ConvergeFlow: Language Flow with Provable Convergence to Token Embeddings

Recent advances in continuous diffusion and flow-based language models (LMs) have achieved performance competitive with discrete LMs. However, existing continuous frameworks still rely on decoders supervised with cross entropy (CE) because the flow trajectories are not guaranteed to terminate at valid token embeddings. Motivated by this limitation, we introduce \textbf{ConvergeFlow}, an embedding-space flow-based LM, which constrains the data predictor to the convex hull of token embeddings and trains it solely with the mean squared error objective induced by flow matching. Under suitable regularity conditions, we prove that the resulting flow converges to valid token embeddings despite errors in the data predictor, enabling direct token prediction without a CE-supervised decoder. We further develop three sampling mechanisms for controlling the trade-off between the generative perplexity and entropy. Experiments on OpenWebText demonstrate that ConvergeFlow achieves performance competitive with existing continuous and discrete diffusion LMs. These findings demonstrate the potential of the flow-based paradigm for language modeling. Our code is available at https://github.com/Na-Li66/ConvergeFlow.

LC-GRPO: Bridging Train-Inference Gap for Flow-Based GRPO with Langevin Correction

Flow-based generative models are typically sampled by solving a deterministic ordinary differential equation (ODE), whereas online reinforcement learning requires stochastic rollouts for policy exploration and optimization. Existing GRPO methods for flow models therefore replace the inference-time ODE with a stochastic differential equation (SDE) during training. Although the ODE and SDE share the same marginal distributions in continuous time, their finite-step discretizations can differ substantially. In particular, SDE rollouts often become blurry as the exploration noise increases, creating a mismatch between the samples used for reinforcement learning and those generated by the test-time ODE sampler. We introduce LC-GRPO, a flow-based GRPO framework with Langevin correction. Each rollout transition first takes an inference-aligned ODE Euler step and then applies a stochastic Langevin correction targeting the marginal distribution at the resulting timestep. The required score is recovered directly from the flow velocity, requiring no additional score model, while the resulting transition remains an isotropic Gaussian with a tractable likelihood for policy optimization. We theoretically show that, under suitable conditions, one Langevin correction step reduces the Wasserstein error of an imperfect ODE Euler step. At a matched randomness level, we further show that the proposed transition can be more accurate than the standard Euler--Maruyama discretization of the reverse SDE. Experiments on SD3.5-Medium, FLUX.1-Dev, and HunyuanVideo demonstrate that LC-GRPO consistently improves reward optimization across text-to-image and text-to-video tasks, preserves generation quality, and substantially narrows the gap between stochastic training rollouts and deterministic test-time ODE inference.

Self-conditioned Flow Map Language Models via Fixed-point Flows

Self-conditioning is a core technique that enhances continuous flow-based language models, where the model learns to denoise generated text by conditioning on its own denoising estimate. While empirically successful, its performance improvements are poorly understood. Moreover, there is growing interest in the use of few-step generators based on flow maps, for which how to leverage self-conditioning is unclear. Here, we show that flow language models with self-conditioning solve a fixed-point iteration that bootstraps the performance of the learned denoiser. We use this viewpoint to formulate fixed-point flows, a two-dimensional class of self-conditioned flows, where the first dimension represents the flow process and the second represents the fixed-point iteration. We show that fixed-point flows define valid flow maps, and show that they can be distilled from self-conditioned flow models by compressing both fixed-point iterations and the flow process, the former with fixed-point distillation and the latter with flow map distillation. Our resulting flow map language model, FMLM$^\star$, outperforms state-of-the-art self-conditioned models and few-step models in one- and few-step generation on OpenWebText. Code is available at https://github.com/Ugness/self-conditioned-fmlm.

Pave-GRPO: Beyond Instantaneous Guidance through Principled Average Velocity Decomposition

Post-training via Group Relative Policy Optimization (GRPO) has emerged as a powerful paradigm for aligning flow-based generative models with human preferences. However, the iterative denoising nature of flow models incurs substantial costs when generating group rollouts for policy-gradient updates, compelling existing methods to train with extremely few denoising steps. This temporal sparsity severely restricts preference optimization: reward feedback can only reach a handful of stages per trajectory, leaving the vast majority of intermediate denoising steps without direct supervision and thus compromising alignment granularity. To address this, we propose Pave-GRPO, which reformulates the GRPO objective through Principled average velocity decomposition. Rather than generating expensive high-step rollouts, we maintain efficient few-step group sampling but decompose each coarse transition into an equivalent ensemble of finer sub-trajectories spanning multiple intermediate timesteps. This propagates reward feedback to a denser set of temporal stages for more comprehensive preference alignment without additional generation cost. This design offers two benefits: (i) zero-cost horizon expansion: through the direct reuse of piece-wise group samples and their associated rewards, Pave-GRPO significantly broadens the effective optimization scope under fixed sampling budgets; and (ii) comprehensive temporal supervision: by equivalently decomposing an instantaneous velocity target into a multi-timestep ensemble, it distributes reward signals across more intermediate stages of the denoising process, enabling finer-grained and more thorough preference optimization. Extensive experiments validate that Pave-GRPO effectively advances preference alignment across different reward settings, offering comprehensive performance enhancement.