Data Mixing as Mixture Experiment: Response Surface Methodology and Optimal Design for Large Language Model Pretraining

Data mixing is a central design problem in large language model pretraining: given a fixed token budget, practitioners must decide how much data to allocate to each domain. Recent proxy-based methods address this problem by training small models on candidate mixtures, fitting a response model, and using the response to select mixtures for larger-scale training. We show that this workflow has the structure of a classical mixture experiment. Under this view, data domains are mixture components, token shares are component proportions, proxy-training runs are experimental design points, and validation loss defines a response surface over the probability simplex. We develop this formulation using sparse second-order Scheffé response-surface models and construct model-robust $\mathcal{I}$-optimal designs for proxy data-mixing experiments. Using RegMix as an empirical case study, we demonstrate how the framework can both interpret observed mixture responses and design more efficient proxy experiments. The Scheffé analysis shows that domain value is strongly relational: several domains that are weak under additive effects become favourable through pairwise interactions, especially through combinations with web-derived text. The sparse Scheffé model preserves mixture rankings across model scales and remains competitive with a flexible machine-learning predictor while providing an explicit decomposition of additive and interaction effects. In a simulation study calibrated to observed proxy-training responses, model-robust $\mathcal{I}$-optimal designs recover the relevant mixture ordering after removing about 25\% of the original proxy runs. These results suggest that LLM data mixing should be treated not only as a prediction problem, but also as an experimental-design problem in which the proxy mixtures themselves can be chosen to improve statistical efficiency.

Retrospective Orthogonal Design: Response-Surface Reconstruction from Observational Data

Regression estimates from observational data can depend on specification under multicollinearity, while sequential sums of squares (SS) depend on term order. We introduce Retrospective Orthogonal Design (ROD), which reconstructs conditional mean surfaces on a probability-balanced lattice. ROD preserves observed cell means, completes unsupported cells, applies weighted tensor-product contrasts, and evaluates the reconstructed surface through piecewise-affine interpolation over Freudenthal polyhedra. Resolution and completion are selected jointly by validation among rank-admissible candidates, followed by refitting and evaluation on an untouched test set. For an admissible lattice, $\mathbf{X}^{\top}\mathbf{W}\mathbf{X}=c\mathbf{I}$, yielding specification-invariant contrast effects and unique, order-independent SS within the retained contrast space. Response-free projection calibration maps the fixed reconstruction onto a declared scientific basis and corrects finite-resolution recovery loss. Across 6,480 simulation conditions spanning nine data-generating processes, ROD matched or exceeded polynomial regression in five processes and performed strongest on threshold, sign-interaction, and localized surfaces. For the quadratic-interaction process, mean out-of-sample $R^2$ differed by only $0.0001$, while calibrated coefficient bias remained small across prespecified targets. A Rao-based information adjustment provides dependence-aware sample-size guidance for ROD planning. In a weighted Mincer application, ROD produced the highest out-of-sample $R^2$ point estimate, with substantial interval overlap with polynomial regression, and provided exhaustive SS allocations invariant to term-entry order.