Efficiently Estimating Optimal Hyperparameter Scaling Laws through Power-Law Entropy Search

The efficacy of continued pre-training for Large Language Models (LLMs) hinges upon hyperparameter configurations, such as learning rate and batch size. However, current practices often rely on heuristics or grid searches, leading to training instability and excessive costs. In this work, we first empirically discover that optimal hyperparameters follow stable and predictable scaling laws throughout the continued pre-training process. Leveraging these insights, we propose a novel framework to establish quantitative relationships between compute budget and optimal hyperparameters for a given checkpoint. Our approach has two stages: (1) \textit{Empirical Law Discovery}, where we train small-scale proxy models to derive functions mapping compute budget to optimal hyperparameters via standard loss-compute scaling laws; and (2) \textit{State-Aware Hyperparameter Prediction}, where we evaluate an initial checkpoint's validation loss and use the inverse scaling law to estimate its \textit{equivalent pre-training compute} -- the compute needed to achieve the same loss from scratch. Combining this with the planned compute budget, we predict optimal hyperparameters for the target run. Empirical results demonstrate that our method reduces the hyperparameter search overhead by up to 90% while achieving comparable or superior performance relative to baselines. This model-agnostic framework generalizes across architectures, providing a principled and efficient methodology for diverse continued pre-training scenarios starting from any given point.

Hyperparameter transfer allows extrapolating optimal optimization hyperparameters from small to large scales, making it critical for training large language models (LLMs). This is done either by fitting a scaling law to the hyperparameters or by a judicious choice of parameterization, such as Maximal Update ($μ$P), that renders optimal hyperparameters approximately scale invariant. In this paper, we first develop a framework to quantify hyperparameter transfer through three metrics: (1) the quality of the scaling law fit, (2) the robustness to extrapolation errors, and (3) the asymptotic loss penalty due to choice of parameterization. Next, we investigate through a comprehensive series of ablations why $μ$P appears to offer high-quality learning rate transfer relative to standard parameterization (SP), as existing theory is inadequate. We find that the overwhelming benefit of $μ$P relative to SP when training with AdamW arises simply from maximizing the learning rate of the embedding layer. In SP, the embedding layer learning rate acts as a bottleneck that induces training instabilities; increasing it by a factor of width to match $μ$P dramatically smooths out training while improving hyperparameter transfer. We also find that weight decay improves the scaling law fits, while, in the fixed token-per-parameter setting, it hurts the robustness of the extrapolation.

Reinforcement learning (RL) for large language models (LLMs) is highly sensitive to hyperparameter configurations, making hyperparameter optimization (HPO) essential yet computationally expensive. Existing multi-fidelity HPO methods remain inefficient for LLM RL due to the massive model scale and resource-intensive training cycles. In this paper, we propose Joint Fidelity Hyperparameter Optimization (JF-HPO), which simultaneously adapts both model size and training budget as fidelity. JF-HPO is empowered by: (i) it leverages a small proxy model of the target LLM for efficient training and evaluation in each HPO trial; (ii) it integrates carefully designed early-stopping strategies based on training dynamics; (iii) it introduces an efficient checkpointing mechanism to eliminate redundant computations. Compared with existing HPO methods, JF-HPO significantly improves the computational efficiency of each trial (up to 14.9 times), while achieving better or competitive predictive accuracy under the same time budget. Notably, compared with utilizing hyperparameter configurations from the VeRL Recipe, JF-HPO delivers performance improvements ranging from 5.8% to 111.6%.