A Fundamental Trade-off in Aligned Language Models and its Relation to Sampling Adaptors

The relationship between the quality of a string and its probability $p(\boldsymbol{y})$ under a language model has been influential in the development of techniques to build good text generation systems. For example, several decoding algorithms have been motivated to manipulate $p(\boldsymbol{y})$ to produce higher-quality text. In this work, we examine the probability--quality relationship in language models explicitly aligned to human preferences, e.g., through Reinforcement Learning through Human Feedback (RLHF). We find that, given a general language model and its aligned version, for corpora sampled from an aligned language model, there exists a trade-off between the average reward and average log-likelihood of the strings under the general language model. We provide a formal treatment of this issue and demonstrate how a choice of sampling adaptor allows for a selection of how much likelihood we exchange for the reward.