d2: Improved Techniques for Training Reasoning Diffusion Language Models

While diffusion language models (DLMs) have achieved competitive performance in text generation, improving their reasoning ability with reinforcement learning remains an active research area. Here, we introduce d2, a reasoning framework tailored for masked DLMs. Central to our framework is a new policy gradient algorithm that relies on accurate estimates of the sampling trajectory likelihoods. Our likelihood estimator, d2-AnyOrder, achieves exact trajectory likelihood with a single model pass for DLMs that support a sampling algorithm called any-order decoding. Through an empirical study of widely used DLMs, we show that any-order decoding is not universally supported in practice. Consequently, for DLMs that do not naturally support any-order decoding, we propose another estimator, d2-StepMerge, which, unlike d2-AnyOrder, only approximates the trajectory likelihood. d2-StepMerge trades off compute for approximation accuracy in an analytically tractable manner. Empirically, d2 significantly outperforms widely-used RL baselines when applied to popular DLMs, and sets a new state-of-the-art performance for DLMs on logical reasoning tasks (Countdown and Sudoku) and math reasoning benchmarks (GSM8K and MATH500).