TLDR: A new research paper introduces the first theoretical framework to analyze the convergence of adaptive optimizers like Adam and Muon under floating-point quantization of all components (gradients, weights, optimizer states). The study reveals that both optimizers can maintain full-precision convergence rates if mantissa length scales logarithmically with iterations. It highlights Adam’s sensitivity to weight and second-moment quantization, while Muon demonstrates greater robustness, explaining empirical successes in low-precision large language model training.
The rapid growth of large language models (LLMs) has made low-precision training a cornerstone of modern deep learning. This technique, which involves using fewer bits to represent numbers during computation, is crucial for reducing memory consumption, boosting efficiency, and enabling the training of even larger models on advanced hardware. Despite its widespread adoption and empirical success, a comprehensive theoretical understanding of why low-precision training works so effectively, especially with popular adaptive optimizers like Adam and Muon, has been largely missing.
Traditional convergence theories for these optimizers often assume perfect, high-precision arithmetic, overlooking the practical realities of hardware-aware quantization. This gap between practical success and theoretical explanation has been a significant challenge for researchers and practitioners alike.
A groundbreaking new research paper, “A Convergence Analysis of Adaptive Optimizers under Floating-point Quantization”, introduces the first theoretical framework to bridge this divide. Authored by Xuan Tang, Jichu Li, and Difan Zou, the paper provides a rigorous analysis of how adaptive optimizers converge when gradients, weights, and optimizer states (like momentum estimates) are all subjected to floating-point quantization.
The core innovation lies in its analytical framework, which explicitly models the quantization of all key components involved in the training process. Unlike previous works that often relied on simplified assumptions such as unbiased quantization or memory-intensive error-feedback mechanisms, this new framework adopts a more realistic relative error model. This model accurately reflects how standard floating-point formats (like FP32 to BF16 or FP8) behave in real-world scenarios, making the theoretical findings directly applicable to modern LLM training pipelines.
The researchers derived convergence rates for both Adam and Muon optimizers on smooth non-convex objectives, a common setting in deep learning. Their analysis clearly shows how quantization errors from different parts of the system individually impact convergence. A key finding is that both algorithms can maintain convergence rates comparable to their full-precision counterparts, provided that the mantissa length (the part of a floating-point number that represents its significant digits) scales logarithmically with the number of training iterations. This is consistent with the precision capabilities of current hardware.
Interestingly, the study reveals a significant difference in sensitivity between Adam and Muon. Adam, a widely used optimizer, was found to be particularly sensitive to the quantization of weights and second-moment estimates. This is largely due to its reliance on a parameter (beta2) that is typically set very close to 1, which can amplify accumulated quantization errors. This theoretical insight aligns with empirical observations that often require higher precision for weights and second moments in low-bit Adam training.
In contrast, the Muon optimizer demonstrated greater robustness to quantization. It requires weaker error control, meaning it can tolerate lower precision levels while still converging effectively. This resilience stems from Muon’s unique SVD-based sign operator, which helps prevent the amplification of quantization errors that can plague other optimizers. This theoretical explanation supports recent empirical findings suggesting Muon’s superior performance in low-precision training environments.
The findings were not just theoretical; numerical experiments on both synthetic data (using the classic Rosenbrock function) and real-world data (training a neural network on CIFAR-10) corroborated the theory. These experiments showed that while very low mantissa lengths can lead to slow convergence and optimization traps, moderate mantissa lengths yield performance nearly identical to full-precision training. This empirical validation further strengthens the paper’s conclusions.
This research significantly narrows the gap between the observed success of quantized adaptive training and its theoretical foundation. It provides a crucial analytical framework for understanding and designing future low-precision optimization algorithms, paving the way for even more efficient and scalable training of large AI models.
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While the paper offers profound insights, the authors acknowledge certain limitations and future directions. These include extending the framework to handle weaker smoothness conditions, integrating low-precision operations beyond just quantization (like FP8 matrix multiplications), and considering communication efficiency in distributed training setups. Addressing these aspects will provide an even more complete theoretical picture of large-scale low-precision optimization.
