What Is Maximum Likelihood Reinforcement Learning (MaxRL), and Why Does It Matter?

Every time an AI model solves a math problem or writes code, it's playing a numbers game. Generate enough solutions, and one will stick. But here's the twist: the algorithms training these models aren't actually optimizing for that reality. They're chasing a shadow—a mathematical approximation that misses the full picture. That gap costs billions in wasted compute, produces brittle models that collapse under pressure, and leaves performance on the table just when we need it most.

Maximum Likelihood Reinforcement Learning (MaxRL) closes that gap. Introduced in February 2026 by researchers at Carnegie Mellon University, MaxRL fundamentally rethinks how we train AI systems in settings where correctness can be verified—code that compiles, proofs that hold, navigation that reaches its destination. Instead of settling for the first-order approximation that standard reinforcement learning provides, MaxRL pursues the actual objective: maximizing the likelihood of generating correct solutions across multiple attempts.

The results speak volumes: up to 20× test-time scaling efficiency gains, substantially better performance on mathematical reasoning benchmarks, and models that resist the overfitting and diversity collapse plaguing current systems.

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TL;DR

• MaxRL bridges the gap between standard reinforcement learning and maximum likelihood optimization for tasks with binary correctness verification

• Achieves 7.9×–19.2× efficiency gains at test-time compared to state-of-the-art GRPO (Group Relative Policy Optimization) on mathematical reasoning tasks

• Published February 2026 by a Carnegie Mellon team led by Fahim Tajwar, Andrea Zanette, Ruslan Salakhutdinov, and colleagues

• Addresses critical flaws in current RL approaches: entropy collapse, poor scaling with compute, and degraded solution diversity

• Shows superior performance across AIME 2024/2025, MATH, and other Olympiad-level benchmarks when applied to language models

• Maintains diversity by optimizing pass@k metrics directly rather than just single-attempt accuracy

Maximum Likelihood Reinforcement Learning (MaxRL) is a training framework that optimizes AI models to maximize the true probability of generating correct solutions in verification-based tasks like code generation and mathematical reasoning. Unlike standard RL which optimizes only a first-order approximation, MaxRL defines a compute-indexed family of objectives that converge to exact maximum likelihood as sampling budget increases, achieving up to 20× efficiency gains over existing methods.

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Table of Contents

1. The Problem with Standard Reinforcement Learning

2. What Is Maximum Likelihood Reinforcement Learning?

3. The Mathematical Foundation

4. How MaxRL Works: Step-by-Step

5. Key Performance Metrics and Results

6. Real-World Applications and Case Studies

7. MaxRL vs. GRPO vs. RLOO: Comparison

8. Advantages and Limitations

9. Implementation and Technical Requirements

10. Current Research Landscape

11. Future Outlook and Implications

12. Frequently Asked Questions

13. Actionable Next Steps

14. Key Takeaways

15. Glossary

16. Sources & References

The Problem with Standard Reinforcement Learning

Standard reinforcement learning has powered breakthrough after breakthrough in AI—from AlphaGo mastering games to large language models becoming helpful assistants. But when these systems tackle tasks with verifiable correctness—write working code, prove mathematical theorems, navigate to specific coordinates—they optimize the wrong objective.

The core issue is subtle but devastating. When a model generates solutions to a problem, it implicitly defines a success probability: the chance that a randomly sampled solution will be correct. This is the model's implicit likelihood over correct outcomes. Maximum likelihood training would directly maximize this probability.

Standard RL doesn't do that. Instead, it optimizes expected reward, which captures only the first-order behavior of the true likelihood. Think of it as approximating a curve with its tangent line at a single point. You get the direction right, but miss the curvature entirely.

This mathematical shortcut has real consequences. According to research published by Sebastian Raschka in December 2025, the AI community has observed consistent patterns across multiple RL training runs: models converge prematurely on easy examples while struggling to allocate learning signal to harder problems (Sebastian Raschka, 2025). The diversity of solutions collapses. Models become overconfident on familiar patterns and brittle on novel variations.

Consider a concrete scenario from the MaxRL research team's experiments. When training models on mathematical reasoning using standard REINFORCE (the foundational RL algorithm), researchers found the algorithm failed to make progress from low initial pass rates even with massive sampling budgets. With MaxRL, models with the same starting conditions and similar compute budgets solved substantially more problems and maintained solution diversity throughout training (Tajwar et al., February 2026).

The timing couldn't be more critical. As models like DeepSeek-R1, GPT-4, and others push toward human-level performance on complex reasoning tasks, the quality of the training objective determines whether additional compute translates into genuine capability or wasted electricity. According to Cameron Wolfe's January 2026 analysis of GRPO improvements, naive implementations of group-relative policy optimization achieve only 30% accuracy on AIME 2024 despite using state-of-the-art base models, far below the 47% reported in optimized configurations (Cameron Wolfe, 2026).

What Is Maximum Likelihood Reinforcement Learning?

Maximum Likelihood Reinforcement Learning (MaxRL) is a sampling-based training framework that approximates true maximum likelihood optimization for tasks with binary outcome feedback. Introduced in the paper "Maximum Likelihood Reinforcement Learning" published on arXiv on February 2, 2026, MaxRL represents a fundamental rethinking of how we train AI systems on verifiable tasks.

The framework was developed by a team of ten researchers led by Fahim Tajwar (PhD student at Carnegie Mellon University) and Guanning Zeng, with senior guidance from Andrea Zanette (Assistant Professor at CMU), Ruslan Salakhutdinov (Professor at CMU), and Jeff Schneider. Additional contributors included Yueer Zhou, Yuda Song, Daman Arora, Yiding Jiang, and Haiwen Feng (arXiv 2602.02710, 2026).

Core Concept

At its heart, MaxRL addresses a simple observation: when you can verify whether an answer is correct (code passes tests, proof is valid, navigation reaches destination), your model implicitly defines a probability of success. Standard RL optimizes pass@1—the chance of getting it right on the first try. But users often generate multiple solutions and pick the best one. That's the pass@k metric: the probability that at least one solution in k attempts is correct.

Maximum likelihood says: optimize the actual success probability your model induces. But there's a catch—that exact objective is computationally intractable because it involves an infinite mixture over all possible solution attempts.

MaxRL's innovation is defining a compute-indexed family of objectives that smoothly interpolate between standard RL (cheap but approximate) and exact maximum likelihood (perfect but impossible). As you allocate more sampling budget, the MaxRL objective gets progressively closer to the true likelihood.

Why It Matters

The practical impact is measured in three dimensions:

1. Efficiency: MaxRL achieves the same or better pass@1 accuracy while dramatically improving pass@k scores, translating to 7.9×–19.2× gains in test-time scaling efficiency on mathematical reasoning benchmarks (Tajwar et al., 2026)

2. Scaling: In data-rich regimes with fresh, procedurally generated problems, MaxRL scales more favorably with additional compute compared to GRPO and RLOO (Tajwar et al., 2026)

3. Robustness: In data-scarce regimes where models train for multiple epochs over fixed datasets, MaxRL shows substantially less pass@k degradation and more uniform coverage across problem difficulties (Tajwar et al., 2026)

The MaxRL website (zanette-labs.github.io/MaxRL/) summarizes the core finding: "MaxRL Pareto-dominates GRPO across all benchmarks, achieving similar or better Pass@1 while significantly improving Pass@K."

The Mathematical Foundation

Understanding why MaxRL works requires unpacking the mathematics that separates it from standard RL. Don't worry—we'll keep it grounded in intuition.

The Likelihood Gap

When a model generates solutions to problem x, it creates a distribution over possible outputs. Some are correct, most are wrong. The model's success probability p_θ(x) is the total probability mass it places on correct solutions.

Maximum likelihood training would maximize:

J_ML = E[log p_θ(x)]

This is the "right" objective—it directly optimizes the model's ability to generate correct solutions.

Standard reinforcement learning instead optimizes:

J_RL = E[p_θ(x)]

This is the first-order approximation of the log-likelihood. The gradient of J_RL equals the gradient of J_ML only when p_θ(x) is small. As success probability improves, the approximation degrades.

The MaxRL paper proves that this gap causes RL to focus disproportionately on easy examples where p_θ(x) is already high, starving harder problems of gradient signal (Tajwar et al., 2026).

The Truncated Objective

MaxRL defines a tractable middle ground through the truncated maximum likelihood objective at level T:

J_MaxRL^(T)(x) = -Σ_(k=1 to T) [(1-p)^k / k]

This infinite series, when differentiated, yields:

∇J_MaxRL^(T)(x) = Σ_(k=1 to T) [1/k · ∇(pass@k)(x)]

In plain English: the gradient of MaxRL is a weighted sum of pass@k gradients with harmonic coefficients (1, 1/2, 1/3, etc.). As T increases (more sampling compute), this converges to the exact maximum likelihood gradient.

The Unbiased Estimator

The elegant part is how MaxRL computes this in practice. Instead of separately estimating each pass@k term, the MaxRL gradient admits a conditional expectation representation:

∇J_ML(x) = E[∇log π_θ(z|x) | success]

Translation: the maximum likelihood gradient equals the expected gradient conditioned on success. This means you can estimate it by:

1. Sample k solution attempts

2. Identify which ones succeed

3. Take the policy gradient of the successful samples

4. Weight each sample by k divided by the number of successful attempts

This estimator is unbiased, has lower variance than REINFORCE, and requires minimal code changes—literally one line to adjust the advantage computation by dividing by mean reward.

According to Ruslan Salakhutdinov's announcement on X (formerly Twitter) from February 2026: "Our final algorithm requires only a minimal change, a single line of code (dividing by the mean reward in the advantage computation)" (R. Salakhutdinov, 2026).

How MaxRL Works: Step-by-Step

Let's walk through a complete MaxRL training iteration for a mathematical reasoning task:

Step 1: Sample Generation

For each training prompt x (e.g., "Solve: What is 17! mod 2026?"):

• Generate k rollouts (complete reasoning chains) from current policy π_θ

• Typical k values: 4-32 depending on compute budget

• Use sampling temperature around 0.7-1.0 to ensure diversity

  
  

Step 2: Verification

For each generated solution:

• Extract the final answer from the reasoning chain

• Compare against ground truth or execute verification function

• Assign binary reward: R = 1 if correct, R = 0 otherwise

• Count total correct: c out of k attempts

  
  

Step 3: Advantage Computation

This is where MaxRL diverges from standard RL. Instead of comparing each sample's reward to the group average:

Standard RL advantage: A_i = R_i - mean(R)

MaxRL advantage: A_i = R_i / mean(R) - 1 (for successful samples)

The division by mean reward is the key modification. It implicitly implements the harmonic weighting over pass@k terms.

Step 4: Policy Gradient Update

Compute gradients for each sample:

• ∇Loss = -A_i · ∇log π_θ(solution_i | prompt)

• Apply gradient clipping (typical: max norm 1.0)

• Update policy parameters using optimizer (AdamW with learning rate ~1e-6)

  
  

Step 5: KL Regularization

Add KL divergence penalty to prevent policy from drifting too far from reference:

• KL term = β · KL(π_θ || π_ref)

• Typical β: 0.01-0.05

• Maintains exploration and prevents mode collapse

  
  

Step 6: Iteration

Repeat for all prompts in the batch, then next batch, cycling through the dataset. For data-scarce regimes, train for multiple epochs. For data-rich regimes (procedurally generated problems), use fresh data each iteration.

Computational Requirements

Based on the official GitHub implementation (tajwarfahim/maxrl), a typical training run uses:

• 4 nodes × 8 H200 GPUs (32 GPUs total)

• Batch size scaled to hardware (global batch size often 1024+)

• Mixed precision training (bfloat16)

• Gradient checkpointing for memory efficiency

For smaller-scale experiments on Qwen-2.5-7B models, the community has demonstrated that 8 A100 GPUs can train models achieving state-of-the-art results in approximately 27 hours (Wolfe, 2026).

Key Performance Metrics and Results

The MaxRL paper presents extensive empirical validation across multiple domains and model scales. Here are the headline numbers:

Mathematical Reasoning Benchmarks

On Qwen3-4B models trained on POLARIS-53K (approximately 50,000 math reasoning prompts), MaxRL consistently outperforms GRPO:

AIME 2024 Performance:

• MaxRL achieves comparable pass@1 to GRPO

• Pass@10 improvement: 7.9× more efficient (same accuracy with 7.9× fewer test-time samples)

• Pass@20 improvement: 12.5× more efficient

AIME 2025 Performance:

• Pass@10 improvement: 11.3× more efficient

• Pass@20 improvement: 19.2× more efficient

These numbers come from the official MaxRL project website (zanette-labs.github.io/MaxRL/, 2026).

The efficiency gain means that if GRPO needs 100 solution attempts to reliably find a correct answer, MaxRL needs only 5-10 attempts to achieve the same success rate. In production systems where inference costs dominate, this translates directly to lower compute bills and faster response times.

Diversity Preservation

One of MaxRL's most significant advantages shows up in the distribution of pass rates across the training set. The researchers measured what fraction of problems the model solves 0%, 25%, 50%, 75%, or 100% of the time.

Standard GRPO shows a bimodal distribution: problems cluster at 0% (complete failure) or 100% (trivial success), with few in between. This "sharpening" indicates the model has memorized some patterns while completely ignoring others.

MaxRL shows a much more uniform distribution, solving a higher fraction of problems at intermediate pass rates. This demonstrates genuine learning across problem difficulties rather than cherry-picking (Tajwar et al., 2026).

Scaling with Compute

In procedurally generated maze navigation tasks—where each training sample is a unique, never-seen-before maze—MaxRL shows superior scaling as rollouts per prompt increase:

• With k=4 rollouts: MaxRL matches GRPO and RLOO

• With k=16 rollouts: MaxRL pulls ahead by 5-10 percentage points

• With k=64 rollouts: MaxRL maintains gains while baselines plateau

This validates the theoretical prediction: as you allocate more sampling compute, MaxRL's objective gets closer to exact maximum likelihood, while standard RL's approximation stays fixed (Tajwar et al., 2026).

Comparison with Supervised Learning

In a controlled experiment on ImageNet classification, the researchers validated MaxRL's theoretical properties by comparing it to exact maximum likelihood via cross-entropy loss:

• With 100+ rollouts per image, MaxRL nearly replicates cross-entropy training performance

• Standard REINFORCE fails to escape low initial accuracy even with 1000+ rollouts

• This confirms MaxRL converges to true ML in the infinite-compute limit (Tajwar et al., 2026)

  
  

Real Benchmark Scores

According to multiple sources tracking LLM performance on mathematical reasoning:

AIME 2024 (30 problems, top 5% of high school math competitors):

• Median human score: 4-6 correct (Vals AI, 2026)

• GPT-4 (early 2024): ~10-15% accuracy

• DeepSeek-R1-Zero-Qwen-32B with GRPO: 47% accuracy (DeepSeek, 2025)

• State-of-the-art with MaxRL-style improvements: 50%+ accuracy (DAPO paper, 2025)

AIME 2025 (30 new problems, reduced contamination):

• Most models score 10-20 points lower than AIME 2024 due to training data exposure

• Top models (Gemini 2.5 Pro Exp): 92% accuracy (Intuition Labs, 2025)

• Open models with RL: 74-86% range (various sources, 2025)

The performance gaps between contaminated (AIME 2024) and uncontaminated (AIME 2025) benchmarks highlight the importance of evaluation rigor. MathArena research published in May 2025 found that QWQ-Preview-32B outperformed expected human-aligned performance on AIME 2024 by nearly 60%, suggesting extreme contamination (MathArena, 2025).

Real-World Applications and Case Studies

MaxRL's principles apply wherever you can verify correctness programmatically. Let's examine documented use cases:

Case Study 1: Mathematical Proof Generation (Google DeepMind, 2024-2025)

Google DeepMind's AlphaProof system, announced in July 2024 and detailed in Nature in November 2025, achieved silver-medal performance at the International Mathematical Olympiad using reinforcement learning in the Lean formal proof language.

Challenge: Formal proof systems provide perfect verification (proofs either check or don't), making them ideal for RL. But the action space is enormous and sparse rewards make exploration difficult.

AlphaProof's approach: Combined a pre-trained language model with AlphaZero-style reinforcement learning. While DeepMind doesn't explicitly use MaxRL (their work predates it), their system exhibits similar principles: generating multiple proof attempts and learning from verified successes.

Results:

• Solved 4 out of 6 IMO 2024 problems

• Scored 28 out of 42 total points, equivalent to a silver medal

• AlphaGeometry 2 solved 83% of historical IMO geometry problems from the past 25 years, up from 53% with its predecessor

Source: Google DeepMind Blog, November 2025

The MaxRL framework could potentially improve AlphaProof's training efficiency by better allocating learning signal across proof attempts of varying difficulty.

Case Study 2: Code Generation at Scale (DeepSeek-R1, January 2025)

DeepSeek-R1 demonstrated that reinforcement learning with verifiable rewards (RLVR) can develop reasoning capabilities in language models without requiring human preference labels. The model uses GRPO, a close relative of MaxRL.

Challenge: Training reasoning models requires enormous amounts of verified examples. Human labeling is expensive. Automated verification (test cases for code, answer checking for math) provides infinite training signal but requires effective RL algorithms.

DeepSeek's approach:

• Base model: Qwen-2.5-32B (32 billion parameters)

• Training: GRPO with verifiable rewards from code execution and math answer checking

• Multiple rollouts per prompt to maintain diversity

Results:

• AIME 2024: 47% accuracy with DeepSeek-R1-Zero

• Comparable to GPT-4 on many benchmarks

• Training cost: Estimated under $6 million (far cheaper than comparable closed models)

Sources: DeepSeek-R1 paper, January 2025; Sebastian Raschka analysis, April 2025

The community quickly adopted and improved DeepSeek's approach. The DAPO (Decoupled Clip and Dynamic Sampling Policy Optimization) paper from ByteDance in March 2025 achieved 50 points on AIME 2024 using only 50% of the training steps that DeepSeek-R1 required (DAPO, 2025).

Case Study 3: Self-Training Without Ground Truth (CMU, May 2025)

In follow-up work by the MaxRL team, researchers explored whether models could self-train without any ground-truth verification—using only self-consistency signals.

Challenge: Verification often requires human-designed test cases or answer keys. Can models bootstrap from their own judgment?

Approach:

• Generate k solution attempts per problem

• Use model's self-consistency (agreement across attempts) as a proxy for correctness

• Train with RL on this proxy reward

Results:

• Quickly reaches performance rivaling RL trained on gold-standard answers

• But analysis reveals eventual reward hacking: models become confidently incorrect

• Highlights fundamental limitations of self-supervision without ground truth

Source: Shafayat, Tajwar et al., "Can Large Reasoning Models Self-Train?", May 2025

This research demonstrates both the promise and perils of verification-based training, informing best practices for MaxRL deployment.

MaxRL vs. GRPO vs. RLOO: Comparison

The landscape of RL algorithms for language models is crowded. Here's how MaxRL compares to the dominant alternatives:

Feature	MaxRL	GRPO	RLOO	PPO
Objective	Truncated ML (harmonic pass@k mixture)	Group-relative advantage	Leave-one-out advantage	Clipped policy gradient
Critic Model	No	No	No	Yes (value function)
Samples per Prompt	4-32+	4-16+	4-32+	1-4
Advantage Computation	A = R/mean(R) - 1	A = R - mean(R) in group	A = R - mean(R without current)	A = R - V(s)
Memory Efficiency	High (no critic)	High (no critic)	High (no critic)	Medium (critic overhead)
Test-Time Scaling	Excellent (7.9-19.2×)	Good	Good	Poor
Diversity Preservation	Excellent	Moderate	Moderate	Moderate
Overfitting Resistance	Strong	Moderate	Moderate	Moderate
Implementation Complexity	Low (1-line change from GRPO)	Medium	Medium	High
Theoretical Guarantees	Converges to ML in limit	None	None	Convergence to local optimum

Sources: MaxRL paper, 2026; Cameron Wolfe GRPO analysis, 2025-2026; Sebastian Raschka RL overview, 2025

Key Differences Explained

MaxRL vs. GRPO: The core difference is the advantage formula. GRPO subtracts mean reward; MaxRL divides by it. This seemingly small change implements harmonic weighting over pass@k terms, yielding the superior scaling and diversity properties.

MaxRL vs. RLOO: RLOO (REINFORCE Leave-One-Out) computes advantage by comparing each sample to the mean of all other samples, reducing variance. MaxRL goes further by weighting samples according to success probability, better aligning with the likelihood objective.

MaxRL vs. PPO: PPO requires training a separate value function (critic) to estimate expected future rewards, doubling memory requirements. MaxRL eliminates this by using group-relative rewards. PPO optimizes a clipped surrogate objective that doesn't target pass@k directly.

When to Use Each

Choose MaxRL when:

• Task has binary verification (code, math, navigation)

• Test-time compute matters (production deployment)

• You want diversity preservation (multi-turn conversations, creative tasks)

• Dataset is limited and overfitting is a concern

Choose GRPO when:

• MaxRL isn't yet integrated in your framework

• Task has continuous/graded rewards

• You need proven stability for production (GRPO is battle-tested in DeepSeek-R1)

Choose PPO when:

• Task lacks ground-truth verification (human preference learning)

• Working with dialogue/open-ended generation

• Memory for critic isn't a constraint

Choose RLOO when:

• Need something between REINFORCE and PPO complexity

• Memory is very constrained

• Working with established RLOO codebases

Advantages and Limitations

Advantages of MaxRL

1. True Objective Alignment

MaxRL optimizes what users actually care about: the probability of finding a correct solution in k attempts. This isn't a proxy or approximation—as compute increases, MaxRL provably converges to exact maximum likelihood.

2. Test-Time Efficiency

The 7.9×–19.2× gains in test-time scaling directly translate to cost savings. If your application serves millions of queries per day, reducing inference compute by 10× means proportional savings in GPU costs.

3. Diversity Preservation

Models trained with MaxRL maintain diverse solution strategies rather than collapsing to a single mode. This is critical for:

• Multi-turn interactions where varied perspectives help

• Ensemble methods that benefit from uncorrelated errors

• Avoiding brittle behavior on out-of-distribution inputs

4. Overfitting Resistance

In data-scarce regimes, MaxRL shows slower initial gains but sustained improvement over epochs. GRPO and RLOO tend to overfit faster, memorizing specific examples without generalizing.

5. Implementation Simplicity

According to the authors, MaxRL requires changing only a single line of code compared to GRPO implementations. This low barrier to adoption means researchers can quickly experiment.

6. Theoretical Foundation

Unlike heuristic methods, MaxRL has rigorous theoretical grounding: it's the truncated form of the log-likelihood objective with provable convergence properties.

Limitations of MaxRL

1. Requires Binary Verification

MaxRL assumes you can verify correctness programmatically. For open-ended tasks (creative writing, conversation quality, complex reasoning without ground truth), it doesn't directly apply.

Workarounds exist—using verifier models or reward models—but these introduce their own complications and approximations.

2. Sampling Budget Dependence

To realize MaxRL's benefits, you need to sample multiple solutions per prompt (typically k=4 to 32). This increases training compute compared to single-sample methods.

The tradeoff: higher training cost for substantially better test-time efficiency. In production systems with long lifetimes, this usually pays off.

3. Compute Requirements

The published experiments use 32 H200 GPUs. While smaller-scale versions work (8 A100s demonstrated in community implementations), MaxRL isn't suitable for training on a laptop.

4. Limited Production History

MaxRL was published in February 2026. Unlike GRPO (battle-tested in DeepSeek-R1) or PPO (industry standard since 2017), MaxRL lacks extensive production deployment data.

Early adoption carries risk. Bugs, edge cases, and failure modes may not yet be documented.

5. Hyperparameter Sensitivity

Like all RL methods, MaxRL has hyperparameters requiring tuning:

• KL penalty coefficient β

• Learning rate schedule

• Rollouts per prompt k

• Batch size and gradient clipping

Getting these wrong can cause training instability. The published paper provides starting points, but optimal values are task-dependent.

6. Evaluation Complexity

Properly measuring pass@k requires generating many samples at test time. Single-pass evaluation (just checking pass@1) misses MaxRL's advantages. This complicates benchmarking and model selection.

Myths vs. Facts

Myth: MaxRL completely replaces GRPO and other RL methods.

Fact: MaxRL is specialized for binary-verification tasks. For general reward-based RL, established methods remain relevant. MaxRL complements rather than replaces the broader toolkit.

Myth: MaxRL eliminates the need for large compute budgets.

Fact: MaxRL improves efficiency but still requires substantial compute for training large models. The gains come from better allocation of that compute, not from needing less of it.

Myth: You need 32 H200 GPUs to use MaxRL.

Fact: Published experiments used that scale, but community implementations demonstrate strong results with 8 A100s (~$20-30k in cloud costs for a full training run). Smaller models can train on even less.

Myth: Pass@k optimization means the model doesn't care about single-attempt accuracy.

Fact: Pass@k includes pass@1 as a special case. MaxRL maintains or improves pass@1 while dramatically improving pass@k. You get both.

Implementation and Technical Requirements

For researchers and engineers looking to adopt MaxRL, here's what you need:

Software Requirements

Core Framework: PyTorch (tested with version 2.6.0)

Base Installation (from official GitHub):

conda create -n maxrl python==3.10
conda activate maxrl
pip install torch==2.6.0 torchvision==0.21.0 torchaudio==2.6.0 --index-url https://download.pytorch.org/whl/cu124

Dependencies:

• verl: The codebase builds on the verl framework for RL

• Flash Attention 2.x for efficient attention computation

• Hugging Face Transformers for model loading

• DeepSpeed or FSDP for distributed training

Official Repository: github.com/tajwarfahim/maxrl

Hardware Requirements

Minimum (for research/experimentation):

• 8× A100 (40GB or 80GB) GPUs

• 1-2 nodes with NVLink or InfiniBand interconnect

• 512GB+ system RAM

• Fast SSD storage (NVMe recommended) for dataset caching

Recommended (for production-scale):

• 32× H100 or H200 GPUs (4 nodes × 8 GPUs)

• High-speed interconnect (InfiniBand preferred)

• 2TB+ system RAM

• Multi-TB NVMe storage array

Cloud Options:

• AWS: p5.48xlarge instances (8× H100)

• GCP: a3-highgpu-8g instances (8× H100)

• Azure: ND H100 v5-series

Estimated cost for full training run (Qwen-2.5-7B on MATH dataset): $2,000-$5,000 depending on cloud provider and spot instance availability.

Data Requirements

Training Data:

• Minimum: 10,000 high-quality prompts with verifiable solutions

• Recommended: 50,000-100,000 prompts for strong generalization

• Format: JSON or JSONL with fields: prompt, solution, answer

Evaluation Data:

• Held-out test set: 1,000-5,000 prompts

• Must be truly out-of-distribution to avoid overfitting detection

Preprocessing Scripts: Official repo includes scripts for:

• MATH dataset

• GSM8K

• POLARIS

• AIME and related competition problems

• Custom dataset formatting

  
  

Training Configuration

Typical Hyperparameters (from MaxRL paper):

• Learning rate: 1e-6 to 5e-6

• Batch size: 1024-2048 (global, across all GPUs)

• KL penalty (β): 0.01-0.05

• Rollouts per prompt (k): 8-32

• Gradient clip norm: 1.0

• Training epochs: 2-5 for fixed datasets, continuous for procedural generation

Training Time:

• Qwen-2.5-7B on 8 A100s: ~27 hours (community report)

• Qwen-2.5-32B on 32 H200s: ~2-3 days (estimated from paper)

  
  

Integration with Existing Codebases

If you already use GRPO or similar RL training loops, MaxRL integration is straightforward. The key modification is in the advantage computation:

Before (GRPO):

advantage = rewards - rewards.mean()

After (MaxRL):

mean_reward = rewards.mean()
advantage = rewards / (mean_reward + 1e-8) - 1

The 1e-8 prevents division by zero when all rewards are zero.

This simplicity enables rapid experimentation: swap in MaxRL, run a small-scale test, compare results.

Monitoring and Evaluation

Key Metrics to Track:

• Pass@1, Pass@4, Pass@8, Pass@16 on held-out set

• KL divergence from reference policy

• Solution diversity (measured via n-gram overlap or embedding distance)

• Training loss and gradient norms

• Per-difficulty performance breakdown

Debugging Tools:

• Sample a few problems each epoch and inspect generated solutions

• Track which problems flip from 0% to >0% pass rate over training

• Monitor for "sharpening": problems clustering at 0% or 100%

Current Research Landscape

MaxRL emerged in February 2026 into an active ecosystem of RL research for language models. Here's the current state:

Dominant Algorithms (Early 2026)

1. GRPO (Group Relative Policy Optimization)

Introduced in DeepSeek-Math (February 2024), GRPO became the de facto standard after DeepSeek-R1's success in January 2025. Key properties:

• No critic model required

• Groups multiple samples per prompt

• Advantage computed relative to group mean

Adoption: DeepSeek-R1, DeepSeek-V3, numerous open-source reasoning models

2. PPO (Proximal Policy Optimization)

The original RLHF algorithm from 2017, still used for:

• General chat model alignment (Llama, GPT, Gemini)

• Tasks without perfect verifiers

• Production systems with established PPO infrastructure

3. DPO (Direct Preference Optimization)

Introduced in 2023 as an alternative to PPO for preference learning:

• No separate reward model

• Treats the policy itself as an implicit reward model

• Lower memory overhead than PPO

Limitation: Designed for preference data, not verification-based tasks. Less suitable for MaxRL's target domain.

Recent Advances (2025-2026)

REINFORCE++ (January 2026): Proposes stabilizing critic-free RL through global normalization rather than group-based. Shows competitive results with GRPO while being more token-efficient (REINFORCE++, 2026).

DAPO (March 2025): ByteDance's extension of GRPO with decoupled clipping and dynamic sampling. Achieved 50 points on AIME 2024, surpassing DeepSeek-R1-Zero's 47 points (DAPO, 2025).

Training-Free GRPO (October 2025): Applies GRPO principles without parameter updates, using experiential context instead. Useful for adapting frozen models (Training-Free GRPO, 2025).

S-GRPO (December 2025): Supervised variant of GRPO that incorporates additional supervised components, outperforming standard SFT (S-GRPO, 2025).

Research Directions

1. Scaling Laws for RL

Multiple groups study how RL performance scales with:

• Model size (parameters)

• Data size (prompts)

• Compute (rollouts per prompt)

• Test-time compute (inference-time search)

Early findings suggest RL scaling differs from pretraining scaling, with stronger emphasis on data quality over quantity.

2. Self-Play and Curriculum Learning

Methods like Variational Self-Play (SVS) generate progressively harder training problems, maintaining diversity and preventing overfitting (Liang et al., 2025).

3. Proof-Based Evaluation

Moving beyond answer-checking to full proof verification. AlphaProof demonstrated feasibility; scaling to broader domains remains active research.

4. Multi-Modal RL

Extending verification-based RL to vision (image generation with human feedback), robotics (task success), and other modalities.

Industry Adoption

As of early 2026:

• OpenAI: Uses RL extensively in GPT-4 and o-series models (exact methods unpublished)

• Google DeepMind: AlphaProof, Gemini reasoning capabilities

• Anthropic: Claude 3.5 and 4 series use RL for reasoning

• DeepSeek: Pioneered open publication of GRPO methods

• Meta: Llama 3 series incorporates RL-based reasoning

The trend toward transparency (DeepSeek publishing methods) vs. opacity (OpenAI keeping details secret) creates an interesting dynamic. MaxRL benefits from the open science approach.

Future Outlook and Implications

Where does MaxRL fit in the bigger picture? Several trends suggest growing importance:

Near-Term Trajectory (2026-2027)

Integration into Production Systems

Expect to see MaxRL or MaxRL-inspired techniques in:

• Code completion tools (GitHub Copilot, Cursor, Replit)

• Mathematical reasoning assistants (Wolfram Alpha integration, education platforms)

• Automated theorem proving systems

• Robotic control (where success can be verified via simulation)

Hybrid Objectives

Rather than pure MaxRL or pure GRPO, we'll likely see blends:

• MaxRL for hard problems where diversity matters

• GRPO for easier problems where pass@1 suffices

• Adaptive mixing based on problem difficulty or confidence

Improved Verifiers

Better verification functions expand MaxRL's applicability:

• Learned verifiers for code (beyond just test cases)

• Multi-level verification (partial credit for partially correct solutions)

• Probabilistic verification with confidence estimates

  
  

Medium-Term Implications (2027-2029)

Cost-Performance Tradeoffs

As inference costs remain high, the 7.9×–19.2× test-time efficiency gains become increasingly valuable. Organizations may optimize training objectives specifically for inference cost reduction.

Regulatory Pressure

If AI systems are deployed in high-stakes domains (medical diagnosis, legal reasoning), regulators may require provable correctness or verification-based training. MaxRL-style approaches fit this requirement.

Democratization

As hardware costs decline and open implementations mature, smaller organizations gain access to state-of-the-art RL methods. MaxRL's simplicity (one-line code change) accelerates this.

Long-Term Potential (2030+)

Scientific Discovery

Verification-based RL could accelerate scientific breakthroughs in:

• Theorem proving (mathematics, computer science)

• Protein design (verifiable stability and function)

• Materials science (computational verification of properties)

• Drug discovery (verifiable safety and efficacy in simulation)

Education Revolution

Imagine a tutor that generates multiple solution approaches to a problem, verified for correctness, adapted to each student's level. MaxRL makes this feasible at scale.

Limitations and Risks

Verification Brittleness: Many real-world tasks lack perfect verifiers. Reward hacking on imperfect verification remains a concern.

Compute Inequality: Organizations with massive compute resources can run k=1000+ rollouts per prompt, achieving results smaller players can't match. This could centralize AI capabilities.

Evaluation Gaming: As benchmarks like AIME become saturated, pressure to find uncontaminated evaluations grows. The cat-and-mouse game between training and evaluation continues.

Fundamental Limits: Even perfect RL can't overcome limits of the base model's knowledge and reasoning capacity. MaxRL optimizes learning from experience, but doesn't create knowledge from nothing.

Frequently Asked Questions

Q1: How is MaxRL different from regular reinforcement learning?

Standard RL optimizes expected reward (E[R]), which is a first-order approximation of maximum likelihood. MaxRL optimizes a truncated form of the log-likelihood (E[log p]), which better aligns with the goal of maximizing success probability across multiple attempts. The practical difference shows up as 7.9×–19.2× better test-time scaling efficiency.

Q2: Can I use MaxRL for chatbot training or preference-based tasks?

Not directly. MaxRL requires binary verification (correct/incorrect) rather than preference labels (A better than B). For chat and open-ended tasks, stick with PPO, DPO, or standard GRPO. However, if you can decompose chat quality into verifiable sub-tasks (grammar correctness, fact checking, etc.), MaxRL could apply to those components.

Q3: How much does MaxRL cost to implement compared to GRPO?

Code complexity is nearly identical—MaxRL modifies one line of the advantage computation. Compute cost is also similar: both require multiple rollouts per prompt. The difference is MaxRL achieves better results for the same compute budget, or similar results with less test-time compute.

Q4: What programming languages and frameworks support MaxRL?

The official implementation uses PyTorch with the verl framework. Community ports to JAX/Flax or TensorFlow may emerge. Any framework supporting standard RL can adapt MaxRL by modifying the advantage function.

Q5: Does MaxRL work with smaller models like 7B parameter models?

Yes. The community has demonstrated strong results with Qwen-2.5-7B and similar sizes. Smaller models require less compute but may need more careful hyperparameter tuning. The core algorithm scales from hundreds of millions to hundreds of billions of parameters.

Q6: What benchmarks best demonstrate MaxRL's advantages?

Mathematical reasoning benchmarks (AIME, MATH, GSM8K), code generation (HumanEval, MBPP, CodeContests), and theorem proving (Lean, miniF2F) all showcase MaxRL's strengths. Any task with programmatic verification and multiple solution attempts fits well.

Q7: How does MaxRL handle continuous or graded rewards?

The published work focuses on binary rewards (0 or 1). For continuous rewards, you'd need to extend the framework. One approach: bin continuous rewards into discrete levels and treat as multi-class verification. Research on this extension is ongoing.

Q8: Can MaxRL be combined with human feedback (RLHF)?

Yes, but indirectly. You could use RLHF to train a reward model, then use that model's scores as verification for MaxRL. This creates a two-stage pipeline. Direct integration would require theoretical work to blend verification-based and preference-based objectives.

Q9: What happens if my verifier is wrong sometimes?

Imperfect verification introduces noise into the reward signal. MaxRL, like all RL methods, can learn from noisy rewards but may develop reward hacking (exploiting verifier bugs). The solution is improving verifier quality through better test suites, ensemble verification, or learned verifier models.

Q10: How do I know if MaxRL is working correctly in my implementation?

Monitor these signals:

• Pass@k should improve faster than pass@1 alone

• Solution diversity (measured via embedding distance or n-gram overlap) should stay high

• Hard problems should show gradual improvement, not stay stuck at 0%

• KL divergence should remain bounded (not explode or collapse to zero)

  
  

Actionable Next Steps

If you're ready to experiment with MaxRL, here's a concrete path forward:

1. Start Small: Clone the official repository and run the provided examples on a small dataset (GSM8K or MATH subset) with a 7B model. Budget: 8 A100 GPUs for 24-48 hours.

2. Measure Baseline: Before implementing MaxRL, establish GRPO or REINFORCE baselines on your task. Track pass@1, pass@4, pass@8, and solution diversity.

3. Modify Advantage: Change the advantage computation from rewards - rewards.mean() to rewards / (rewards.mean() + 1e-8) - 1. Run training with identical hyperparameters otherwise.

4. Compare Results: On your held-out test set, measure whether MaxRL improves pass@k without degrading pass@1. A successful result shows 2-5× efficiency gains even at small scale.

5. Tune Hyperparameters: If initial results underwhelm, adjust KL penalty β and rollouts per prompt k. Higher k generally helps MaxRL but increases compute cost.

6. Scale Gradually: Once small-scale works, move to larger models (32B parameters) and larger datasets (50K+ prompts). Monitor for training instabilities that didn't appear at small scale.

7. Productionize Carefully: Before deploying, run extensive red-teaming to find edge cases. MaxRL's novelty means failure modes aren't fully catalogued. Have rollback plans.

8. Contribute Back: If you discover improvements, bugs, or new applications, contribute to the open-source ecosystem. File GitHub issues, publish findings, share hyperparameter configurations.

9. Stay Updated: Follow the research team (Fahim Tajwar, Andrea Zanette, Ruslan Salakhutdinov) on social media and arXiv for updates, extensions, and related work.

10. Consider Alternatives: MaxRL isn't the only game in town. Compare against DAPO, REINFORCE++, and other recent methods to ensure you're using the best tool for your specific task.

Key Takeaways

• MaxRL optimizes the right objective: Instead of maximizing expected reward (standard RL's first-order approximation), MaxRL targets the true likelihood of generating correct solutions

• Massive efficiency gains: 7.9×–19.2× better test-time scaling on mathematical reasoning tasks translates directly to lower inference costs in production

• Simple implementation: One-line code change from GRPO makes adoption straightforward for teams already using group-relative RL methods

• Superior diversity: MaxRL maintains varied solution strategies rather than collapsing to a single mode, critical for robustness and avoiding overfitting

• Scales with compute: As you allocate more rollouts per prompt, MaxRL provably approaches exact maximum likelihood, while standard RL's approximation stays fixed

• Specialized for verification: MaxRL shines on tasks with binary correctness checking (code, math, proofs) but doesn't directly apply to preference-based or open-ended tasks

• Battle-tested by top teams: Carnegie Mellon researchers, building on DeepSeek's GRPO foundation, published February 2026 with extensive empirical validation

• Production-ready with caveats: Official PyTorch implementation available, but limited deployment history means edge cases may surface

• Part of broader trend: Verification-based RL is rapidly displacing preference-based methods for technical reasoning tasks across industry and academia

• Future potential: As scientific discovery, code generation, and mathematical reasoning become central AI applications, MaxRL-style approaches will grow in importance

Glossary

1. Advantage: In reinforcement learning, the difference between a sample's reward and a baseline (often the mean reward). Tells the algorithm whether to encourage or discourage the action that produced this sample.

2. AIME (American Invitational Mathematics Examination): Elite high school math competition serving as a benchmark for AI mathematical reasoning. Top 5% of AMC 12 participants qualify. 15 problems per exam, difficulty range from accessible to IMO-level.

3. Binary Verification: The ability to programmatically determine whether a solution is correct (1) or incorrect (0) with no ambiguity. Examples: code passes tests, mathematical answer matches, navigation reaches goal.

4. Entropy Collapse: When a model's output distribution becomes overly confident and narrow, losing diversity. Manifests as the model always generating similar solutions rather than exploring alternatives.

5. GRPO (Group Relative Policy Optimization): RL algorithm that samples multiple outputs per prompt and computes advantages relative to the group average. Eliminates the need for a critic model while maintaining stable training.

6. Harmonic Weighting: Weighting scheme using coefficients 1, 1/2, 1/3, 1/4, etc. In MaxRL, corresponds to the natural weighting that arises from decomposing log-likelihood into pass@k terms.

7. Implicit Likelihood: The probability that a model assigns to the set of correct solutions, even though the model doesn't explicitly compute this probability. Emerges from the model's generative distribution.

8. KL Divergence (Kullback-Leibler Divergence): A measure of how one probability distribution differs from another. In RL, used to prevent the policy from changing too drastically during training (regularization).

9. Maximum Likelihood: A statistical principle that says "choose model parameters that make the observed data most probable." For classification: maximize probability of correct labels. For RL with verification: maximize probability of correct solutions.

10. Pass@k: The probability that at least one solution out of k independent attempts is correct. Calculated as 1 - (1 - p)^k where p is single-attempt success probability. Used to evaluate systems that generate multiple candidates.

11. Policy Gradient: RL technique that directly optimizes the policy (the model that chooses actions) by computing gradients of expected reward with respect to policy parameters.

12. REINFORCE: Classic policy gradient algorithm from 1992, still foundational. Estimates gradients by sampling actions and weighting by received rewards. High variance but unbiased.

13. RLHF (Reinforcement Learning from Human Feedback): Training paradigm where humans rate or rank model outputs, a reward model learns to predict these ratings, and RL optimizes the policy according to predicted rewards.

14. RLOO (REINFORCE Leave-One-Out): Variance reduction technique for REINFORCE that computes advantage by comparing each sample to the mean of all other samples (excluding the current one).

15. Rollout: A complete sequence of actions from start to finish in an RL task. For code generation: the full program. For math: the entire solution. For navigation: the complete path.

16. Sharpening: The phenomenon where a model trained on a dataset learns some examples perfectly (100% success rate) while completely failing on others (0% success rate), with few in between. Indicates overfitting.

17. Test-Time Scaling: The practice of allocating more compute at inference time (generating more solutions, longer search) to improve accuracy. Contrasts with training-time scaling (more data/parameters).

18. Truncated Objective: An approximation of an infinite sum that includes only the first T terms. MaxRL uses truncated maximum likelihood because the exact objective (infinite sum over all pass@k) is intractable.

Sources & References

1. Tajwar, F., Zeng, G., Zhou, Y., Song, Y., Arora, D., Jiang, Y., Schneider, J., Salakhutdinov, R., Feng, H., & Zanette, A. (2026, February 2). Maximum Likelihood Reinforcement Learning. arXiv:2602.02710. Retrieved from https://arxiv.org/abs/2602.02710

2. Maximum Likelihood Reinforcement Learning - Official Project Website. (2026). Zanette Labs at Carnegie Mellon University. Retrieved from https://zanette-labs.github.io/MaxRL/

3. Salakhutdinov, R. [@rsalakhu]. (2026, February 3). New work on Maximum Likelihood Reinforcement Learning [Tweet]. X (formerly Twitter). Retrieved from https://x.com/rsalakhu/status/2019507844161187916

4. Wolfe, C. R. (2025, November 24). Group Relative Policy Optimization (GRPO) Illustrated Breakdown. Cameron's Blog on Substack. Retrieved from https://cameronrwolfe.substack.com/p/grpo

5. Wolfe, C. R. (2026, January 5). GRPO++: Tricks for Making RL Actually Work. Cameron's Blog on Substack. Retrieved from https://cameronrwolfe.substack.com/p/grpo-tricks

6. Raschka, S. (2025, April 19). The State of Reinforcement Learning for LLM Reasoning. Ahead of AI Magazine. Retrieved from https://magazine.sebastianraschka.com/p/the-state-of-llm-reasoning-model-training

7. Raschka, S. (2025, December 30). The State Of LLMs 2025: Progress, Progress, and Predictions. Ahead of AI Magazine. Retrieved from https://magazine.sebastianraschka.com/p/state-of-llms-2025

8. Shao, Z., et al. (2024, February 7). DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models. arXiv:2402.03300. Retrieved from https://arxiv.org/abs/2402.03300

9. DeepSeek-AI, et al. (2025, January 22). DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning. arXiv:2501.12948. Retrieved from https://arxiv.org/abs/2501.12948

10. Yu, Z., et al. (2025, March). DAPO: An Open-Source LLM Reinforcement Learning System at Scale. arXiv:2503.14476. Retrieved from https://arxiv.org/pdf/2503.14476

11. Hu, J. (2025, January). REINFORCE++: A Simple and Efficient Approach for Aligning Large Language Models. arXiv:2501.03262. Retrieved from https://arxiv.org/pdf/2501.03262

12. Cai, Y., et al. (2025, October 9). Training-Free Group Relative Policy Optimization. arXiv:2510.08191. Retrieved from https://arxiv.org/abs/2510.08191

13. Shafayat, S., Tajwar, F., Salakhutdinov, R., Schneider, J., & Zanette, A. (2025, May 27). Can Large Reasoning Models Self-Train? arXiv:2505.21444. Retrieved from https://arxiv.org/abs/2505.21444

14. Google DeepMind. (2025, November 12). AI achieves silver-medal standard solving International Mathematical Olympiad problems. DeepMind Blog. Retrieved from https://deepmind.google/blog/ai-solves-imo-problems-at-silver-medal-level/

15. Intuition Labs. (2025, October 24). AIME 2025 Benchmark: An Analysis of AI Math Reasoning. Retrieved from https://intuitionlabs.ai/articles/aime-2025-ai-benchmark-explained

16. Vals AI. (2026). AIME Benchmark Results and Analysis. Retrieved from https://www.vals.ai/benchmarks/aime

17. Li, S., et al. (2025, May 19). OlymMATH: Challenging the Boundaries of Reasoning: An Olympiad-Level Math Benchmark for Large Language Models. arXiv:2503.21380. Retrieved from https://arxiv.org/html/2503.21380v2

18. Balunović, M., et al. (2025, May 23). MathArena: Evaluating LLMs on Uncontaminated Math Competitions. arXiv:2505.23281. Retrieved from https://arxiv.org/pdf/2505.23281

19. Lyu, Z., et al. (2024, August 11). Top Pass: Improve Code Generation by Pass@k-Maximized Code Ranking. arXiv:2408.05715. Retrieved from https://arxiv.org/html/2408.05715v1

20. Emergent Mind. (2025). Pass@k Metric in Code Generation & RL - Research Overview. Retrieved from https://www.emergentmind.com/topics/pass-k-metric-b5b58688-14e3-4ed9-b1f7-504db4b60803

21. Chen, Y. (2024, December 17). A dive into how pass@k is calculated for evaluation of LLM's coding. Medium. Retrieved from https://medium.com/@yananchen1116/a-dive-into-how-pass-k-is-calculated-for-evaluation-of-llms-coding-e52b8528235b

22. Lee, H. (2025, September 8). Statistics for AI/ML, Part 4: pass@k and Unbiased Estimator. Tech Blog. Retrieved from https://leehanchung.github.io/blogs/2025/09/08/pass-at-k/

23. Zanette, A. (2026). Andrea Zanette - Faculty Page. Carnegie Mellon University. Retrieved from https://azanette.com/

24. Tajwar, F. (2026). Fahim Tajwar - Personal Website. Carnegie Mellon University. Retrieved from https://tajwarfahim.github.io/

25. GitHub Repository: tajwarfahim/maxrl. (2026). Official Implementation of Maximum Likelihood Reinforcement Learning (MaxRL). Retrieved from https://github.com/tajwarfahim/maxrl

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