Asymptotics of Language Model Alignment

• URL: http://arxiv.org/abs/2404.01730v1
• Date: Tue, 2 Apr 2024 08:40:07 GMT
• Title: Asymptotics of Language Model Alignment
• Authors: Joy Qiping Yang, Salman Salamatian, Ziteng Sun, Ananda Theertha Suresh, Ahmad Beirami,
• Abstract summary: We show that the optimal KL-constrained RL solution satisfies a large deviation principle. We also show that the rate of growth of the scaled cumulants of the reward is characterized by proper Renyi cross entropy.
• Score: 27.37118975691123
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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