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In response to the inability of simple monotonic functions to capture all aspects of LLM learning, researchers explore more sophisticated and diverse mathematical functions to model training curves. This approach, exemplified in studies by Alabdulmohsin et al. [2022] and Caballero et al. [2023], aims to find better fits for complex phenomena that standard scaling laws miss.

Fitting LLM Learning Curves with Diverse Functions

When modeling the performance of a Large Language Model, the loss, denoted as $$\mathcal{L}(x)$$, is often expressed as a function of a variable of interest, $$x$$. This variable $$x$$ can represent factors like the number of model parameters, and the loss $$\mathcal{L}(x)$$ is typically measured using metrics such as the cross-entropy loss on test data. The most straightforward mathematical form used to model this relationship is a power law.

Power Law Formulation for LLM Loss

Based on the scenario described, evaluate the suitability of using a simple, monotonically decreasing function to model this learning curve. Justify your evaluation and describe the key characteristic that a more appropriate mathematical function must possess to accurately capture the observed behavior.

Modeling an Unconventional Learning Curve

Critique the exclusive use of simple, monotonically decreasing functions for modeling the performance of Large Language Models as they scale. In your response, identify at least one significant limitation of this approach and explain why exploring more diverse and complex mathematical functions is a necessary step for researchers in the field.

Evaluating Modeling Approaches for LLM Learning Curves

A research team observes the performance of a new model as they increase the amount of training data. They plot the test error and notice a distinct pattern: the error first decreases, then temporarily increases, before starting to decrease again. When attempting to create a mathematical formula to predict this error curve, which approach would be most appropriate?

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