A Scalable Approach to Performing Multiplication and Matrix Dot-Products in Unary (2307.03204v1)

Abstract: Stochastic computing is a paradigm in which logical operations are performed on randomly generated bit streams. Complex arithmetic operations can be executed by simple logic circuits, resulting in a much smaller area footprint compared to conventional binary counterparts. However, the random or pseudorandom sources required for generating the bit streams are costly in terms of area and offset the advantages. Additionally, due to the inherent randomness, the computation lacks precision, limiting the applicability of this paradigm. Importantly, achieving reasonable accuracy in stochastic computing involves high latency. Recently, deterministic approaches to stochastic computing have been proposed, demonstrating that randomness is not a requirement. By structuring the computation deterministically, exact results can be obtained, and the latency greatly reduced. The bit stream generated adheres to a "unary" encoding, retaining the non-positional nature of the bits while discarding the random bit generation of traditional stochastic computing. This deterministic approach overcomes many drawbacks of stochastic computing, although the latency increases quadratically with each level of logic, becoming unmanageable beyond a few levels. In this paper, we present a method for approximating the results of the deterministic method while maintaining low latency at each level. This improvement comes at the cost of additional logic, but we demonstrate that the increase in area scales with the square root of n, where n represents the equivalent number of binary bits of precision. Our new approach is general, efficient, composable, and applicable to all arithmetic operations performed with stochastic logic. We show that this approach outperforms other stochastic designs for matrix multiplication (dot-product), which is an integral step in nearly all machine learning algorithms.

• The paper presents a novel deterministic unary encoding method that approximates multiplication and dot-product results with efficient scaling.
• It eliminates the need for random bit streams, reducing area overhead and latency while controlling increased logic complexity.
• The approach outperforms traditional stochastic designs in matrix multiplication, offering a general, efficient, and composable solution for machine learning.

The paper discusses advancements in performing multiplication and matrix dot-products within a unary encoding paradigm. This approach is rooted in stochastic computing, a computational framework where operations are executed on bit streams instead of traditional binary representations. Stochastic computing is known for allowing complex arithmetic functions to be performed with relatively simple logic circuits, reducing area footprint.

However, stochastic computing's reliance on random or pseudorandom bit streams often negates its benefits due to the high area cost of generating these streams. Additionally, the inherent randomness hinders computational precision and imposes high latency to achieve acceptable accuracy.

To address these issues, recent research has shifted towards deterministic methods that forgo randomness, enabling precise computation and significantly reduced latency. The research presented in this paper builds on these deterministic approaches by generating bit streams through unary encoding. Unary encoding preserves the non-positional bit structure while eliminating the need for random bit generation, thus overcoming many limitations of traditional stochastic computing. However, one challenge is that the latency tends to increase quadratically with each level of logic, eventually becoming a bottleneck.

The paper introduces a novel method that approximates results from these deterministic computations while maintaining low latency. This new approach increases the logic requirements, but the growth in additional logic scales efficiently with the square root of the equivalent number of binary bits of precision denoted as $n$.

This approach is touted for being general, efficient, and composable. It promises applicability across a range of arithmetic operations that are traditionally performed using stochastic logic. The method is particularly effective for matrix multiplication, especially in the context of machine learning, where matrix operations are crucial. The authors demonstrate that their method surpasses existing stochastic designs in performance for matrix multiplication tasks, highlighting its potential utility in machine learning algorithms.