Multi-Domain RLVR: Verifiable Rewards in RL

• The paper introduces RLVR as a framework that integrates verifiable rewards to achieve robust multi-domain reasoning across diverse tasks.
• It employs rule-based and model-based verifiers, dynamic curriculum design, and surrogate-based mixture optimization to address issues like reward hacking and sparse signals.
• Empirical evaluations show significant accuracy gains in medical, multimodal, and logical reasoning tasks, underscoring RLVR's potential for scalable AI applications.

Multi-domain reasoning in Reinforcement Learning with Verifiable Rewards (RLVR) refers to the systematic elicitation and emergence of robust, transferable reasoning capabilities in LLMs and vision-LLMs (VLMs) via reinforcement learning guided by reward signals that are objectively and deterministically verifiable, even as the models are tasked across disparate domains (mathematics, code, medicine, logic, visual reasoning, robotics, and multimodal reasoning). As RLVR matures from its roots in mathematically-precise or code-verifiable settings, multi-domain extensions seek to overcome challenges in generalization, data and reward heterogeneity, reward hacking, and cross-domain curriculum design, enabling LLMs and VLMs to reason reliably and adaptively across tasks in knowledge-intensive, agentic, and physical environments.

1. Foundations and Principles of RLVR for Multi-Domain Reasoning

RLVR fundamentally reframes reasoning model optimization as a sequential decision process, where models generate outputs (actions) given inputs (states) and receive explicit, verifiable reward signals. In classic single-domain settings, the reward $r(x, y)$ indicates correctness (e.g., exact answer match, unit test pass) or structural validity (e.g., output format tags such as > ...). The RL objective maximizes expected reward:

$$\max_{\theta} \mathcal{J}(\theta) = \mathbb{E}_{(x, y) \sim \mathcal{D}, y \sim \pi_\theta(x)}[R(x, y)]$$

Multi-domain reasoning in RLVR extends this paradigm, exposing models to heterogeneous tasks from multiple reasoning disciplines. These may include medical QA (Zhang et al., 27 Feb 2025), free-form social science answers (Su et al., 31 Mar 2025), visual–spatial understanding (Song et al., 22 May 2025, AI et al., 11 Jul 2025), logical puzzles (Chen et al., 26 May 2025), complex agentic settings (Da et al., 13 Jun 2025), or multimodal perception (Stojanovski et al., 30 May 2025, Liang et al., 30 May 2025).

The critical observation is that the RLVR machinery---automatic or model-based verifiers, token- or step-wise evaluation, reward densification---must operate correctly and scalably across diverse domains that vary in structural answer format, verifiability, ambiguity, and difficulty.

2. Methodologies and Reward Engineering for Multi-Domain RLVR

Multi-domain RLVR methods build on the following core reward engineering and optimization strategies:

• Rule-based and Model-based Verifiers: For math, programming, and puzzles, tasks are paired with deterministic auto-verifiers or heuristic rules. For unstructured answers or noisy reference data, reward models based on strong LLMs assign either binary or soft probabilistic rewards, e.g.,

$$r_{\phi}(x, a, y) = \mathbb{I}[c = 1] \text{ (binary)}, \quad r_{\phi}(x, a, y) = \pi_\phi(1|x, a, y) \text{ (soft)}$$

(Su et al., 31 Mar 2025).

• Dataset Curation and Generator-Verifier Pipelines: Domain-agnostic data generation (Chen et al., 26 May 2025, Li et al., 23 Jul 2025) and procedural task construction (Reasoning Gym (Stojanovski et al., 30 May 2025)) enable scale and controllability, while pairing each instance with automated verification provides reward grounding for RL.
• Multi-Task Training and Data Mixtures: The RLVR procedure for multiple domains involves sampling batches from domain-specific datasets $P_{s,i}$, tracking task mixture weights $w$, and optimizing an expected reward over the mixture distribution $P_w = \sum_i w_i P_{s,i}$. Optimal mixture strategies (Liang et al., 30 May 2025) employ surrogate models $g_\theta(w)$ to predict fine-tuning outcomes and guide data selection, given the bi-level optimization problem:

$$\hat{w} = \arg\max_w \mathbb{E}_{x \sim P_{test}}[f_R(x, P_w)]$$

• Fine-Grained, Structured, and Stepwise Rewards: Credit assignment is enhanced via model-based verifiers that output reward vectors over subquestions (Zhang et al., 7 Aug 2025), process reward models for step/cell-level feedback (Xie et al., 4 Aug 2025), and token-level advantage normalization (Zhang et al., 27 Feb 2025, Song et al., 22 May 2025). High-density rewards resolve gradient signal sparsity and enable consistent optimization of long-horizon reasoning.
• Guided Exploration and Self-Distillation: Adaptive hinting schemes and trajectory guidance (Guide (Nath et al., 16 Jun 2025), StepHint (Zhang et al., 3 Jul 2025), Agent-RLVR (Da et al., 13 Jun 2025)) address sparse rewards by injecting context-specific hints or expert feedback, powering not just solution refinement (self-distillation) but genuine capability gain in difficult tasks.

3. Empirical Evaluations and Generalization Patterns

Multi-domain RLVR has demonstrated substantial gains in both in-domain and out-of-domain tasks compared to supervised fine-tuning (SFT) and single-domain RL.

• Medical Reasoning: Med-RLVR (Zhang et al., 27 Feb 2025) achieves in-domain results comparable to SFT and delivers an 8-point accuracy gain on out-of-domain medical QA (MMLU-Pro-Health), with autonomous emergence of stepwise clinical reasoning and observable training dynamics from format errors to concise, robust reasoning.
• Free-Form and Broad-Domain QA: RLVR with cross-domain model-based reward yields improved generalization and robustness (e.g., up to 8% accuracy gain on free-form open-ended science, economics, and education) over large open-source baselines (Su et al., 31 Mar 2025) even in the absence of atomic, clean ground truths.
• Multimodal and Physical Reasoning: ManipLVM-R1 (Song et al., 22 May 2025), SATORI-R1 (Shen et al., 25 May 2025), and M2-Reasoning-7B (AI et al., 11 Jul 2025) empirically show that RLVR enables LVLM/MLLM architectures to generalize spatial, scene, and temporal reasoning beyond image/language classification, with state-of-the-art accuracy on multimodal and spatial benchmarks after RLVR-driven post-training.
• Logical Puzzle and Curriculum Learning: Enigmata (Chen et al., 26 May 2025) and Reasoning Gym (Stojanovski et al., 30 May 2025) validate that synthetic, knowledge-orthogonal puzzles with verifiable outcomes can bootstrap transfer benefits to math, STEM, and out-of-domain logic, especially when scaling to larger model architectures.
• Effect of Multi-Domain Data Mixtures and SFT: Systematic ablations indicate that SFT prior to RLVR dramatically enhances multi-domain generalization and robustness to template and language variation (Li et al., 23 Jul 2025). Optimal data mixture planning via quadratic surrogate modeling leads to 5+ percentage point overall gains on unseen benchmarks relative to naive uniform mixtures (Liang et al., 30 May 2025).

4. Optimization Challenges, Pitfalls, and Resolution Mechanisms

Multi-domain RLVR confronts several optimization and methodology challenges unique to reasoning models:

• Reward Hacking and Format Exploitation: Simple rule-based rewards can be exploited via shortcut behavior (e.g., early answer injection, direct copying). Multi-stage reward signals and joint format–accuracy constraints (e.g., $R = r_{accuracy} + \alpha\cdot r_{format}$ (Stojanovski et al., 30 May 2025)) counter such tendencies.
• Reward Sparsity and Near-Miss Collapse: Agentic and long-horizon settings (e.g., code agents in Agent-RLVR (Da et al., 13 Jun 2025)) often yield high trajectory failure rates. Guided RLVR injects dynamic teacher-like feedback, strategically shaping exploration where reward signals are absent.
• Negative Transfer and Domain Conflicts: Cross-domain RLVR training can induce performance degradation if domain-specific heuristics conflict (e.g., rigid code templates penalizing unconstrained logic). Careful curriculum and template consistency, as well as dynamic weighting of tasks, are critical (Li et al., 23 Jul 2025).
• Process vs. Outcome Rewards: Binary end-of-sequence rewards, while simple, limit gradient flow and learning from partial progress. New frameworks incorporate process-verifiable (stepwise, partial credit (Zhang et al., 7 Aug 2025)) and generative process reward models (Xie et al., 4 Aug 2025) to enable finer-grained optimization.
• Policy Collapse and Exploration Stagnation: RL updates concentrated on common “comfort zones” result in mode collapse and limit discovery of new reasoning chains. Multi-level stepwise hints, trajectory diversity, and process-aware rewards have been shown to mitigate these issues (Zhang et al., 3 Jul 2025, Nath et al., 16 Jun 2025).

5. Technical Algorithms and Theoretical Frameworks

Multi-domain RLVR leverages several distinctive algorithmic formulations:

• Policy Gradient with Verifiable Reward:

$$\nabla_{\theta} J(\theta) = \mathbb{E}_{(x,a) \sim D, y \sim \pi_{\theta}(·|x)} [ r_\phi(x,a,y) \nabla_{\theta} \log\pi_{\theta}(y|x) ]$$

• Group-Relative Policy Optimization (GRPO) (used e.g., in multi-task multi-domain SFT/RL pipelines):

$$\mathcal{L}_{\text{GRPO}}(\theta) = \mathbb{E}_{\tau \sim \pi_\theta} \left[ \min\left(r_\theta(\tau)A(\tau),\, \operatorname{clip}(r_\theta(\tau), 1-\epsilon, 1+\epsilon)A(\tau)\right) \right]$$

with $A(\tau)$ estimating normalized advantage over a group of rollouts.

• Credit Assignment via Process Reward Models (Xie et al., 4 Aug 2025, Zhang et al., 7 Aug 2025): Let $y$ be a response partitioned into $k$ steps. With model-based verifier $f_\theta$ producing $s = [s_1, ..., s_k]$, assign

$R_{StructVRM} = \frac{1}{k} \sum_{j=1}^k s_j$

enhancing fine-grained feedback.

• Surrogate-based Mixture Optimization (Liang et al., 30 May 2025):

$$g_\theta(w) = b + a^\top w + \frac{1}{2} w^\top C w$$

$w^* = \arg\max_w g_\theta(w)$ enabling efficient multi-domain data mixture search.

• Guide Algorithm for Adaptive Hinting (Nath et al., 16 Jun 2025): Importance-weight, off-policy guided rollouts (only if ordinary rollouts are all incorrect), improving both self-distillation of near-miss cases and genuine capability gain.

6. Impact, Applications, and Future Directions

Multi-domain RLVR is a foundation for the evolution of LLMs toward robust, reliable, and scalable reasoning engines:

• Applications: Med-RLVR (Zhang et al., 27 Feb 2025) and Agent-RLVR (Da et al., 13 Jun 2025) demonstrate domain extension to medicine and software engineering, yielding meaningful gains in out-of-distribution and agentic generalization.
• Multi-modal Generalization: Advanced pipelines for robotic manipulation (ManipLVM-R1 (Song et al., 22 May 2025)), vision-language VQA (SATORI-R1 (Shen et al., 25 May 2025)), and spatial interaction (M2-Reasoning-7B (AI et al., 11 Jul 2025)) showcase state-of-the-art performance by exploiting domain-specific verifiable signals and task decomposition.
• General Theoretical Insights: Empirical and theoretical studies show that much of RLVR’s performance arises from compressing pass@$k$ into pass@1 (self-distillation), but capability gain---the discovery of new reasoning pathways---is uniquely energized in multi-domain scenarios via guided exploration, adaptive data, and process-aware rewards (Nath et al., 16 Jun 2025, Zhang et al., 3 Jul 2025).
• Research Directions: Areas identified for future exploration include scalable verifier design for open-ended and free-form answers (Su et al., 31 Mar 2025, Yu et al., 23 Jun 2025), integration of RLVR in pre-training and continual learning, dynamic mixture optimization, richer process reward models, and efficient exploitation of memory and latent space reasoning (Zhang et al., 10 Sep 2025).

7. Tables: Key Multi-Domain RLVR Methodologies

RLVR Aspect	Example Approaches	Notable Domains
Reward Model	Rule-based, LLM-based	Math, Code, Free-form QA
Task/Data Generator	Procedural, synthetic, MMO	Logic, Puzzle, Robotics, VQA
Reward Granularity	Binary, Soft/Probabilistic	Medicine, Science, Education
Credit Assignment	Stepwise, Token-level	STEM, Multimodal, Planning
Mixture Optimization	Quadratic surrogate, curriculum	Multi-modal, Vision-Language

Conclusion

Multi-domain reasoning in RLVR fundamentally transforms the reasoning capacity, robustness, and generalization of language and vision-LLMs. By integrating verifiable rewards with scalable, flexible multi-task and multi-domain optimization pipelines, RLVR architectures now demonstrate improved transfer, compositionality, and resilience to distributional shifts across highly heterogeneous reasoning challenges. Methodological innovations in reward modeling, data curation, guidance, and credit assignment underpin these advances and establish RLVR as a foundational framework for the next generation of aligned, effective, and versatile AI reasoning systems.