Inspired by DeepSeek R1’s success in reasoning via reinforcement learning without human feedback, we train a 3B language model using the Countdown Game with pure reinforcement learning. Our model outperforms baselines on four of five benchmarks, demonstrating improved generalization beyond its training data. Notably, response length does not correlate with reasoning quality, and while "aha moments" emerge, they do not always yield correct answers. These findings highlight the potential of RL-only training for reasoning enhancement and suggest future work on refining reward structures to bridge emergent insights with accuracy.

1. Bibliographic Information

• Title: REINFORCEMENTLEARNING IS ALL YOU NEED
• Authors: Yongsheng Lian, from the Mechanical Engineering Department at the University of Louisville.
• Journal/Conference: The paper does not specify a publication venue. Its formatting and the presence of future-dated references (e.g., 2025) suggest it is a preprint, possibly submitted to a conference or an online repository like arXiv.
• Publication Year: The content references papers from 2024 and cites future works in 2025, suggesting this paper was written in late 2024 or early 2025. The current date is October 7, 2025.
• Abstract: The paper explores training a 3B language model for reasoning tasks using pure reinforcement learning (RL) without human feedback, inspired by the success of DeepSeek R1. The model is trained on the Countdown Game. The results show that the trained model surpasses baseline models on four out of five evaluation benchmarks, indicating strong generalization. Key findings include the lack of correlation between response length and reasoning quality and the emergence of "aha moments" (sudden insights) that, while human-like, do not guarantee correct answers. The author concludes that RL-only training is a promising direction for enhancing reasoning and suggests that future work should focus on improving reward structures to align these emergent insights with factual accuracy.
• Original Source Link: /files/papers/68e4c7a25f732f8abc81b78f/paper.pdf (This is a preprint document).

2. Executive Summary

• Background & Motivation (Why):

  • Core Problem: Enhancing the reasoning capabilities of Large Language Models (LLMs) is a critical challenge. Traditional methods often rely on extensive supervised fine-tuning (SFT) with human-annotated data, which can be costly and resource-intensive.
  • Gap in Prior Work: While reinforcement learning has shown promise (e.g., AlphaGo, DeepSeek R1), a key question is whether pure RL, without any human feedback or preference data, can effectively teach a model to reason and generalize from a narrow, rule-based task.
  • Innovation: This paper investigates a minimalist approach: using a simple, self-contained numerical reasoning task (the Countdown Game) with a rule-based reward signal to train a 3B parameter model entirely through reinforcement learning. The goal is to see if reasoning skills learned in this environment can transfer to a wide range of unseen, complex benchmarks.

• Main Contributions / Findings (What):

  • Primary Contribution: The paper demonstrates that a 3B LLM trained exclusively with reinforcement learning on a numerical puzzle game can achieve significant improvements in general reasoning capabilities.
  • Key Findings:
    1. Improved Generalization: The RL-trained model outperformed its baseline counterparts on four out of five diverse benchmarks (GSM8K, MATH, BBH, MMLU-Pro), proving that the reasoning skills were not limited to the training task.
    2. Emergent Human-like Thinking: During training, the model developed complex problem-solving behaviors, including systematic exploration, backtracking, and "aha moments" where it appears to recognize and correct its own mistakes mid-thought.
    3. Insights are Not Always Correct: A crucial observation is that these "aha moments," while indicative of a more sophisticated reasoning process, do not always lead to the correct final answer. This highlights a gap between recognizing a flawed path and finding the correct one.
    4. No Correlation Between Length and Quality: The study found that the length of the model's generated reasoning process (its "thought process") does not correlate with the quality or correctness of the final answer. This challenges the common assumption that longer, more detailed chain-of-thought responses are inherently better.

3. Prerequisite Knowledge & Related Work

• Foundational Concepts:

  • Reinforcement Learning (RL): A type of machine learning where an agent learns to make decisions by performing actions in an environment to maximize a cumulative reward. Unlike supervised learning, it doesn't require labeled data but instead learns from the consequences of its actions.
  • Supervised Fine-Tuning (SFT): The process of further training a pre-trained language model on a smaller, specific dataset of high-quality examples (e.g., instruction-response pairs) to align its behavior with desired tasks or styles.
  • Chain-of-Thought (CoT) Prompting: A technique that encourages LLMs to solve complex problems by prompting them to generate a step-by-step reasoning process before giving the final answer. This often improves performance on tasks requiring logic and calculation.
  • Policy Optimization Algorithms: These are RL algorithms that directly optimize the agent's decision-making policy.
    • Proximal Policy Optimization (PPO): A popular and stable RL algorithm that prevents the learning policy from changing too drastically from the previous policy in each update step, ensuring more reliable training.
    • Group Relative Policy Optimization (GRPO): An alternative to PPO, used in this paper. Instead of learning a separate value function to estimate rewards, it generates a group of responses, calculates their actual rewards, and normalizes them to determine which responses were "better" or "worse" relative to the group average. This simplifies the process and can be more computationally efficient.

• Previous Works:

  • AlphaGo [12]: A landmark achievement where DeepMind used a combination of supervised learning (from human expert games) and reinforcement learning (from self-play) to create a program that defeated the world's best Go players. It demonstrated the power of RL in achieving superhuman performance in a complex strategic domain.
  • DeepSeek R1 [5, 13]: A powerful open-source LLM that was trained using a multi-stage process involving SFT followed by reinforcement learning. It notably used GRPO to incentivize reasoning capabilities without direct human feedback, serving as the primary inspiration for this paper's RL-only approach.
  • Process-based Reward Models (PRM) [10, 11]: An alternative reward modeling technique where a model provides feedback on each step of the reasoning process, rather than just the final answer. This paper uses a simpler, final-answer-based reward but acknowledges PRMs as a more advanced alternative.

• Differentiation: This paper distinguishes itself by adopting a pure RL-only post-training approach. While DeepSeek R1 used RL after SFT stages, this work investigates if RL alone, applied to a base model, is sufficient to cultivate generalizable reasoning. It uses a simple, verifiable game (Countdown) and a rule-based reward, eliminating the need for human-annotated SFT data or complex preference models.

4. Methodology (Core Technology & Implementation)

The core of the methodology is to fine-tune a pre-trained 3B language model using reinforcement learning on a specific numerical reasoning task.

• Principles: The underlying principle is that by repeatedly rewarding a model for solving a structured, logical puzzle, it will internalize the underlying principles of systematic reasoning, which can then be applied to other, unrelated problems. The entire process is automated, relying on a rule-based system for rewards instead of human feedback.

• Steps & Procedures:

  1. Dataset: The model is trained on the Countdown Game. This is a numerical puzzle where the goal is to use a given set of numbers and basic arithmetic operations ($+,-, \times, \div$) to reach a target number. For example, given numbers 6, 7, 8, 9 and target 24, a solution could be $8 \times (9 - 6) = 24$. This game is ideal because solutions are easily and unambiguously verifiable, providing a perfect source for a clear reward signal.
  2. Reward Modeling: A rule-based reward model is created to score the model's generated solutions. It consists of two parts:
     • Format Reward: This reward checks if the model's output adheres to the required structure. The prompt asks the model to place its reasoning within tags and the final equation in tags. The Format Reward ensures this structure is followed and penalizes violations, such as nested tags. This enforces a clean, parseable output.
     • Answer Reward: This is the primary reward signal. It evaluates the equation within the tags. If the equation is mathematically correct and produces the target number, a positive reward is given. Otherwise, the reward is zero or negative. This directly incentivizes correctness.
  3. Reinforcement Learning Algorithm: The paper uses Group Relative Policy Optimization (GRPO). Instead of the more common PPO, which requires a learned value function to estimate future rewards, GRPO simplifies the process:
     • For a given problem, the current policy generates a group of G possible responses.
     • The rule-based reward model scores each of the G responses.
     • The rewards within this group are normalized (by subtracting the mean and dividing by the standard deviation) to calculate an "advantage" for each response. A response with a higher-than-average reward gets a positive advantage, and one with a lower-than-average reward gets a negative advantage.
     • The policy is then updated to increase the probability of generating responses with high advantage and decrease the probability of those with low advantage.

• Mathematical Formulas & Key Details: The GRPO objective function to be maximized is given by: $$\mathcal { I } _ { G R P O } ( \theta ) = \mathbb { E } \Bigg [ \sum _ { i = 1 } ^ { G } \left( \operatorname* { m i n } \left( \frac { \pi _ { \theta } ( o _ { i } ) } { \pi _ { \theta _ { \mathrm { o d } } } ( o _ { i } ) } A _ { i } , \mathrm { c l i p } \Big ( \frac { \pi _ { \theta } ( o _ { i } ) } { \pi _ { \theta _ { \mathrm { o d } } } ( o _ { i } ) } , 1 - \varepsilon , 1 + \varepsilon \Big ) A _ { i } \right) - \beta \mathbb { D } _ { K L } \left( \pi _ { \theta } \parallel \pi _ { \mathrm { r e f } } \right) \right) \Bigg ] \mathrm { ~ , ~ }$$

  • $\pi_{\theta}(o_i)$: The probability of generating response $o_i$ with the new policy (with parameters $\theta$).

  • $\pi_{\theta_{old}}(o_i)$: The probability of generating response $o_i$ with the old policy before the update.

  • $A_i$: The advantage of response $o_i$, calculated by normalizing its reward relative to the other responses in the group.

  • $\varepsilon$: A hyperparameter that defines the clipping range, preventing the policy from changing too aggressively.

  • $\beta$: A hyperparameter that controls the strength of the KL divergence penalty.

  • $\mathbb{D}_{KL}(\pi_{\theta} \parallel \pi_{ref})$: The Kullback-Leibler (KL) divergence, a term that penalizes the new policy $\pi_{\theta}$ for straying too far from a reference policy $\pi_{ref}$ (often the original pre-trained model), ensuring stability.

    The advantage $A_i$ is calculated as: $$A _ { i } = { \frac { r _ { i } - \operatorname * { m e a n } ( \{ r _ { 1 } , r _ { 2 } , \cdots , r _ { G } \} ) } { \operatorname * { s t d } ( \{ r _ { 1 } , r _ { 2 } , \cdots , r _ { G } \} ) } }$$

  • $r_i$: The reward for the $i$-th response in a group of $G$ responses.

  • This formula standardizes the rewards, making the learning process more stable and less dependent on the absolute scale of rewards.

5. Experimental Setup

• Datasets: The model's performance was evaluated on five diverse and challenging benchmarks to test for generalization:

  • Grade School Math 8K (GSM8K): A dataset of grade-school level math word problems that require multi-step reasoning.
  • Instruction-Following Eval (IFEval): Assesses an LLM's ability to follow complex and constrained instructions precisely.
  • BIG-Bench Hard (BBH): A difficult subset of the BIG-Bench benchmark, containing tasks that demand advanced multi-step reasoning.
  • Mathematics Aptitude Test of Heuristics (MATH): A dataset of challenging math competition problems that require more than just rote calculation.
  • A More Robust and Challenging Multi-Task Language Understanding Benchmark (MMLU-Pro): An enhanced version of MMLU focused on complex reasoning across various professional-level subjects.

• Evaluation Metrics:

  • Strict-Match / Strict Accuracy: The generated answer must exactly match the reference answer.
  • Flexible-Extract / Loose Accuracy: The metric allows for minor formatting differences, extracting the final answer from the response and comparing it to the reference.
  • math_verify: A metric for the MATH benchmark that checks if the final numerical answer is mathematically correct, regardless of the reasoning steps or formatting.

• Baselines:

  • Base Model: The original 3B parameter model (Qwen1.5-3B-Chat) without any RL fine-tuning, tested with a default prompt.
  • Base Model + R1 Prompt: The same base model, but prompted with the structured prompt used during the RL training (which asks for and tags). This isolates the effect of the prompt from the effect of the training.

• Training Setup: The RL fine-tuning was conducted using the HuggingFace TRL library with the following key hyperparameters:

  • Total training steps: 850
  • Batch size: 2
  • Learning rate: $1.0 \times 10^{-6}$
  • GRPO samples per step: 2 (meaning the advantage was calculated from a group of two generated responses)
  • KL regularization coefficient ($\beta$): 0.04

6. Results & Analysis

The paper presents both qualitative observations from the training process and quantitative results from the benchmark evaluations.

Qualitative Analysis of the Learning Process

• Early Training Stages: Initially, the model struggled with basic rules, such as generating nested tags, violating the required format.
• Emergence of Heuristics: As training progressed, the model began to show signs of human-like thinking. Instead of brute-forcing solutions, it started using heuristics, such as estimating outcomes ("too high," "too low") and systematically trying combinations of operations.
• The "Aha Moment": In later stages, the model exhibited "aha moments," where it would identify a mistake in its own reasoning chain and attempt to correct it, often signaled by phrases like "But wait...". For example, it might realize it used the wrong number from the prompt and backtrack. However, a key finding is that this self-correction did not always lead to a correct final answer, suggesting the model's verification mechanism was still imperfect.

Response Length vs. Reasoning Capability

The paper found no direct correlation between the length of the model's generated response and its reasoning quality.

该图像为折线图，展示随训练步数（Steps）增加，模型生成文本的完成长度（Completion Length）变化趋势。图中观察到完成长度起初较长，随后整体下降并波动，说明生成文本长度随训练进展有显著波动，无明显线性关系。该结果对应论文中提到的生成长度与推理质量无直接相关性的结论。

As shown in Image 2, the completion length fluctuates significantly throughout training, with a general downward trend followed by stabilization. This suggests that as the model becomes more efficient at reasoning, it may produce more concise and direct solutions rather than longer, exploratory ones. This contradicts the common assumption that longer chain-of-thought is always better.

Core Quantitative Results

The RL-trained model showed significant improvements over the baselines on four of the five benchmarks.

该图像为柱状图，比较了三种模型（基础模型、基础模型加R1提示、训练后模型）在五个基准测试（GSM8K、MATH、IFEEval、BBH、MMLU-Pro）上的性能表现。训练后模型在大多数基准上表现优于其他两种，尤其在GSM8K和MATH测试中提升明显，显示纯强化学习训练带来的性能提升与泛化能力增强。

Image 1 provides a summary of the performance comparison. The green bar (Trained Model) is highest in GSM8K, MATH, BBH, and MMLU-Pro.

• GSM8K (Math Word Problems): The trained model achieved a Strict-Match score of 64.8%, a significant improvement over the Base Model (29.3%) and Base Model + R1 Prompt (55.0%). This shows RL training improved both reasoning depth and the consistency of the output format.
• MATH (Competition Math): On the math_verify metric, the trained model scored 27.4%, more than doubling the Base Model's score of 13.0%. An analysis of a sample problem shows the trained model developed a more sophisticated mathematical understanding (e.g., relating the number of digits to powers of the base) compared to the base model's procedural but flawed approach.
• BBH (Hard Reasoning Tasks): The trained model achieved an accuracy of 44.1%, outperforming the Base Model (37.5%). The largest gains were seen in tasks requiring logical deduction and tracking shuffled objects, indicating an improvement in core reasoning and state-tracking abilities.
• MMLU-Pro (Professional Knowledge): The trained model's score rose to 22.4% from the Base Model's 16.3%. The improvements were spread across diverse fields like Biology, Psychology, and Mathematics, suggesting the enhanced reasoning ability is domain-general.
• IFEval (Instruction Following): This was the only benchmark where the trained model did not improve. Its performance was slightly lower than the base model (60.1% vs. 60.3% Strict Accuracy). This may suggest that narrowly focusing on numerical reasoning might slightly degrade performance on tasks that require strict, non-mathematical instruction following, or that the training task did not cover this capability.

7. Conclusion & Reflections

• Conclusion Summary: The study successfully demonstrates that pure reinforcement learning, using a simple, rule-based reward signal from a numerical game, is a viable and effective method for enhancing the general reasoning capabilities of a language model. The model not only improved on its training domain but generalized its learned skills to a wide array of complex reasoning tasks. The emergence of human-like thought processes like "aha moments" is a promising sign, though the fact they don't always lead to correct answers points to areas for future improvement.

• Limitations & Future Work: The author identifies several limitations and directions for future research:

  • Response Format Violations: Even after training, the model sometimes failed to adhere to the specified output format.
  • Evaluation Harness Limitations: Standard evaluation tools sometimes misjudged responses that deviated from the expected format, suggesting a need for more robust evaluation methods.
  • Reward Function Imperfections: The simple binary (correct/incorrect) reward function could be improved. A more nuanced reward system (like a Process-based Reward Model) could provide partial credit for logically sound steps, even if the final answer is wrong.
  • GRPO Sample Size: The study used a small group size (G=2) for GRPO. Future work should investigate how varying this sample size affects training performance and stability.

• Personal Insights & Critique:

  • Impressive Generalization from a Narrow Task: It is remarkable that training on a single, highly structured numerical puzzle can lead to broad improvements in logical deduction, professional Q&A, and math word problems. This supports the idea that there is a core, transferable "reasoning skill" that models can learn.
  • "Aha Moments" as a Double-Edged Sword: The discovery that "aha moments" can be misleading is a critical insight. It suggests that while RL can foster more complex internal reasoning, it doesn't automatically equip the model with a reliable self-verification mechanism. Future research could focus on specifically rewarding correct self-correction.
  • Scalability Questions: The experiment was conducted on a relatively small 3B parameter model. It would be valuable to see if these findings hold for larger, more capable models, and whether the training becomes more or less efficient at scale.
  • The Power of Simplicity: This work is a testament to the power of a simple, well-designed training loop. By removing the need for expensive human data and relying on a clear, verifiable task, the author provides a blueprint for a more scalable and efficient way to enhance LLM reasoning.