Learning to chain-of-thought with Jensen's evidence lower bound

We propose a way to optimize chain-of-thought with reinforcement learning, but without external reward function. Our algorithm relies on viewing chain-of-thought as latent variable as part of a probabilistic inference problem. Contrary to the full evidence lower bound, we propose to apply a much simpler Jensen's lower bound, which derives tractable objectives with simple algorithmic components (e.g., without the need for parametric approximate posterior), making it more conducive to modern large-scale training. The lower bound approach naturally interpolates other methods such as supervised fine-tuning and online reinforcement learning, whose practical trade-offs we will illustrate. Finally, we show that on mathematical reasoning problems, optimizing with Jensen's lower bound is as effective as policy gradient with external reward. Taken together, our results showcase as a proof of concept to this new algorithmic paradigm's potential to more generic applications.