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.

Restoring Calibration for Aligned Large Language Models: A Calibration-Aware Fine-Tuning Approach

One of the key technologies for the success of Large Language Models (LLMs) is preference alignment. However, a notable side effect of preference alignment is poor calibration: while the pre-trained models are typically well-calibrated, LLMs tend to become poorly calibrated after alignment with human preferences. In this paper, we investigate why preference alignment affects calibration and how to address this issue. For the first question, we observe that the preference collapse issue in alignment undesirably generalizes to the calibration scenario, causing LLMs to exhibit overconfidence and poor calibration. To address this, we demonstrate the importance of fine-tuning with domain-specific knowledge to alleviate the overconfidence issue. To further analyze whether this affects the model's performance, we categorize models into two regimes: calibratable and non-calibratable, defined by bounds of Expected Calibration Error (ECE). In the calibratable regime, we propose a calibration-aware fine-tuning approach to achieve proper calibration without compromising LLMs' performance. However, as models are further fine-tuned for better performance, they enter the non-calibratable regime. For this case, we develop an EM-algorithm-based ECE regularization for the fine-tuning loss to maintain low calibration error. Extensive experiments validate the effectiveness of the proposed methods.

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.

Magnetic Preference Optimization: Achieving Last-iterate Convergence for Language Models Alignment

Self-play methods have demonstrated remarkable success in enhancing model capabilities across various domains. In the context of Reinforcement Learning from Human Feedback (RLHF), self-play not only boosts Large Language Model (LLM) performance but also overcomes the limitations of traditional Bradley-Terry (BT) model assumptions by finding the Nash equilibrium (NE) of a preference-based, two-player constant-sum game. However, existing methods either guarantee only average-iterate convergence, incurring high storage and inference costs, or converge to the NE of a regularized game, failing to accurately reflect true human preferences. In this paper, we introduce Magnetic Preference Optimization (MPO), a novel approach capable of achieving last-iterate convergence to the NE of the original game, effectively overcoming the limitations of existing methods. Building upon Magnetic Mirror Descent (MMD), MPO attains a linear convergence rate, making it particularly suitable for fine-tuning LLMs. To ensure our algorithm is both theoretically sound and practically viable, we present a simple yet effective implementation that adapts the theoretical insights to the RLHF setting. Empirical results demonstrate that MPO can significantly enhance the performance of LLMs, highlighting the potential of self-play methods in alignment.

Entropic Distribution Matching in Supervised Fine-tuning of LLMs: Less Overfitting and Better Diversity

Large language models rely on Supervised Fine-Tuning (SFT) to specialize in downstream tasks. Cross Entropy (CE) loss is the de facto choice in SFT, but it often leads to overfitting and limited output diversity due to its aggressive updates to the data distribution. This paper aim to address these issues by introducing the maximum entropy principle, which favors models with flatter distributions that still effectively capture the data. Specifically, we develop a new distribution matching method called GEM, which solves reverse Kullback-Leibler divergence minimization with an entropy regularizer. For the SFT of Llama-3-8B models, GEM outperforms CE in several aspects. First, when applied to the UltraFeedback dataset to develop general instruction-following abilities, GEM exhibits reduced overfitting, evidenced by lower perplexity and better performance on the IFEval benchmark. Furthermore, GEM enhances output diversity, leading to performance gains of up to 7 points on math reasoning and code generation tasks using best-of-n sampling, even without domain-specific data. Second, when fine-tuning with domain-specific datasets for math reasoning and code generation, GEM also shows less overfitting and improvements of up to 10 points compared with CE.

Fine-Tuning Linear Layers Only Is a Simple yet Effective Way for Task Arithmetic

Task arithmetic has recently emerged as a cost-effective and scalable approach to edit pre-trained models directly in weight space, by adding the fine-tuned weights of different tasks. The performance has been further improved by a linear property which is illustrated by weight disentanglement. Yet, conventional linearization methods (e.g., NTK linearization) not only double the time and training cost but also have a disadvantage on single-task performance. We propose a simple yet effective and efficient method that only fine-tunes linear layers, which improves weight disentanglement and efficiency simultaneously. Specifically, our study reveals that only fine-tuning the linear layers in the attention modules makes the whole model occur in a linear regime, significantly improving weight disentanglement. To further understand how our method improves the disentanglement of task arithmetic, we present a comprehensive study of task arithmetic by differentiating the role of representation model and task-specific model. In particular, we find that the representation model plays an important role in improving weight disentanglement whereas the task-specific models such as the classification heads can degenerate the weight disentanglement performance. Overall, our work uncovers novel insights into the fundamental mechanisms of task arithmetic and offers a more reliable and effective approach to editing pre-trained models.

Bridging the Gap: Rademacher Complexity in Robust and Standard Generalization

Training Deep Neural Networks (DNNs) with adversarial examples often results in poor generalization to test-time adversarial data. This paper investigates this issue, known as adversarially robust generalization, through the lens of Rademacher complexity. Building upon the studies by Khim and Loh (2018); Yin et al. (2019), numerous works have been dedicated to this problem, yet achieving a satisfactory bound remains an elusive goal. Existing works on DNNs either apply to a surrogate loss instead of the robust loss or yield bounds that are notably looser compared to their standard counterparts. In the latter case, the bounds have a higher dependency on the width $m$ of the DNNs or the dimension $d$ of the data, with an extra factor of at least $\mathcal{O}(\sqrt{m})$ or $\mathcal{O}(\sqrt{d})$. This paper presents upper bounds for adversarial Rademacher complexity of DNNs that match the best-known upper bounds in standard settings, as established in the work of Bartlett et al. (2017), with the dependency on width and dimension being $\mathcal{O}(\ln(dm))$. The central challenge addressed is calculating the covering number of adversarial function classes. We aim to construct a new cover that possesses two properties: 1) compatibility with adversarial examples, and 2) precision comparable to covers used in standard settings. To this end, we introduce a new variant of covering number called the \emph{uniform covering number}, specifically designed and proven to reconcile these two properties. Consequently, our method effectively bridges the gap between Rademacher complexity in robust and standard generalization.

On the Algorithmic Bias of Aligning Large Language Models with RLHF: Preference Collapse and Matching Regularization

Accurately aligning large language models (LLMs) with human preferences is crucial for informing fair, economically sound, and statistically efficient decision-making processes. However, we argue that reinforcement learning from human feedback (RLHF) -- the predominant approach for aligning LLMs with human preferences through a reward model -- suffers from an inherent algorithmic bias due to its Kullback--Leibler-based regularization in optimization. In extreme cases, this bias could lead to a phenomenon we term preference collapse, where minority preferences are virtually disregarded. To mitigate this algorithmic bias, we introduce preference matching (PM) RLHF, a novel approach that provably aligns LLMs with the preference distribution of the reward model under the Bradley--Terry--Luce/Plackett--Luce model. Central to our approach is a PM regularizer that takes the form of the negative logarithm of the LLM's policy probability distribution over responses, which helps the LLM balance response diversification and reward maximization. Notably, we obtain this regularizer by solving an ordinary differential equation that is necessary for the PM property. For practical implementation, we introduce a conditional variant of PM RLHF that is tailored to natural language generation. Finally, we empirically validate the effectiveness of conditional PM RLHF through experiments on the OPT-1.3B and Llama-2-7B models, demonstrating a 29% to 41% improvement in alignment with human preferences, as measured by a certain metric, compared to standard RLHF.

Uniformly Stable Algorithms for Adversarial Training and Beyond

In adversarial machine learning, neural networks suffer from a significant issue known as robust overfitting, where the robust test accuracy decreases over epochs (Rice et al., 2020). Recent research conducted by Xing et al.,2021; Xiao et al., 2022 has focused on studying the uniform stability of adversarial training. Their investigations revealed that SGD-based adversarial training fails to exhibit uniform stability, and the derived stability bounds align with the observed phenomenon of robust overfitting in experiments. This motivates us to develop uniformly stable algorithms specifically tailored for adversarial training. To this aim, we introduce Moreau envelope-$\mathcal{A}$, a variant of the Moreau Envelope-type algorithm. We employ a Moreau envelope function to reframe the original problem as a min-min problem, separating the non-strong convexity and non-smoothness of the adversarial loss. Then, this approach alternates between solving the inner and outer minimization problems to achieve uniform stability without incurring additional computational overhead. In practical scenarios, we show the efficacy of ME-$\mathcal{A}$ in mitigating the issue of robust overfitting. Beyond its application in adversarial training, this represents a fundamental result in uniform stability analysis, as ME-$\mathcal{A}$ is the first algorithm to exhibit uniform stability for weakly-convex, non-smooth problems.

PAC-Bayesian Spectrally-Normalized Bounds for Adversarially Robust Generalization

Deep neural networks (DNNs) are vulnerable to adversarial attacks. It is found empirically that adversarially robust generalization is crucial in establishing defense algorithms against adversarial attacks. Therefore, it is interesting to study the theoretical guarantee of robust generalization. This paper focuses on norm-based complexity, based on a PAC-Bayes approach (Neyshabur et al., 2017). The main challenge lies in extending the key ingredient, which is a weight perturbation bound in standard settings, to the robust settings. Existing attempts heavily rely on additional strong assumptions, leading to loose bounds. In this paper, we address this issue and provide a spectrally-normalized robust generalization bound for DNNs. Compared to existing bounds, our bound offers two significant advantages: Firstly, it does not depend on additional assumptions. Secondly, it is considerably tighter, aligning with the bounds of standard generalization. Therefore, our result provides a different perspective on understanding robust generalization: The mismatch terms between standard and robust generalization bounds shown in previous studies do not contribute to the poor robust generalization. Instead, these disparities solely due to mathematical issues. Finally, we extend the main result to adversarial robustness against general non-$\ell_p$ attacks and other neural network architectures.