MIRROR: Learning from the Other View for Multi-Modal Reasoning

Unlike large language models (LLMs) that exhibit strong reasoning capabilities, vision-language models (VLMs) struggle with visual reasoning, even on geometry problems that admit equivalent text, diagram, and combined diagram+text views. We show that these views often elicit different behaviors: a model may solve a problem from text but fail on the corresponding diagram, or succeed visually while failing textually. This inconsistency suggests that different views expose complementary reasoning paths and failure modes that standard multimodal post-training does not fully exploit. To study and exploit this phenomenon, we construct ODA-Data, a high-quality paired multimodal geometry dataset with text-dominant, image-dominant, and combined image+text views of the same problems, together with splits for training and evaluating modality-dependent reasoning behaviors. We then develop Modality-Informed Reciprocal Reasoning Optimization (MIRROR), a reinforcement learning approach for improving multimodal reasoning via self supervision. For each problem, MIRROR evaluates the model under all views, selects the best-performing view as a teacher, and trains other views with a reverse-KL objective towards the teacher. Across reasoning benchmarks that evaluate on geometry problems, MIRROR improves over standard RL and yields more accurate and consistent behavior across modalities

Addressing Over-Refusal in LLMs with Competing Rewards

Safety training on language models often induces over-refusal: improved safety on harmful prompts at the cost of increased refusal on harmless ones. Though this trade-off can be mitigated by training models with reinforcement learning (RL) to reason before answering, it does not remove the underlying problem that reasoning can often be a "rubber stamp" for a predetermined response. In this paper, we address the safety-refusal trade-off by rethinking how models are trained to reason about safety. Our key insight is that unsafe reasoning can itself serve as a useful exploratory signal. Rather than preemptively blocking harmful thoughts, we encourage the model to sufficiently explore unsafe reasoning but produce a safe response. The harmful exploration improves the model's ability to distinguish harmful from harmless prompts by resolving ambiguity, allowing it to remain safe while complying only when appropriate. We cast this as an adversarial optimization problem in which a reasoning player explores strategies for producing an unsafe response and an answer player ensures that the final output is safe. We train a single model with dense rewards to play both roles within one chain-of-thought, across different segments. To achieve this, we find that process rewards are crucial for stable optimization of competing objectives. Our resulting model SEAR deliberately engages in harmful reasoning as exploration while reliably flipping back to a safe answer. We demonstrate that this behavior helps mitigate over-refusal and defend against attacks that directly manipulate the reasoning to be harmful.

ExpRL: Exploratory RL for LLM Mid-Training

Sparse reward reinforcement learning (RL) has become a standard tool for improving LLM reasoning, but its success depends critically on the coverage present in the base model. In practice, models are often primed for RL through \emph{mid-training} on curated reasoning traces that teach useful primitive skills such as decomposition, verification, or self-correction. Although effective, this strategy requires manually specifying what the model should learn, and it remains unclear whether such primitive coverage is enough for much harder problems, which require combining these skills into broader solution strategies. We study a more automated approach: \emph{RL-based mid-training} using large corpora of human-written question-answer data. Rather than treating reference solutions as targets to imitate, our method, ExpRL, uses them as \emph{reward scaffolds}: references are hidden from the policy and used only to construct problem-specific grading rubrics for judging on-policy reasoning traces. The policy samples from the original problem prompt, while an LLM judge compares the sampled reasoning trace against the reference solution and assigns outcome-level or process-level dense rewards. This lets ExpRL reinforce partial progress, useful intermediate reductions, and productive reasoning behaviors that sparse final-answer rewards often fail to upweight. On challenging math reasoning tasks, ExpRL yields stronger RL priming than SFT, sparse-reward GRPO, and self-distillation, and provides a better initialization for subsequent sparse-reward RL. Additional mixed-domain experiments further suggest that ExpRL can extend beyond the original math-only setting.

AsyncWebRL: Efficient Multi-Step RL for Visual Web Agents

Training vision-language web agents with multi-step RL is compute-intensive, with two dominant forms of inefficiency: idle GPUs in synchronous RL, and trajectories that use more steps and tokens than necessary. We present AsyncWebRL, which addresses both. On the system side, an asynchronous design overlaps rollout, gradient update, and policy refresh across iterations, paired with two web-agent-specific adaptations, namely an everlasting rollout pool and lightweight screenshot handling, that together deliver up to a $2.9\times$ end-to-end training-throughput speedup over the previously fastest open synchronous pipeline (WebGym). On the algorithmic side, we identify the per-trajectory normalizer $1/|τ_i|$ in multi-step GRPO as the root cause of trajectory-level and token-level inefficiency: because failures are systematically longer than successes, it down-weights the negative gradient on failed tokens, so the policy keeps producing verbose memory schemas. Replacing $1/|τ_i|$ with a constant $1/k$ breaks this coupling, contracting trajectories while preserving aggregate success. Together, these contributions set a new open-source state of the art on the WebGym out-of-distribution test split (+5.8% relative over the 42.9% prior best), with the largest gains on the harder slices (+42% relative on Medium, +48% relative on Hard).

Recursive Agent Optimization

We introduce Recursive Agent Optimization (RAO), a reinforcement learning approach for training recursive agents: agents that can spawn and delegate sub-tasks to new instantiations of themselves recursively. Recursive agents implement an inference-time scaling algorithm that naturally allows agents to scale to longer contexts and generalize to more difficult problems via divide-and-conquer. RAO provides a method to train models to best take advantage of such recursive inference, teaching agents when and how to delegate and communicate. We find that recursive agents trained in this way enjoy better training efficiency, can scale to tasks that go beyond the model's context window, generalize to tasks much harder than the ones the agent was trained on, and can enjoy reduced wall-clock time compared to single-agent systems.

QED-Nano: Teaching a Tiny Model to Prove Hard Theorems

Proprietary AI systems have recently demonstrated impressive capabilities on complex proof-based problems, with gold-level performance reported at the 2025 International Mathematical Olympiad (IMO). However, the training pipelines behind these systems remain largely undisclosed, and their reliance on large "internal" models and scaffolds makes them expensive to run, difficult to reproduce, and hard to study or improve upon. This raises a central question: can small, open models also be trained to achieve competitive reasoning performance on difficult Olympiad-level math? In this paper, we answer this question by building QED-Nano, a 4B model post-trained for Olympiad-level proofs. Our training recipe has three stages: (1) supervised fine-tuning to imbue good proof-writing styles by distilling from DeepSeek-Math-V2, (2) reinforcement learning (RL) with rubric-based rewards, and (3) expanding RL with a reasoning cache, which decomposes long proofs into iterative summarize-and-refine cycles and enables stronger test-time reasoning. QED-Nano surpasses the proof-generation performance of much larger open models, including Nomos-1 and GPT-OSS-120B, and approaches the performance of proprietary models like Gemini 3 Pro, at a fraction of the inference cost. To support further research on open mathematical reasoning, we release the full QED-Nano pipeline, including the QED-Nano and QED-Nano-SFT models, the FineProofs-SFT and FineProofs-RL datasets, and the training and evaluation code.

IsoCompute Playbook: Optimally Scaling Sampling Compute for LLM RL

While scaling laws guide compute allocation for LLM pre-training, analogous prescriptions for reinforcement learning (RL) post-training of large language models (LLMs) remain poorly understood. We study the compute-optimal allocation of sampling compute for on-policy RL methods in LLMs, framing scaling as a compute-constrained optimization over three resources: parallel rollouts per problem, number of problems per batch, and number of update steps. We find that the compute-optimal number of parallel rollouts per problem increases predictably with compute budget and then saturates. This trend holds across both easy and hard problems, though driven by different mechanisms: solution sharpening on easy problems and coverage expansion on hard problems. We further show that increasing the number of parallel rollouts mitigates interference across problems, while the number of problems per batch primarily affects training stability and can be chosen within a broad range. Validated across base models and data distributions, our results recast RL scaling laws as prescriptive allocation rules and provide practical guidance for compute-efficient LLM RL post-training.

What Does Flow Matching Bring To TD Learning?

Recent work shows that flow matching can be effective for scalar Q-value function estimation in reinforcement learning (RL), but it remains unclear why or how this approach differs from standard critics. Contrary to conventional belief, we show that their success is not explained by distributional RL, as explicitly modeling return distributions can reduce performance. Instead, we argue that the use of integration for reading out values and dense velocity supervision at each step of this integration process for training improves TD learning via two mechanisms. First, it enables robust value prediction through \emph{test-time recovery}, whereby iterative computation through integration dampens errors in early value estimates as more integration steps are performed. This recovery mechanism is absent in monolithic critics. Second, supervising the velocity field at multiple interpolant values induces more \emph{plastic} feature learning within the network, allowing critics to represent non-stationary TD targets without discarding previously learned features or overfitting to individual TD targets encountered during training. We formalize these effects and validate them empirically, showing that flow-matching critics substantially outperform monolithic critics (2$\times$ in final performance and around 5$\times$ in sample efficiency) in settings where loss of plasticity poses a challenge e.g., in high-UTD online RL problems, while remaining stable during learning.

BPP: Long-Context Robot Imitation Learning by Focusing on Key History Frames

Many robot tasks require attending to the history of past observations. For example, finding an item in a room requires remembering which places have already been searched. However, the best-performing robot policies typically condition only on the current observation, limiting their applicability to such tasks. Naively conditioning on past observations often fails due to spurious correlations: policies latch onto incidental features of training histories that do not generalize to out-of-distribution trajectories upon deployment. We analyze why policies latch onto these spurious correlations and find that this problem stems from limited coverage over the space of possible histories during training, which grows exponentially with horizon. Existing regularization techniques provide inconsistent benefits across tasks, as they do not fundamentally address this coverage problem. Motivated by these findings, we propose Big Picture Policies (BPP), an approach that conditions on a minimal set of meaningful keyframes detected by a vision-language model. By projecting diverse rollouts onto a compact set of task-relevant events, BPP substantially reduces distribution shift between training and deployment, without sacrificing expressivity. We evaluate BPP on four challenging real-world manipulation tasks and three simulation tasks, all requiring history conditioning. BPP achieves 70% higher success rates than the best comparison on real-world evaluations.

Reasoning Cache: Continual Improvement Over Long Horizons via Short-Horizon RL

Large Language Models (LLMs) that can continually improve beyond their training budgets are able to solve increasingly difficult problems by adapting at test time, a property we refer to as extrapolation. However, standard reinforcement learning (RL) operates over fixed problem distributions and training budgets, which limits extrapolation amidst distribution shift at test time. To address this, we introduce RC, an iterative decoding algorithm that replaces standard autoregressive decoding during both training and inference. RC exploits an asymmetry between the response generation and summarization capabilities of LLMs to construct reasoning chains that consistently improve across iterations. Models trained to use RC can extrapolate and continually improve over reasoning horizons more than an order of magnitude longer than those seen during training. Empirically, training a 4B model with RC using a 16k-token training budget improves performance on HMMT 2025 from 40% to nearly 70% with 0.5m tokens at test time, outperforming both comparably sized models and many larger reasoning LLMs. Finally, we also show that models trained with RC can more effectively leverage existing scaffolds to further scale test-time performance, due to the improved summary-conditioned generation abilities learned through training.