Large language models (LLMs) have made fundamental contributions over the last a few years. To train an LLM, one needs to alternatingly run `forward' computations and backward computations. The forward computation can be viewed as attention function evaluation, and the backward computation can be viewed as a gradient computation. In previous work by [Alman and Song, NeurIPS 2023], it was proved that the forward step can be performed in almost-linear time in certain parameter regimes, but that there is no truly sub-quadratic time algorithm in the remaining parameter regimes unless the popular hypothesis $\mathsf{SETH}$ is false. In this work, we show nearly identical results for the harder-seeming problem of computing the gradient of loss function of one layer attention network, and thus for the entire process of LLM training. This completely characterizes the fine-grained complexity of every step of LLM training.

Although LLM training is typically centralized with high-bandwidth interconnects and large compute budgets, emerging methods target communication-constrained training in distributed environments. The model trade-offs introduced by this shift remain underexplored, and our goal is to study them. We use the open-source nanochat project, a compact 8K-line full-stack ChatGPT-like implementation containing tokenization, pretraining, fine-tuning, and serving, as a controlled baseline. We implement the DiLoCo algorithm as a lightweight wrapper over nanochat's training loop, performing multiple local steps per worker before synchronization with an outer optimizer, effectively reducing communication by orders of magnitude. This inner-outer training is compared against a standard data-parallel (DDP) setup. Because nanochat is small and inspectable, it enables controlled pipeline adaptations and allows direct comparison with the conventional centralized baseline. DiLoCo achieves stable convergence and competitive loss in pretraining but yields worse MMLU, GSM8K, and HumanEval scores after mid-training and SFT. We discover that using DiLoCo-pretrained weights and running mid- and post-training with DDP fails to recover performance, revealing irreversible representation drift from asynchronous updates that impairs downstream alignment. We provide this implementation as an official fork of nanochat on GitHub.

The training scale of large language models (LLMs) has reached tens of thousands of GPUs and is still continuously expanding, enabling faster learning of larger models. Accompanying the expansion of the resource scale is the prevalence of failures (CUDA error, NaN values, job hang, etc.), which poses significant challenges to training stability. Any large-scale LLM training infrastructure should strive for minimal training interruption, efficient fault diagnosis, and effective failure tolerance to enable highly efficient continuous training. This paper presents ByteRobust, a large-scale GPU infrastructure management system tailored for robust and stable training of LLMs. It exploits the uniqueness of LLM training process and gives top priorities to detecting and recovering failures in a routine manner. Leveraging parallelisms and characteristics of LLM training, ByteRobust enables high-capacity fault tolerance, prompt fault demarcation, and localization with an effective data-driven approach, comprehensively ensuring continuous and efficient training of LLM tasks. ByteRobust is deployed on a production GPU platform and achieves 97% ETTR for a three-month training job on 9,600 GPUs.

Large language models (LLMs) have made a profound impact across various fields due to their advanced capabilities. However, training these models at unprecedented scales requires extensive AI accelerator clusters and sophisticated parallelism strategies, which pose significant challenges in maintaining system reliability over prolonged training periods. A major concern is the substantial loss of training time caused by inevitable hardware and software failures. To address these challenges, we present FlashRecovery, a fast and low-cost failure recovery system comprising three core modules: (1) Active and real-time failure detection. This module performs continuous training state monitoring, enabling immediate identification of hardware and software failures within seconds, thus ensuring rapid incident response; (2) Scale-independent task restart. By employing different recovery strategies for normal and faulty nodes, combined with an optimized communication group reconstruction protocol, our approach ensures that the recovery time remains nearly constant, regardless of cluster scale; (3) Checkpoint-free recovery within one step. Our novel recovery mechanism enables single-step restoration, completely eliminating dependence on traditional checkpointing methods and their associated overhead. Collectively, these innovations enable FlashRecovery to achieve optimal Recovery Time Objective (RTO) and Recovery Point Objective (RPO), substantially improving the reliability and efficiency of long-duration LLM training. Experimental results demonstrate that FlashRecovery system can achieve training restoration on training cluster with 4, 800 devices in 150 seconds. We also verify that the time required for failure recovery is nearly consistent for different scales of training tasks.

Learning rate (LR) schedules in large language model (LLM) training often follow empirical templates: warm-up, constant plateau/stable phase, and decay (WSD). However, the mechanistic explanation for this strategy remains underexplored, and the choice of plateau height and decay schedule is largely heuristic. In this paper, we connect training dynamics to a thermodynamic analogy via the Mpemba effect - a phenomenon in which a hotter system cools faster than a colder one when quenched into the same bath. We analyze a class of "valley-river" loss landscapes, where sharp (valley) directions equilibrate quickly, while flatter (river) directions govern global descent. The Mpemba effect provides an explanation for the necessity of the warm-up phase and motivates a high plateau - rather than a low one - for accelerating loss decrease during decay. We show that for certain loss landscapes, there exists an optimal plateau learning rate - the "strong Mpemba point" - at which the slowest mode vanishes, resulting in faster convergence during the decay phase. We derive analytical conditions for its existence and estimate decay dynamics required to preserve the Mpemba advantage. Our minimal model and analysis offer a principled justification for plateau-based schedulers and provide guidance for tuning LR in LLMs with minimal hyperparameter sweep.

Training large language models (LLMs) increasingly relies on geographically distributed accelerators, causing prohibitive communication costs across regions and uneven utilization of heterogeneous hardware. We propose HALoS, a hierarchical asynchronous optimization framework that tackles these issues by introducing local parameter servers (LPSs) within each region and a global parameter server (GPS) that merges updates across regions. This hierarchical design minimizes expensive inter-region communication, reduces straggler effects, and leverages fast intra-region links. We provide a rigorous convergence analysis for HALoS under non-convex objectives, including theoretical guarantees on the role of hierarchical momentum in asynchronous training. Empirically, HALoS attains up to 7.5x faster convergence than synchronous baselines in geo-distributed LLM training and improves upon existing asynchronous methods by up to 2.1x. Crucially, HALoS preserves the model quality of fully synchronous SGD-matching or exceeding accuracy on standard language modeling and downstream benchmarks-while substantially lowering total training time. These results demonstrate that hierarchical, server-side update accumulation and global model merging are powerful tools for scalable, efficient training of new-era LLMs in heterogeneous, geo-distributed environments.

Online batch selection methods offer an adaptive alternative to static training data selection by dynamically selecting data batches during training. However, existing methods either rely on impractical reference models or simple heuristics that may not capture true data informativeness. To address these limitations, we propose \emph{GREedy Approximation Taylor Selection} (GREATS), a principled and efficient online batch selection method that applies greedy algorithm to optimize the data batch quality approximated by Taylor expansion. We develop a series of techniques to scale GREATS to large-scale model training. Extensive experiments with large language models (LLMs) demonstrate that GREATS significantly improves training convergence speed and generalization performance.

Large language models (LLMs) have made fundamental contributions over the last a few years. To train an LLM, one needs to alternatingly run forward' computations and backward computations. The forward computation can be viewed as attention function evaluation, and the backward computation can be viewed as a gradient computation. In previous work by [Alman and Song, NeurIPS 2023], it was proved that the forward step can be performed in almost-linear time in certain parameter regimes, but that there is no truly sub-quadratic time algorithm in the remaining parameter regimes unless the popular hypothesis \mathsf{SETH} is false. In this work, we show nearly identical results for the harder-seeming problem of computing the gradient of loss function of one layer attention network, and thus for the entire process of LLM training.

Recent years have witnessed a clear trend towards language models with an ever-increasing number of parameters, as well as the growing training overhead and memory usage. Distributed training, particularly through Sharded Data Parallelism (ShardedDP) which partitions optimizer states among workers, has emerged as a crucial technique to mitigate training time and memory usage. Yet, a major challenge in the scalability of ShardedDP is the intensive communication of weights and gradients. While compression techniques can alleviate this issue, they often result in worse accuracy. Driven by this limitation, we propose SDP4Bit (Toward 4Bit Communication Quantization in Sharded Data Parallelism for LLM Training), which effectively reduces the communication of weights and gradients to nearly 4 bits via two novel techniques: quantization on weight differences, and two-level gradient smooth quantization.

Beyond neural scaling laws, little is known about the laws underlying large language models (LLMs). We introduce Neural Thermodynamic Laws (NTL) -- a new framework that offers fresh insights into LLM training dynamics. On the theoretical side, we demonstrate that key thermodynamic quantities (e.g., temperature, entropy, heat capacity, thermal conduction) and classical thermodynamic principles (e.g., the three laws of thermodynamics and the equipartition theorem) naturally emerge under river-valley loss landscape assumptions. On the practical side, this scientific perspective yields intuitive guidelines for designing learning rate schedules.