Differential Voting: Loss Functions For Axiomatically Diverse Aggregation of Heterogeneous Preferences
Zhiyu An, Duaa Nakshbandi, Wan Du
TL;DR
The paper addresses how RLHF-style preference aggregation implicitly selects a voting rule and argues that this choice encodes normative assumptions. It introduces Differential Voting, a framework of differentiable, instance-wise losses whose population optima correspond to classical voting rules such as BTL/majority, Copeland, and Kemeny, and analyzes their gradient fields and limiting behavior. The contributions include differentiable surrogates for Copeland and Kemeny, rigorous consistency results, axiomatic analyses, and controlled experiments that reveal how loss geometry shapes aggregation under heterogeneous preferences. This work enables explicit design choices over normative guarantees and optimization stability in alignment pipelines, offering principled trade-offs between axioms and practical learnability. The results demonstrate that loss design is not merely a statistical surrogate but a mechanism that determines the social-choice behavior of the learned reward function in RLHF contexts.
Abstract
Reinforcement learning from human feedback (RLHF) implicitly aggregates heterogeneous human preferences into a single utility function, even though the underlying utilities of the participants are in practice diverse. Hence, RLHF can be viewed as a form of voting, where the aggregation mechanism is defined by the loss function. Although Arrow's Impossibility Theorem suggests that different mechanisms satisfy different sets of desirable axioms, most existing methods rely on a single aggregation principle, typically the Bradley-Terry-Luce (BTL) model, which corresponds to Borda count voting. This restricts the axiomatic properties of the learned reward and obscures the normative assumptions embedded in optimization. In this work, we introduce Differential Voting, a unifying framework that constructs instance-wise, differentiable loss functions whose population-level optima provably correspond to distinct classical voting rules. We develop differentiable surrogates for majority-based aggregation (BTL), Copeland, and Kemeny rules, and formally analyze their calibration properties, gradient fields, and limiting behavior as smoothing parameters vanish. For each loss, we establish consistency with the corresponding social choice rule and characterize the axioms it satisfies or violates. Our analysis shows how design choices in loss geometry-such as margin sensitivity and boundary concentration-directly translate into normative aggregation behavior. Differential Voting makes preference aggregation an explicit and controllable design choice in RLHF, enabling principled trade-offs between axiomatic guarantees and optimization stability. Code to reproduce our experiments is open-sourced.