Rethinking Bradley-Terry Models in Preference-Based Reward Modeling: Foundations, Theory, and Alternatives

The Bradley-Terry (BT) model is a common and successful practice in reward modeling for Large Language Model (LLM) alignment. However, it remains unclear why this model -- originally developed for multi-player stochastic game matching -- can be adopted to convert pairwise response comparisons to reward values and make predictions. Especially given the fact that only a limited number of prompt-response pairs are sparsely compared with others. In this paper, we first revisit the foundations of using BT models in reward modeling, and establish the convergence rate of BT reward models based on deep neural networks using embeddings, providing a theoretical foundation for their use. Despite theoretically sound, we argue that the BT model is not a necessary choice from the perspective of downstream optimization. This is because a reward model only needs to preserve the correct ranking predictions through a monotonic transformation of the true reward. We highlight the critical concept of order consistency in reward modeling and demonstrate that the BT model possesses this property. Consequently, we propose a simple and straightforward upper-bound algorithm, compatible with off-the-shelf binary classifiers, as an alternative order-consistent reward modeling objective. To offer practical insights, we empirically evaluate the performance of these different reward modeling approaches across more than 12,000 experimental setups, using $6$ base LLMs, $2$ datasets, and diverse annotation designs that vary in quantity, quality, and pairing choices in preference annotations.

Multi-marginal Schrödinger Bridges with Iterative Reference

Practitioners frequently aim to infer an unobserved population trajectory using sample snapshots at multiple time points. For instance, in single-cell sequencing, scientists would like to learn how gene expression evolves over time. But sequencing any cell destroys that cell. So we cannot access any cell's full trajectory, but we can access snapshot samples from many cells. Stochastic differential equations are commonly used to analyze systems with full individual-trajectory access; since here we have only sample snapshots, these methods are inapplicable. The deep learning community has recently explored using Schr\"odinger bridges (SBs) and their extensions to estimate these dynamics. However, these methods either (1) interpolate between just two time points or (2) require a single fixed reference dynamic within the SB, which is often just set to be Brownian motion. But learning piecewise from adjacent time points can fail to capture long-term dependencies. And practitioners are typically able to specify a model class for the reference dynamic but not the exact values of the parameters within it. So we propose a new method that (1) learns the unobserved trajectories from sample snapshots across multiple time points and (2) requires specification only of a class of reference dynamics, not a single fixed one. In particular, we suggest an iterative projection method inspired by Schr\"odinger bridges; we alternate between learning a piecewise SB on the unobserved trajectories and using the learned SB to refine our best guess for the dynamics within the reference class. We demonstrate the advantages of our method via a well-known simulated parametric model from ecology, simulated and real data from systems biology, and real motion-capture data.

Consistent Validation for Predictive Methods in Spatial Settings

Spatial prediction tasks are key to weather forecasting, studying air pollution, and other scientific endeavors. Determining how much to trust predictions made by statistical or physical methods is essential for the credibility of scientific conclusions. Unfortunately, classical approaches for validation fail to handle mismatch between locations available for validation and (test) locations where we want to make predictions. This mismatch is often not an instance of covariate shift (as commonly formalized) because the validation and test locations are fixed (e.g., on a grid or at select points) rather than i.i.d. from two distributions. In the present work, we formalize a check on validation methods: that they become arbitrarily accurate as validation data becomes arbitrarily dense. We show that classical and covariate-shift methods can fail this check. We instead propose a method that builds from existing ideas in the covariate-shift literature, but adapts them to the validation data at hand. We prove that our proposal passes our check. And we demonstrate its advantages empirically on simulated and real data.