Predicting Human-Gain Curves under Local Proxy Optimization from Repeated Ratings

Direct Preference Optimization (DPO) learns from pairwise preferences through a logistic objective whose score-difference parameterization follows the Bradley--Terry model. We study how failure of this modeling assumption limits DPO, distinguishing structural misspecification from limited model capacity and finite-sample estimation. Our main contribution is a characterization of structural approximation error through the triangle-curl of the preference log-odds, which vanishes exactly when preferences admit a Bradley--Terry representation. We upper bound the achievable excess logistic risk by a score-class approximation term and a triangle-curl term, and establish a lower bound proportional to triangle-curl under additional boundedness assumptions. For ReLU score models, including clipped models, we derive explicit approximation rates showing that increasing capacity can eliminate the score-approximation term while leaving the structural obstruction unchanged. We further establish finite-sample bounds for clipped empirical risk minimizers that separate network approximation, structural misspecification, and estimation error. We place these results within a surrogate-risk framework by identifying contextual preference accuracy (CPA) as a natural measure of agreement between preference judgments and changes in pairwise probability ratios relative to the reference policy. Standard logistic consistency and calibration results connect DPO's objective to this policy property and transfer our excess-risk upper bounds to CPA, without treating CPA as a complete measure of generated-response quality.

Progress in language model development is often driven by comparative decisions: which architecture to adopt, which pretraining corpus to use, or which training recipe to apply. Making these decisions well requires reliable performance forecasts, yet the two commonly used signals are fundamentally limited. Cross-entropy loss is poorly aligned with downstream capabilities, and direct downstream evaluation is expensive, sparse, and often uninformative at early training stages. Instead, we propose to construct proxy metrics by aggregating token-level statistics, such as entropy, top-k accuracy, and expert token rank, from a candidate model's next token distribution over expert-written solutions. Across three settings, our proxies consistently outperform loss- and compute-based baselines: 1) For cross-family model selection, they rank a heterogeneous population of reasoning models with mean Spearman Rho = 0.81 (vs. Rho = 0.36 for cross-entropy loss); 2) For pretraining data selection, they reliably rank 25 candidate corpora for a target model at roughly $10{,}000\times$ less compute than direct evaluation, pushing the Pareto frontier beyond existing methods; and 3) for training-time forecasting, they extrapolate downstream accuracy across an $18\times$ compute horizon with roughly half the error of existing alternatives. Together, these results suggest that expert trajectories are a broadly useful source of signal for assessing model capabilities, enabling reliable performance forecasting throughout the model development life cycle.

Inference-time scaling methods rely on Process Reward Models (PRMs), which are often poorly calibrated and overestimate success probabilities. We propose, to our knowledge, the first use of conditional optimal transport for calibrating PRMs, modifying conditional OT (CondOT) map learning \cite{bunne2022supervised} to estimate a monotonic conditional quantile function over success probabilities estimated by the PRM, conditioned on PRM hidden states. This yields structurally valid quantile estimates and enables efficient extraction of confidence bounds at arbitrary levels, which we integrate into the instance-adaptive scaling (IAS) framework of \cite{park2025know}. We evaluate on mathematical reasoning benchmarks spanning moderate-difficulty problems (MATH-500) and harder out-of-distribution problems (AIME). For PRMs with reliable ranking signals, our method substantially improves calibration over both uncalibrated PRMs and quantile regression. On downstream Best-of-N IAS performance, our method generally improves over uncalibrated PRMs. These results establish conditional optimal transport as another principled and practical approach to PRM calibration, offering structural guarantees and flexible uncertainty estimation.