How AI settled the complexity of the oldest SGD algorithm

An essential catalyst for the remarkable breakthroughs in AI that led to the modern large language models (LLMs) such as ChatGPT and Gemini has been the algorithms used to train these models on massive datasets. While the LLM architectures have gotten progressively more complex, the training algorithms have stayed relatively simple, and in fact, they have all been based on the decades-old paradigm of stochastic gradient descent (SGD). The key idea behind SGD is that in order to minimize a certain objective function (such as an LLM's error on the training data), it suffices to access only a noisy estimate of that objective at any given time (e.g., based on a small sample of the data) while making incremental progress towards the solution. This is essential for LLM training, as the datasets have become so massive one could not hope to perform computations on everything all at once. Commonly attributed to a 1951 paper by Robbins and Monro [34], SGD has seen a resurgence of interest over the last 20 years by AI researchers and computer scientists striving to understand its effectiveness, leading to numerous variants and extensions used in modern LLMs [12, 9], most notably the Adam algorithm [25]. As a result, we have gained a robust mathematical understanding of the computational complexity of SGD algorithms in a wide range of settings (e.g., see [11, 15, 5, 17]). Yet, despite this progress there is a surprising gap in the understanding of SGD: The complexity of an algorithm proposed by Stefan Kaczmarz in 1937 [24] for solving a system of linear equations - the oldest published example of an SGD algorithm, which predates Robbins and Monro's paper by over a decade - has not been settled.