Scaling Laws for Precision in High-Dimensional Linear Regression

Since scaling laws were introduced for modern LLMs (Kaplan et al., 2020), several works have extended them to account for quantization, tracking the emerging low-precision data types supported on AI accelerators. We develop a more accurate scaling law for quantized pretraining with mixed precision across the forward and backward passes, grounded in optimization theory and validated empirically. Our formulation separates the effects of quantization on model capacity and on optimization efficiency, and explicitly models the persistent optimization error from forward and backward quantization identified through convergence analysis. To select the regimes for fitting and evaluating our law, we first decompose existing 4-bit training recipes into their forward and backward components, isolating those that most affect loss. Our Qwen3 experiments under selected 4-bit regimes, together with published Llama-style experiments, show that our law predicts loss with up to $11\times$ lower MSE than existing laws as model size and token count scale. Finally, we extend our law to layer-wise mixed precision and identify the optimal quantized-pretraining recipe, i.e., model size, training tokens, and per-layer forward/backward precision, under a hardware-aware compute budget. Unlike prior precision scaling laws, our law predicts that the optimal precision increases with the compute budget.

Neural scaling laws are foundational for language model development, yet standard formulations systematically under- and overestimate loss at data-scarce and overtraining extremes. This failure originates in the underlying assumption that model size and training data impact the loss independently. To address this, we introduce the Skaling law, a generalized functional form that couples model capacity and data through a single interaction exponent. This simple extension reduces the Mean Absolute Percentage Error (MAPE) by 1.5-3x across both interpolation and extrapolation regimes. When paired with a sparse grid strategy restricted to low-compute regimes, the Skaling law achieves accurate full-grid extrapolation using approximately 10x less compute than uniform sweeps. By enabling reliable performance prediction from small-scale experiments, the Skaling law provides a more robust and resource-efficient framework for allocating compute budgets in next-generation model training.

Training compute is increasingly outpacing the availability of high-quality data. This shifts the central challenge from optimal compute allocation to extracting maximum value from limited data. The widely adopted Chinchilla scaling law assumes every training token is unique. This limits its ability to guide pretraining decisions in data-constrained regimes. We model the excess loss under repetition with a simple additive overfitting penalty and find that it accurately describes model behavior. Our scaling law yields qualitatively new compute-optimal allocation advice. Beyond a point, further repetition is counterproductive and compute is better spent on model capacity. We show that following our law's recommended configuration improves performance in data-constrained regimes. Finally, because our one-parameter form isolates overfitting in a single coefficient, it enables direct comparison across training configurations. As a case study, we show that strong weight decay ($λ=1.0$) reduces this coefficient by approximately 70%, providing a scaling-law explanation for recent findings that optimal weight decay in data-constrained regimes is an order of magnitude larger than standard practice.