Theoretical Tensions in RLHF: Reconciling Empirical Success with Inconsistencies in Social Choice Theory

Despite its empirical success, Reinforcement Learning from Human Feedback (RLHF) has been shown to violate almost all the fundamental axioms in social choice theory -- such as majority consistency, pairwise majority consistency, and Condorcet consistency. This raises a foundational question: why does RLHF perform so well in practice if it fails these seemingly essential properties? In this paper, we resolve this paradox by showing that under mild and empirically plausible assumptions on the preference profile, RLHF does satisfy pairwise majority and Condorcet consistency. These assumptions are frequently satisfied in real-world alignment tasks, offering a theoretical explanation for RLHF's strong practical performance. Furthermore, we show that a slight modification to the reward modeling objective can ensure pairwise majority or Condorcet consistency even under general preference profiles, thereby improving the alignment process. Finally, we go beyond classical axioms in economic and social choice theory and introduce new alignment criteria -- preference matching, preference equivalence, and group preference matching -- that better reflect the goal of learning distributions over responses. We show that while RLHF satisfies the first two properties, it fails to satisfy the third. We conclude by discussing how future alignment methods may be designed to satisfy all three.

Fundamental Limits of Game-Theoretic LLM Alignment: Smith Consistency and Preference Matching

Nash Learning from Human Feedback is a game-theoretic framework for aligning large language models (LLMs) with human preferences by modeling learning as a two-player zero-sum game. However, using raw preference as the payoff in the game highly limits the potential of the game-theoretic LLM alignment framework. In this paper, we systematically study using what choices of payoff based on the pairwise human preferences can yield desirable alignment properties. We establish necessary and sufficient conditions for Condorcet consistency, diversity through mixed strategies, and Smith consistency. These results provide a theoretical foundation for the robustness of game-theoretic LLM alignment. Further, we show the impossibility of preference matching -- i.e., no smooth and learnable mappings of pairwise preferences can guarantee a unique Nash equilibrium that matches a target policy, even under standard assumptions like the Bradley-Terry-Luce model. This result highlights the fundamental limitation of game-theoretic LLM alignment.

Statistical Impossibility and Possibility of Aligning LLMs with Human Preferences: From Condorcet Paradox to Nash Equilibrium

Aligning large language models (LLMs) with diverse human preferences is critical for ensuring fairness and informed outcomes when deploying these models for decision-making. In this paper, we seek to uncover fundamental statistical limits concerning aligning LLMs with human preferences, with a focus on the probabilistic representation of human preferences and the preservation of diverse preferences in aligned LLMs. We first show that human preferences can be represented by a reward model if and only if the preference among LLM-generated responses is free of any Condorcet cycle. Moreover, we prove that Condorcet cycles exist with probability converging to one exponentially fast under a probabilistic preference model, thereby demonstrating the impossibility of fully aligning human preferences using reward-based approaches such as reinforcement learning from human feedback. Next, we explore the conditions under which LLMs would employ mixed strategies -- meaning they do not collapse to a single response -- when aligned in the limit using a non-reward-based approach, such as Nash learning from human feedback (NLHF). We identify a necessary and sufficient condition for mixed strategies: the absence of a response that is preferred over all others by a majority. As a blessing, we prove that this condition holds with high probability under the probabilistic preference model, thereby highlighting the statistical possibility of preserving minority preferences without explicit regularization in aligning LLMs. Finally, we leverage insights from our statistical results to design a novel, computationally efficient algorithm for finding Nash equilibria in aligning LLMs with NLHF. Our experiments show that Llama-3.2-1B, aligned with our algorithm, achieves a win rate of 60.55\% against the base model.

Diffusion-PINN Sampler

Recent success of diffusion models has inspired a surge of interest in developing sampling techniques using reverse diffusion processes. However, accurately estimating the drift term in the reverse stochastic differential equation (SDE) solely from the unnormalized target density poses significant challenges, hindering existing methods from achieving state-of-the-art performance. In this paper, we introduce the Diffusion-PINN Sampler (DPS), a novel diffusion-based sampling algorithm that estimates the drift term by solving the governing partial differential equation of the log-density of the underlying SDE marginals via physics-informed neural networks (PINN). We prove that the error of log-density approximation can be controlled by the PINN residual loss, enabling us to establish convergence guarantees of DPS. Experiments on a variety of sampling tasks demonstrate the effectiveness of our approach, particularly in accurately identifying mixing proportions when the target contains isolated components.