortion function for matrix multiplication of iid Gaussian matrices, which exhibits an interesting phase-transition at $R\approx 0.906$ bit/entry.
Code (2)
Tasks
QuantizationMethods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
Optimal Scalar Quantization for Matrix Multiplication: Closed-Form Density and Phase Transition
We study entrywise scalar quantization of two matrices prior to multiplication. Given $A\in R^{m\times k}$ and $B\in R^{k\times n}$, we quantize entries of $A$ and $B$ independently using scalar quantizers with $K_X$ and…
A Deep Learning Inference Scheme Based on Pipelined Matrix Multiplication Acceleration Design and Non-uniform Quantization
Matrix multiplication is the bedrock in Deep Learning inference application. When it comes to hardware acceleration on edge computing devices, matrix multiplication often takes up a great majority of the time. To achieve…
Edge-computingQ-LearningQuantizationNestQuant: Nested Lattice Quantization for Matrix Products and LLMs
Post-training quantization (PTQ) has emerged as a critical technique for efficient deployment of large language models (LLMs). This work proposes NestQuant, a novel PTQ scheme for weights and activations that is based on…
QuantizationHigh-Rate Quantized Matrix Multiplication II
This is the second part of the work investigating quantized matrix multiplication (MatMul). In part I we considered the case of calibration-free quantization, whereas here we discuss the setting where covariance matrix $…
High-Rate Quantized Matrix Multiplication I
This paper investigates the problem of quantized matrix multiplication (MatMul), which has become crucial for the efficient deployment of large language models (LLMs). We consider a Generic MatMul setting, where both mat…