Asymptotics of Language Model Alignment
<|reference_start|>Asymptotics of Language Model Alignment: Let $p$ denote a generative language model. Let $r$ denote a reward model that returns a scalar that captures the degree at which a draw from $p$ is preferred. The goal of language model alignment is to alter $p$ to a new distribution $\phi$ that results in a higher expected reward while keeping $\phi$ close to $p.$ A popular alignment method is the KL-constrained reinforcement learning (RL), which chooses a distribution $\phi_\Delta$ that maximizes $E_{\phi_{\Delta}} r(y)$ subject to a relative entropy constraint $KL(\phi_\Delta || p) \leq \Delta.$ Another simple alignment method is best-of-$N$, where $N$ samples are drawn from $p$ and one with highest reward is selected. In this paper, we offer a closed-form characterization of the optimal KL-constrained RL solution. We demonstrate that any alignment method that achieves a comparable trade-off between KL divergence and reward must approximate the optimal KL-constrained RL solution in terms of relative entropy. To further analyze the properties of alignment methods, we introduce two simplifying assumptions: we let the language model be memoryless, and the reward model be linear. Although these assumptions may not reflect complex real-world scenarios, they enable a precise characterization of the asymptotic behavior of both the best-of-$N$ alignment, and the KL-constrained RL method, in terms of information-theoretic quantities. We prove that the reward of the optimal KL-constrained RL solution satisfies a large deviation principle, and we fully characterize its rate function. We also show that the rate of growth of the scaled cumulants of the reward is characterized by a proper Renyi cross entropy. Finally, we show that best-of-$N$ is asymptotically equivalent to KL-constrained RL solution by proving that their expected rewards are asymptotically equal, and concluding that the two distributions must be close in KL divergence.<|reference_end|>
@article{yang2024asymptotics,
    title={Asymptotics of Language Model Alignment},
    author={Joy Qiping Yang and Salman Salamatian and Ziteng Sun and Ananda
  Theertha Suresh and Ahmad Beirami},
    journal={arXiv preprint arXiv:2404.01730},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.01730},
    primaryClass={cs.LG cs.IT math.IT stat.ML}
}