Unlocking the Theory Behind Scaling 1-Bit Neural Networks

Majid Daliri, Zhao Song, Chiwun Yang

Keywords: 1-bit neural network, quantization, neural tangent kernel

Abstract: Recently, 1-bit Large Language Models (LLMs) have emerged, showcasing an impressive combination of efficiency and performance that rivals traditional LLMs. Research by Wang et al. (2023); Ma et al. (2024) indicates that the performance of these 1-bit LLMs progressively improves as the number of parameters increases, hinting at the potential existence of a *Scaling Law for 1-bit Neural Networks*. In this paper, we present the \emph{first theoretical} result that rigorously establishes this scaling law for 1-bit models. We prove that, despite the constraint of weights restricted to $\\{-1, +1\\}$, the dynamics of model training inevitably align with kernel behavior as the network width grows. This theoretical breakthrough guarantees convergence of the 1-bit model to an arbitrarily small loss as width increases. Furthermore, we introduce the concept of the generalization difference, defined as the gap between the outputs of 1-bit networks and their full-precision counterparts, and demonstrate that this difference maintains a negligible level as network width scales. Building on the work of Kaplan et al. (2020), we conclude by examining how the training loss scales as a power-law function of the model size, dataset size, and computational resources utilized for training. Our findings underscore the promising potential of scaling 1-bit neural networks, suggesting that int1 could become the standard in future neural network precision.

Primary Area: learning theory

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Submission Number: 1803