Power Lines: Scaling Laws for Weight Decay and Batch Size in LLM Pre-training

Efficient LLM pre-training requires well-tuned hyperparameters (HPs), including learning rate {\eta} and weight decay {\lambda}. We study scaling laws for HPs: formulas for how to scale HPs as we scale model size N, dataset size D, and batch size B. Recent work suggests the AdamW timescale, B/({\eta}{\lambda}D), should remain constant across training settings, and we verify the implication that optimal {\lambda} scales linearly with B, for a fixed N,D. However, as N,D scale, we show the optimal timescale obeys a precise power law in the tokens-per-parameter ratio, D/N. This law thus provides a method to accurately predict {\lambda}opt in advance of large-scale training. We also study scaling laws for optimal batch size Bopt (the B enabling lowest loss at a given N,D) and critical batch size Bcrit (the B beyond which further data parallelism becomes ineffective). In contrast with prior work, we find both Bopt and Bcrit scale as power laws in D, independent of model size, N. Finally, we analyze how these findings inform the real-world selection of Pareto-optimal N and D under dual training time and compute objectives.

Straight to Zero: Why Linearly Decaying the Learning Rate to Zero Works Best for LLMs

LLMs are commonly trained with a learning rate (LR) warmup, followed by cosine decay to 10% of the maximum (10x decay). In a large-scale empirical study, we show that under an optimal peak LR, a simple linear decay-to-zero (D2Z) schedule consistently outperforms other schedules when training at compute-optimal dataset sizes. D2Z is superior across a range of model sizes, batch sizes, datasets, and vocabularies. Benefits increase as dataset size increases. Leveraging a novel interpretation of AdamW as an exponential moving average of weight updates, we show how linear D2Z optimally balances the demands of early training (moving away from initial conditions) and late training (averaging over more updates in order to mitigate gradient noise). In experiments, a 610M-parameter model trained for 80 tokens-per-parameter (TPP) using D2Z achieves lower loss than when trained for 200 TPP using 10x decay, corresponding to an astonishing 60% compute savings. Models such as Llama2-7B, trained for 286 TPP with 10x decay, could likely have saved a majority of compute by training with D2Z.

Normalization Layer Per-Example Gradients are Sufficient to Predict Gradient Noise Scale in Transformers

Per-example gradient norms are a vital ingredient for estimating gradient noise scale (GNS) with minimal variance. Observing the tensor contractions required to compute them, we propose a method with minimal FLOPs in 3D or greater tensor regimes by simultaneously computing the norms while computing the parameter gradients. Using this method we are able to observe the GNS of different layers at higher accuracy than previously possible. We find that the total GNS of contemporary transformer models is predicted well by the GNS of only the normalization layers. As a result, focusing only on the normalization layer, we develop a custom kernel to compute the per-example gradient norms while performing the LayerNorm backward pass with zero throughput overhead. Tracking GNS on only those layers, we are able to guide a practical batch size schedule that reduces training time by 18% on a Chinchilla-optimal language model.