On a few pitfalls in KL divergence gradient estimation for RL

We point out a few pitfalls in implementing gradient estimation for KL divergence in RL training for LLM, as seen in a number of open source projects and papers. The first major pitfall is to differentiate through the KL estimate as loss functions to minimize KL divergence. We show that such implementations are generally incorrect and do not produce the desired KL gradient. Secondly, we show that some implementations do not account for the sequential nature of the estimation problem and produce a partial gradient at best. We demonstrate the impact of such issues with illustrative tabular and LLM experiments, and show the correct way to implement the KL gradient.