Predictable Scaling: Efficient Model Exploration via Scaling-Law Prediction

Scaling laws guide the design choices for training large foundation models, but deriving them involves training an exhaustive grid over hyperparameters, token budgets, and parameter counts, which is computationally expensive. Fitting a scaling law, however, only requires the best-loss frontier across compute scales, discarding most of the trained configurations. We propose a framework for efficient scaling law construction that formulates data collection as a Bayesian optimization problem, and introduce metrics for comparing scaling law fitting methods under constrained compute budgets. We find that progressively expanding the compute budget during acquisition, mirroring the compute-ordered evaluation of configurations in practice, substantially improves recovery efficiency. Augmenting the observed configurations with surrogate-fantasized evaluations then recovers the broader experimental grid, allowing accurate scaling law fitting without training every configuration. Together, these can closely match scaling law fits over a full dense grid at computational savings of up to $10\text{--}100\times$.

Empirical scaling laws relate loss to parameters $N$, training tokens $D$ and compute $C$. This technical note explains three mechanisms behind their exponents. Local approximation bounds function error at rate $N_e^{-γs/d}$, with effective parameters $N_e$, smoothness $s$, reuse exponent $γ$ and dimension $d$; the loss function determines the risk bound. Independent feature coverage gives $α_D=(β-1)/ζ$ from predictive-weight decay $β$ and occurrence rarity $ζ$. Separable-loss optimization determines training allocations and deployment crossovers. Nonlinear scoring can produce apparent emergent abilities, while grokking concerns delayed generalization and verification constrains test-time scaling. For regular molecular configurations, the $\mathrm{SE}(3)$ quotient gives $d_{\mathrm{int}}=d-6$ and relative approximation-exponent gain $6/d_{\mathrm{int}}$. Optimized dense-to-equivariant excess-loss ratios scale as $C^{p_{\mathrm{eq}}-p_{\mathrm{dense}}}$, where $p_j$ are compute-loss exponents and the prefactor depends on loss amplitudes and FLOP costs. Domain examples also cover visual resolution, conformational coverage and correlated-sample precision.

Scaling laws predict large-model performance from small experiments, but finding a reliable law itself requires costly experiments. SLDBench tests whether language-model agents can discover scaling laws when all experimental results are given upfront. We introduce SLDBench v2 with two extensions. First, we expand the corpus from 8 to 127 scenarios spanning language-model pretraining, downstream performance, and vision-language transfer, and select 30 challenging scenarios on which a reference power-law fit reaches $R^2 \le 0.8$; they represent at least $1.09 \times 10^{23}$ FLOPs of source-reported training compute. Second, we introduce active scaling law discovery: an agent chooses which experiments to run, pays each one's recorded cost (e.g., training FLOPs), and revises its law as outcomes arrive, and is scored on the same hidden extrapolation targets as in the passive setting. Across 8 frontier agent systems, median $R^2$ drops from 0.72-0.75 (passive) to 0.24-0.57 (active), and prediction error rises 1.5-4.7 times, depending on the system. Agents favor cheap experiments, spending 8-44% of the full-pool cost on average; low-cost acquisition can yield strong predictions, but cheap purchases also leave experimental conditions needed for extrapolation unobserved. Experimental planning thus remains a key open challenge for agents that assist model development in the prevailing AI-for-AI study. Project page: https://lingbei-meng.github.io/SLDBench-v2/.