Discrete MeanFlow: One-Step Generation via Conditional Transition Kernels

MeanFlow enables one-step generation in continuous spaces by learning an average velocity over a time interval rather than the instantaneous velocity field of flow matching. However, discrete state spaces do not have smooth trajectories or spatial derivatives, so the continuous formulation does not directly apply. We introduce Discrete MeanFlow, which replaces the motion of a point with the transport of probability mass over finite states. Our key object is the conditional transition kernel of a continuous-time Markov chain (CTMC), from which we define a mean discrete rate that measures the average change in transition probability over a time interval. We prove a Discrete MeanFlow identity that relates this finite-interval rate to the instantaneous CTMC generator at the endpoint, with the Kolmogorov forward equation replacing the spatial chain rule of continuous MeanFlow. Based on this identity, we parameterize the transition kernel directly using a boundary-by-construction design that guarantees valid probability outputs and exact boundary conditions without auxiliary losses. Since the learned kernel is itself a probability distribution, generation reduces to a single forward pass followed by one categorical draw meaning no iterative denoising, ODE integration, or multi-step refinement is required. We validate the framework on exact finite-state Markov chains, where the learned kernel recovers the analytical ground truth to high precision, and on factorized synthetic sequence generation tasks with varying alphabet sizes and sequence lengths.

Discrete Flow Matching for Offline-to-Online Reinforcement Learning

Many reinforcement learning (RL) tasks have discrete action spaces, but most generative policy methods based on diffusion and flow matching are designed for continuous control. Meanwhile, generative policies usually rely heavily on offline datasets and offline-to-online RL is itself challenging, as the policy must improve from new interaction without losing useful behavior learned from static data. To address those challenges, we introduce DRIFT, an online fine-tuning method that updates an offline pretrained continuous-time Markov chain (CTMC) policy with an advantage-weighted discrete flow matching loss. To preserve useful pretrained knowledge, we add a path-space penalty that regularizes the full CTMC trajectory distribution, rather than only the final action distribution. For large discrete action spaces, we introduce a candidate-set approximation that updates the actor over a small subset of actions sampled from reference-policy rollouts and uniform exploration. Our theoretical analysis shows that the candidate-set error is controlled by missing target probability mass, and the induced CTMC generator error decreases as the candidate set covers more high-probability actions. Experiments on prevailing discrete action RL task show that our method provides stable offline-to-online improvement across all tasks, achieving the highest average score on Jericho with a simple GRU encoder while outperforming methods that use pretrained language models. Controlled experiments further confirm that the path-space penalty remains bounded during fine-tuning and that the CTMC generator adapts to shifted rewards faster than deterministic baselines. The candidate-set mechanism is supported by a stability analysis showing that the generator error decreases exponentially with candidate coverage.

Flow Matching for Offline Reinforcement Learning with Discrete Actions

Generative policies based on diffusion models and flow matching have shown strong promise for offline reinforcement learning (RL), but their applicability remains largely confined to continuous action spaces. To address a broader range of offline RL settings, we extend flow matching to a general framework that supports discrete action spaces with multiple objectives. Specifically, we replace continuous flows with continuous-time Markov chains, trained using a Q-weighted flow matching objective. We then extend our design to multi-agent settings, mitigating the exponential growth of joint action spaces via a factorized conditional path. We theoretically show that, under idealized conditions, optimizing this objective recovers the optimal policy. Extensive experiments further demonstrate that our method performs robustly in practical scenarios, including high-dimensional control, multi-modal decision-making, and dynamically changing preferences over multiple objectives. Our discrete framework can also be applied to continuous-control problems through action quantization, providing a flexible trade-off between representational complexity and performance.