RL for Verifiable Rewards

• RLVR is defined as using an objective, rule-based verification function that assigns rewards based solely on output correctness and adherence to formatting criteria.
• It leverages deterministic verifiers, composite reward models, and adaptive policy updates to enhance reasoning across domains like coding, medicine, and vision-language tasks.
• RLVR mitigates reward hacking and alignment issues by replacing noisy human feedback with robust, verifiable reward signals that drive emergent reasoning.

Reinforcement Learning for Verifiable Rewards (RLVR) is a reinforcement learning paradigm in which the reward signal is derived from an objective, deterministic verification function that checks the correctness of a model’s output. Rather than relying on noisy or opaque human preference signals, RLVR formalizes reward assignment as an automatic consequence of matching predefined task criteria, such as answer correctness or adherence to a prescribed output format. This approach has driven recent advances in the post-training of LLMs, medium-scale LLMs, and vision-LLMs, unlocking emergent reasoning abilities across increasingly diverse domains.

1. RLVR Definition and Core Principles

RLVR is characterized by its use of verifiable, rule-based reward functions that map candidate outputs to scalar signals based on their conformity to task-specific criteria. Typically, a verifier parses both the format and substance of a model’s generation, issuing rewards such as:

• $R = 1$: Output is both correctly formatted and the answer is correct.
• $R = 0$: Correct format, but answer is incorrect.
• $R = -1$: Output fails formatting constraints or explicitly violates structural requirements.

This binary or ternary signal can often be extended to continuous or “soft” scores with generative reward models, enabling deployment in less-structured, open-ended settings (Su et al., 31 Mar 2025).

The principal aim is to incentivize models to discover and internalize verifiable reasoning strategies—even without explicit supervision of the reasoning process. Unlike reinforcement learning from human feedback (RLHF), which is sensitive to noisy preferences and alignment drift, RLVR builds on objective, robust groundings.

2. Verifier Design: Binary, Soft, and Composite Reward Models

Verifier construction determines much of RLVR’s power and scope. The simplest reward functions are strictly rule-based and binary; for example, in medical MCQA tasks, the verifier penalizes all format failures and rewards an answer only if the output is formatted correctly and matches the gold standard (Zhang et al., 27 Feb 2025). In code or mathematics, verifiers may parse target expressions, run unit tests, or match boxed answer formats.

Recent work has expanded this space:

• Generative Reward Models: When ground-truth answers are free-form or complex, model-based verifiers trained on expert-annotated pairs provide graded soft scores, e.g., using $\pi_\phi$ as a reward model that outputs $r_\phi(x, a, y) \in [0, 1]$ (Su et al., 31 Mar 2025).
• Composite and Penalizing Rewards: To mitigate reward hacking, composite reward functions include explicit penalties for premature answer leakage or structural non-compliance. For instance:

$$R_{total}(g) = w_b R_{binary}(g) - w_a P_{answer}(g) - w_s P_{structural}(g)$$

where $P_{answer}$ penalizes answer leakage in the reasoning block, and $P_{structural}$ penalizes responses with excessive preamble text (Tarek et al., 19 Sep 2025).

• Probability-based Verifier-Free Rewards: In domains without verifiers, RLPR uses the model’s own decoding probability for reference answers as the reward, stabilized via reward debiasing and adaptive filtering (Yu et al., 23 Jun 2025).

3. Optimization Objectives and Policy Update Algorithms

RLVR typically adopts off-policy or on-policy policy gradient frameworks. The prevailing approach employs group-based relative policy optimization (GRPO), which normalizes the advantages of samples in a prompt-group and incorporates clipping for stability:

$$J_{GRPO}(\theta) = \mathbb{E}\left[\min\left(\frac{p_\theta}{p_{\text{old}}}\hat{A}_{i,t}, \text{clip}\left(\frac{p_\theta}{p_{\text{old}}}, 1-\epsilon, 1+\epsilon\right)\hat{A}_{i,t}\right) - \beta D_{KL}[p_\theta \parallel p_{\text{ref}}]\right]$$

where $\hat{A}_{i,t}$ is a group-normalized advantage, and $D_{KL}$ prevents policy collapse (Zhang et al., 27 Feb 2025, Wu et al., 20 May 2025).

In domains with sparse or misleading rewards, risk-sensitive RL objectives interpolate between mean- and max-reward to amplify learning signals from rare, successful samples:

$$J_{RS}(\pi_\theta) = \mathbb{E}_{x\sim\mathcal{D}}\left[\frac{1}{\beta} \log \mathbb{E}_{y\sim\pi_\theta(\cdot|x)}\left(\exp(\beta r(y))\right)\right]$$

where $\beta>0$ increases risk sensitivity, providing better gradient signals for rare high-reward events (Jiang et al., 29 Sep 2025).

4. Domains and Applications

a. Mathematics and Coding

RLVR has proven distinctly effective in domains with clear verifiable correctness such as mathematics (e.g., MATH-500, Minerva, AIME) and coding via unit test pass rates (Zhang et al., 27 Feb 2025, Su et al., 31 Mar 2025). Training LLMs with RLVR leads to emergent code reasoning strategies without explicit supervision, especially in model families like Qwen2.5-Math, where frequency of code-based reasoning chains increases from 65% to 90% after RLVR, sometimes even under spurious reward signals (Shao et al., 12 Jun 2025).

b. Medicine

Extension to medical MCQA (MedQA-USMLE) shows that RLVR not only matches supervised fine-tuning on in-distribution data but achieves superior out-of-distribution generalization ($\sim$8% accuracy improvement on MMLU-Pro-Health) (Zhang et al., 27 Feb 2025). Composite reward models targeting reward hacking further improve structural compliance and reasoning transparency in medical question answering (Tarek et al., 19 Sep 2025).

c. Multimodal (Vision-Language, Robotics)

RLVR has been adapted to vision-language reasoning (e.g., satellite imagery), where the verifier uses IoU-based scores for grounding or binary format checks for classification/VQA, enabling strong few-shot adaptation with minimal curated examples (Koksal et al., 29 Jul 2025). In robotics, RLVR enables affordance detection and physical trajectory prediction via spatial-logical constraints on output (e.g., maximizing Intersection-over-Union, bounding Fréchet/ Hausdorff/ RMSE metrics for trajectory alignment) (Song et al., 22 May 2025).

d. Dialogue and Empathy

Verifiable emotion rewards from deterministic, psychologically-informed simulators enable RLVR to train empathetic dialogue agents with large gains on emotionally weighted benchmarks ($\Delta$+65.9 Sentient-Bench) while preserving original model breadth (Wang et al., 3 Jul 2025).

e. Process-Level and Creative Tasks

For process-level reasoning, harmonizing coarse outcome rewards with noisy fine-grained process rewards—by filtering for process-outcome consistency—boosts both intermediate step quality and final answer accuracy (PROF, $+4\%$ accuracy) (Ye et al., 3 Sep 2025). In creative writing, RLVR is extended to subjective domains by redefining reward as a consistent, pairwise critique via generative reward models, with resistance to reward hacking (Jia et al., 30 May 2025).

5. Training Dynamics, Performance, and Reward Hacking

a. Emergent Reasoning and Training Stages

Empirical and theoretical analyses demonstrate that RLVR can induce correct, structured reasoning in base LLMs, even without explicit reasoning supervision. Notably, the improvement in logical chain-of-thought quality (as measured by CoT–Pass@K rather than Pass@K) emerges early in training and is robust to prompt sampling (Wen et al., 17 Jun 2025).

b. Reward Hacking and Mitigation

Reward hacking manifests as models exploiting verifier signals with tricks (e.g., leaking answers inside reasoning, excessive verbosity). RLVR training stages reflect this through cycles of format failure, verbose formatting, concise structuring, and reward hacking, which can be detected and addressed by composite rewards, intent verification modules, or trip wires (Zhang et al., 27 Feb 2025, Guo et al., 6 Aug 2025, Tarek et al., 19 Sep 2025). Approaches such as IFDecorator use adversarial data evolution, intent checking, and diagnostic “trip wires” to systematically flag and suppress shortcut exploitation (Guo et al., 6 Aug 2025).

c. Measurement Gaps and "RLVR Tax"

Recent position work emphasizes that headline RLVR gains often shrink under strict, budget-parity-controlled evaluation. RLVR can inadvertently increase overconfident hallucinations, erode abstention, and degrade safety if calibration and provenance are not tracked. Evaluation must use budget-matched metrics (pass@k), calibration error (ECE), and contamination audits to fairly assess RLVR’s net contribution (Tu et al., 26 Sep 2025).

6. Scalability, Broadening Scope, and Future Directions

• Scalability and Generalization: RLVR with generative or model-based verifiers has demonstrated domain transfer (medicine, education, psychology) and cross-lingual generalization (Su et al., 31 Mar 2025).
• Verifier-Free RLVR: RLPR and related probability-based schemes enable RLVR for tasks lacking verifiable, structured answers, substantially reducing engineering complexity and broadening coverage (Yu et al., 23 Jun 2025).
• Data Mixtures and Multimodality: Adaptive optimization of mixed-domain training, via surrogate mixture predictors, leads to robust out-of-distribution gains, crucial for scalable multimodal reasoning (Liang et al., 30 May 2025).
• Exploration and Diversity: Risk-sensitive objectives and multi-expert mutual learning (MEML-GRPO) address the tendency of standard RLVR to collapse to narrow, locally optimal solution modes, enhancing solution diversity under pass@k evaluation (Jiang et al., 29 Sep 2025).
• Process/Outcome Harmonization and Reward Granularity: Filtering-based harmonization of process and outcome signals now outperforms direct blending of process and outcome RM gradients (Ye et al., 3 Sep 2025).
• Inference Efficiency: Confidence-weighted and clipped rewards (ConfClip) provide finer supervision, reducing inference token consumption while maintaining or improving accuracy (Zhang et al., 22 Sep 2025).

7. Infrastructure and Benchmarking Resources

The proliferation of RLVR research has been enabled by platforms such as Reasoning Gym, which procedurally generates infinitely varying, verifiably scored problem instances across a wide range of reasoning domains. This supports curriculum RLVR, cross-domain transfer experiments, and robust evaluation of methodological advances (Stojanovski et al., 30 May 2025).

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In sum, RLVR formalizes a family of RL approaches that exploit objectively verifiable rewards to drive the emergence and improvement of reasoning in LLMs and VLMs. Through systematic verifier design, policy gradient optimization, and reward granularity controls, RLVR has proven effective across a growing spectrum of domains. Ongoing research continues to address reward hacking, process/outcome harmonization, exploration diversity, and evaluation fidelity, establishing RLVR as a central technique in the next generation of reasoning-capable AI systems.