Enforcing Axioms for AI Alignment under Loss-Based Rules

arXiv:2611.13369v1 Announce Type: cross Abstract: Many alignment methods, most notably reinforcement learning from human feedback (RLHF), train reward models by minimizing a loss on pairwise comparisons of model responses. Inspired by principle-guided methods such as Constitutional AI, we treat each principle as a voter that compares responses according to that principle, for example, preferring the more helpful or less toxic response. We represent these voters as linear directions in a fixed embedding space, a simplifying assumption motivated by the Linear Representation Hypothesis. In this linear social choice model, Ge et al. (2024) showed that an optimal linear reward model can violate Pareto optimality (PO): From the principles-as-voters lens, this means a response A can be less helpful and more toxic than B, yet still receive a higher reward. We ask whether richer reward models or changes to comparison data can recover PO within loss-based training. We show that PO can still fail for polynomials of any fixed degree and one-hidden-layer ReLU multilayer perceptrons (MLPs) of any fixed width $ d\ge 2 $ with suitable finite parameter bounds. We then recover PO for loss-based trai