Recently, Large Language Models (LLMs) have been widely adopted in a wide range of tasks, leading to increasing attention towards the research on how scaling LLMs affects their performance. Existing works, termed Scaling Laws, have discovered that the final test loss of LLMs scales as power-laws with model size, computational budget, and dataset size. However, the temporal change of the test loss of an LLM throughout its pretraining process remains unexplored, though it is valuable in many aspects, such as selecting better hyperparameters *directly* on the target LLM. In this paper, we propose the novel concept of Temporal Scaling Law, studying how the test loss of an LLM evolves as the training steps scale up. In contrast to modeling the test loss as a whole in a coarse-grained manner, we break it down and dive into the fine-grained test loss of each token position, and further develop a dynamic hyperbolic-law. Afterwards, we derive the much more precise temporal scaling law by studying the temporal patterns of the parameters in the dynamic hyperbolic-law. Results on both in-distribution (ID) and out-of-distribution (OOD) validation datasets demonstrate that our temporal scaling law accurately predicts the test loss of LLMs across training steps. Our temporal scaling law has broad practical applications. First, it enables direct and efficient hyperparameter selection on the target LLM, such as data mixture proportions. Secondly, viewing the LLM pretraining dynamics from the token position granularity provides some insights to enhance the understanding of LLM pretraining.

Large Language Models (LLMs) have achieved impressive performance across a range of natural language processing tasks. However, recent advances demonstrate that further gains—particularly in complex reasoning tasks—require more than merely scaling up model sizes or training data. One promising direction is to enable models to “think” during the reasoning process. Recently, Quiet-STaR significantly improves reasoning by generating token-level thought traces, but incurs substantial inference overhead. In this work, we propose Fast Quiet-STaR, a more efficient reasoning framework that preserves the benefits of token-level reasoning while reducing computational cost. Our method introduces a curriculum-learning-based training strategy that gradually reduces the number of thought tokens, enabling the model to internalize more abstract and concise reasoning processes. We further extend this approach to the standard Next Token Prediction (NTP) setting through reinforcement learning-based fine-tuning, resulting in Fast Quiet-STaR NTP, which eliminates the need for explicit thought token generation during inference. Experiments on four benchmark datasets with Mistral 7B and Qwen2.5 7B demonstrate that Fast Quiet-STaR consistently outperforms Quiet-STaR in terms of average accuracy under the same inference time budget. Notably, Fast Quiet-STaR NTP achieves an average accuracy improvement of 9% on Mistral 7B and 5.7% on Qwen2.5 7B, while maintaining the same inference latency.