Chinchilla Approach 2 is among the most widely used methods for fitting neural scaling laws. Its parabolic approximation introduces systematic biases in compute-optimal allocation estimates, even on noise-free synthetic data. Applied to published Llama 3 IsoFLOP data at open frontier compute scales, these biases imply a parameter underallocation corresponding to 6.5% of the $3.8\times10^{25}$ FLOP training budget and \$1.4M (90% CI: \$412K-\$2.9M) in unnecessary compute at 50% H100 MFU. Simulated multimodal model misallocations show even greater opportunity costs due to higher loss surface asymmetry. Three sources of this error are examined: IsoFLOP sampling grid width (Taylor approximation accuracy), uncentered IsoFLOP sampling, and loss surface asymmetry ($α\neq β$). Chinchilla Approach 3 largely eliminates these biases but is often regarded as less data-efficient, numerically unstable, prone to local minima, and harder to implement. Each concern is shown to be unfounded or addressable, especially when the partially linear structure of the objective is exploited via Variable Projection, enabling unbiased inference on all five loss surface parameters through a two-dimensional optimization that is well-conditioned, analytically differentiable, and amenable to dense, or even exhaustive, grid search. It may serve as a more convenient replacement for Approach 2 or a more scalable alternative for adaptations of Approach 3 to richer scaling law formulations. See https://github.com/Open-Athena/vpnls for details and https://openathena.ai/scaling-law-analysis for other results from this study.

However, practical model deployment often operates in the gap between these two limit cases, as real-world applications demand adaptation to specific subdomains, tasks or concepts -- spread over the entire, varying life cycle of a model.

We explain the discrepancy by reproducing the Kaplan et al. scaling law on two datasets (OpenWebText2 and RefinedWeb)

Do feature attribution methods correctly attribute features?

Test-time scaling has emerged as a critical avenue for enhancing the reasoning capabilities of Large Language Models (LLMs). Though the straight-forward ''best-of-$N$'' (BoN) strategy has already demonstrated significant improvements in performance, it lacks principled guidance on the choice of $N$, budget allocation, and multi-stage decision-making, thereby leaving substantial room for optimization. While many works have explored such optimization, rigorous theoretical guarantees remain limited. In this work, we propose new methodologies to predict and improve scaling properties via tail-guided search. By estimating the tail distribution of rewards, our method predicts the scaling law of LLMs without the need for exhaustive evaluations. Leveraging this prediction tool, we introduce Scaling-Law Guided (SLG) Search, a new test-time algorithm that dynamically allocates compute to identify and exploit intermediate states with the highest predicted potential. We theoretically prove that SLG achieves vanishing regret compared to perfect-information oracles, and achieves expected rewards that would otherwise require a polynomially larger compute budget required when using BoN. Empirically, we validate our framework across different LLMs and reward models, confirming that tail-guided allocation consistently achieves higher reward yields than Best-of-$N$ under identical compute budgets. Our code is available at https://github.com/PotatoJnny/Scaling-Law-Guided-search.