  Scaling law principles indicate a power-law correlation between loss and
variables such as model size, dataset size, and computational resources
utilized during training. These principles play a vital role in optimizing
various aspects of model pre-training, ultimately contributing to the success
of large language models such as GPT-4, Llama and Gemini. However, the original
scaling law paper by OpenAI did not disclose the complete details necessary to
derive the precise scaling law formulas, and their conclusions are only based
on models containing up to 1.5 billion parameters. Though some subsequent works
attempt to unveil these details and scale to larger models, they often neglect
the training dependency of important factors such as the learning rate, context
length and batch size, leading to their failure to establish a reliable formula
for predicting the test loss trajectory. In this technical report, we confirm
that the scaling law formulations proposed in the original OpenAI paper remain
valid when scaling the model size up to 33 billion, but the constant
coefficients in these formulas vary significantly with the experiment setup. We
meticulously identify influential factors and provide transparent, step-by-step
instructions to estimate all constant terms in scaling-law formulas by training
on models with only 1M~60M parameters. Using these estimated formulas, we
showcase the capability to accurately predict various attributes for models
with up to 33B parameters before their training, including (1) the minimum
possible test loss; (2) the minimum required training steps and processed
tokens to achieve a specific loss; (3) the critical batch size with an optimal
time/computation trade-off at any loss value; and (4) the complete test loss
trajectory with arbitrary batch size.
