Scaling laws for very large neural nets

2.3 Observational

Ruan, Maddison, and Hashimoto (2025):

Understanding how language model performance varies with scale is critical to benchmark and algorithm development. Scaling laws are one approach to building this understanding, but the requirement of training models across many different scales has limited their use. We propose an alternative, observational approach that bypasses model training and instead builds scaling laws from ~100 publicly available models. Building a single scaling law from multiple model families is challenging due to large variations in their training compute efficiencies and capabilities. However, we show that these variations are consistent with a simple, generalized scaling law where language model performance is a function of a low-dimensional capability space, and model families only vary in their efficiency in converting training compute to capabilities. Using this approach, we show the surprising predictability of complex scaling phenomena: we show that several emergent phenomena follow a smooth, sigmoidal behaviour and are predictable from small models; we show that the agent performance of models such as GPT-4 can be precisely predicted from simpler non-agentic benchmarks; and we show how to predict the impact of post-training interventions like Chain-of-Thought and Self-Consistency as language model capabilities continue to improve.

Standard compute scaling In compute scaling laws, there is a hypothesized power-law relationship between models’ compute measures $C_m$ (e.g., training FLOPs) and their errors $E_m$ (e.g., perplexity). Specifically, for a model $m$ within a family $f$ (e.g., Llama-2 7B, 13B, and 70B) we hypothesize

$$\log \left(E_m\right)\approx {\beta }_f\log \left(C_m\right)+{\alpha }_f$$

and if this linear fit is sufficiently accurate, we draw inferences about the performance of a model at future compute scales $C^{\prime }>C$ by extrapolating this relationship. However, fitting such a scaling law can be tricky, as each model family $f$ and downstream benchmark has its own scaling coefficients ${\beta }_f$ and ${\alpha }_f$. This means that scaling experiments, especially for post-training analysis, are often fitted on very few (3-5) models sharing the same model family, and any predictions are valid only for a specific scaling strategy used within a model family. Several studies […] have generalized the functional form to analyse the scaling of LMs’ downstream performance (where $E_m$ is normalised to $[0,1]$) with a sigmoidal link function $\sigma$ :

$${\sigma }^{-1}\left(E_m\right)\approx {\beta }_f\log \left(C_m\right)+{\alpha }_f$$

Observational scaling In our work, we hypothesize the existence of a low-dimensional capability measure for LMs that relate compute to more complex LM capabilities and can be extracted from observable standard LM benchmarks, as illustrated in Figure 2. Specifically, given $T$ simple benchmarks and $B_{i,m}$ the error of a model $m$ on benchmark $i\in [T]$, we hypothesize that there exists some capability vector $S_m\in {\mathbb{R}}^K$ such that,

$$\begin{array}{rl}{\sigma }^{-1}\left(E_m\right)& \approx {\beta }^{\mathrm{\top }}{S}_m+\alpha \\ S_m& \approx {\theta }_f\log \left(C_m\right)+{\nu }_f\\ B_{i,m}& \approx {\gamma }_i^{\mathrm{\top }}{S}_m\end{array}$$

for ${\theta }_f,{\nu }_f,\beta \in {\mathbb{R}}^K,\alpha \in \mathbb{R}$, and orthonormal vectors ${\gamma }_i\in {\mathbb{R}}^K$.