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.

Orthogonal Bootstrap: Efficient Simulation of Input Uncertainty

Bootstrap is a popular methodology for simulating input uncertainty. However, it can be computationally expensive when the number of samples is large. We propose a new approach called \textbf{Orthogonal Bootstrap} that reduces the number of required Monte Carlo replications. We decomposes the target being simulated into two parts: the \textit{non-orthogonal part} which has a closed-form result known as Infinitesimal Jackknife and the \textit{orthogonal part} which is easier to be simulated. We theoretically and numerically show that Orthogonal Bootstrap significantly reduces the computational cost of Bootstrap while improving empirical accuracy and maintaining the same width of the constructed interval.

The Local Landscape of Phase Retrieval Under Limited Samples

In this paper, we provide a fine-grained analysis of the local landscape of phase retrieval under the regime with limited samples. Our aim is to ascertain the minimal sample size necessary to guarantee a benign local landscape surrounding global minima in high dimensions. Let $n$ and $d$ denote the sample size and input dimension, respectively. We first explore the local convexity and establish that when $n=o(d\log d)$, for almost every fixed point in the local ball, the Hessian matrix must have negative eigenvalues as long as $d$ is sufficiently large. Consequently, the local landscape is highly non-convex. We next consider the one-point strong convexity and show that as long as $n=\omega(d)$, with high probability, the landscape is one-point strongly convex in the local annulus: $\{w\in\mathbb{R}^d: o_d(1)\leqslant \|w-w^*\|\leqslant c\}$, where $w^*$ is the ground truth and $c$ is an absolute constant. This implies that gradient descent initialized from any point in this domain can converge to an $o_d(1)$-loss solution exponentially fast. Furthermore, we show that when $n=o(d\log d)$, there is a radius of $\widetilde\Theta\left(\sqrt{1/d}\right)$ such that one-point convexity breaks in the corresponding smaller local ball. This indicates an impossibility to establish a convergence to exact $w^*$ for gradient descent under limited samples by relying solely on one-point convexity.