GPassK: Are Your LLMs Capable of Stable Reasoning?

August 5, 2025 · View on GitHub

🚀 News

[2025.6] 🔥 We release the v202505 version of LiveMathBench, please refer to HuggingFace
[2025.2.28] 🔥 We provide Python Implementation and Evalution Framework using Lighteval.
[2025.2.13] 🔥 We release new results on LiveMathBench, MATH, and AIME24/25.
[2025.1.10] 🔥 We release a small-scale judge model LiveMath-Judge.
[2025.1.6] 🔥 LiveMathBench now can be accessed through huggingface, and you can now evaluate your LLMs on it using G-Pass@k in OpenCompass. We have addressed potential errors in LiveMathBench and inconsistencies in the sampling parameters. Please also refer to our updated version of the Paper for further details.
[2024.12.18] 🎉 We release the ArXiv Paper of G-Pass@k.

G-Pass@k is a novel evaluation metric that provides a continuous assessment of model performance across multiple sampling attempts, quantifying both the model’s peak performance potential and its stability. In addition, it comes with LiveMathBench, a dynamic benchmark comprising challenging, contemporary mathematical problems designed to minimize data leakage risks during evaluation. In order to track the latest performance and stability of LLMs, we will continue updating the benchmark with new comptition level mathmatical problems and provide the latest results of the models on the benchmark with G-Pass@k.

🌲 Definition of G-Pass@k

$\text{G-Pass@}k = \mathbb{E}_{\text{Questions}} \left[ \frac{{c \choose k}}{{n \choose k}} \right]$

where $n$ represents the total number of generations per question, and $c$ denotes the number of generations resulting in correct solutions.

$\text{G-Pass@}k_{\tau} = E_{\text{Questions}} \left[ \sum_{j = \lceil \tau \cdot k \rceil}^{c} \frac{\binom{c}{j} \cdot \binom{n - c}{k - j}}{\binom{n}{k}} \right]$

where $\lceil \tau \cdot k \rceil$ denotes the smallest integer greater than or equal to $\tau \cdot k$ .

$\text{mG-Pass@}k_{\tau} = 2\int_{0.5}^{1.0} \text{G-Pass@}k_{\tau} d \tau = \frac{2}{k} \sum_{i= \lceil 0.5 \cdot k \rceil + 1}^{k} \text{G-Pass@}k_{\frac{i}{k}}$

Intuitively, $\text{mG-Pass@}k$ provides an interpolated estimate of the area under the curve of $\text{mG-Pass@}k_{[0.5:1.0]}$ , serving as a comprehensive metric that integrates all $\text{G-Pass@}k_{\tau}$ values where $\tau \in [0.5, 1.0]$ .

📚 Main Results

⚽: General Models
🏐: Math Models
🏀: o1-like Models

LiveMathBench-202412

LLMs	Greedy	G-Pass@16_0.5	G-Pass@16_0.75	G-Pass@16_1.0	mG-Pass@16
Llama-3.1-8B-Instruct ⚽	24.0	18.2	11.3	4.5	10.4
Qwen2.5-7B-Instruct ⚽	37.0	36.5	27.2	16.0	25.8
Llama-3.3-70B-Instruct ⚽	40.3	36.2	28.9	19.1	27.5
InternLM3-8B-Instruct ⚽	44.5	43.0	35.4	23.0	33.6
Claude-3.5-Sonnet ⚽	46.7	44.1	36.2	26.6	35.3
Mistral-Large-Instruct-2411 ⚽	41.6	39.4	37.1	32.9	36.4
Qwen2.5-Math-7B-Instruct 🏐	68.4	44.1	38.3	28.1	36.6
Qwen2.5-32B-Instruct ⚽	50.8	47.3	39.6	29.0	37.8
Qwen2.5-Max ⚽	52.9	52.7	44.3	31.1	42.2
Qwen2.5-Math-72B-Instruct 🏐	57.6	52.7	45.4	27.9	42.3
DeepSeek-Distill-Llama-8B 🏀	58.4	67.8	56.8	31.9	52.2
QwQ-32B-Preview 🏀	72.7	74.9	65.8	40.1	61.2
DeepSeek-Distill-Qwen-7B 🏀	65.6	73.0	66.4	48.4	63.1
OpenAI-o1-mini 🏀	74.1	76.3	67.3	48.3	64.8
DeepSeek-Distill-Qwen-32B 🏀	67.7	81.2	72.3	54.5	69.7
DeepSeek-Distill-Llama-70B 🏀	74.8	80.8	73.0	53.0	69.7
OpenAI-o3-mini 🏀	84.7	85.7	78.8	65.3	76.8
DeepSeek-R1 🏀	81.1	83.6	79.1	69.5	77.6

LiveMathBench-Hard-202412

LLMs	Greedy	G-Pass@16_0.5	G-Pass@16_0.75	G-Pass@16_1.0	mG-Pass@16
Llama-3.1-8B-Instruct ⚽	2.2	0.8	0.0	0.0	0.0
Qwen2.5-7B-Instruct ⚽	13.3	6.2	3.2	2.2	3.3
Qwen2.5-Math-7B-Instruct 🏐	15.6	8.2	3.3	2.2	3.8
QwQ-32B-Preview 🏀	15.6	5.9	4.4	2.4	4.0
Llama-3.3-70B-Instruct ⚽	4.4	7.8	4.8	2.4	4.6
DeepSeek-Distill-Llama-8B 🏀	8.9	16.1	5.6	2.4	6.2
Llama-3.1-70B-Instruct ⚽	4.4	12.3	7.4	2.7	6.9
InternLM3-8B-Instruct ⚽	11.1	10.7	8.2	2.7	7.0
Qwen2.5-Math-72B-Instruct 🏐	11.1	11.8	7.9	5.9	7.9
DeepSeek-Distill-Qwen-7B 🏀	17.8	13.9	8.8	3.3	8.1
OpenAI-o1-mini 🏀	18.4	21.0	10.1	0.5	8.5
Qwen2.5-32B-Instruct ⚽	13.3	14.1	10.5	3.5	9.1
Qwen2.5-72B-Instruct ⚽	17.8	15.3	11.3	5.4	10.5
DeepSeek-Distill-Qwen-32B 🏀	22.2	29.9	16.9	3.3	15.1
DeepSeek-Distill-Llama-70B 🏀	35.6	33.1	19.0	5.8	17.3
OpenAI-o3-mini 🏀	43.3	47.4	32.5	7.7	28.6
DeepSeek-R1 🏀	42.2	46.6	33.6	9.8	29.6

MATH500-L5

LLMs	Greedy	G-Pass@16_0.5	G-Pass@16_0.75	G-Pass@16_1.0	mG-Pass@16
Llama-3.1-8B-Instruct ⚽	26.1	17.8	10.7	3.5	9.7
Llama-3.1-70B-Instruct ⚽	39.6	41.8	32.1	16.1	29.3
InternLM3-8B-Instruct ⚽	51.5	49.9	40.3	26.9	38.3
Qwen2.5-7B-Instruct ⚽	56.0	54.9	43.3	28.0	41.5
Llama-3.3-70B-Instruct ⚽	54.5	55.4	49.5	35.0	47.3
Qwen2.5-72B-Instruct ⚽	63.4	62.5	54.4	44.9	53.1
Qwen2.5-Max ⚽	63.4	65.8	57.3	38.9	54.5
Qwen2.5-32B-Instruct ⚽	64.2	66.6	59.4	41.0	55.6
Qwen2.5-Math-72B-Instruct 🏐	71.6	64.9	59.4	46.0	57.4
Qwen2.5-Math-7B-Instruct 🏐	65.7	65.0	62.2	57.6	61.5
DeepSeek-Distill-Llama-8B 🏀	65.7	79.5	70.0	39.5	64.5
QwQ-32B-Preview 🏀	82.8	87.2	78.8	57.4	75.6
DeepSeek-Distill-Qwen-7B 🏀	78.4	87.9	80.5	62.6	77.6
DeepSeek-Distill-Qwen-32B 🏀	83.6	89.9	83.8	70.4	81.9
DeepSeek-Distill-Llama-70B 🏀	87.3	89.6	85.5	66.8	81.9

AIME2024-45

LLMs	Greedy	G-Pass@16_0.5	G-Pass@16_0.75	G-Pass@16_1.0	mG-Pass@16
Llama-3.1-8B-Instruct ⚽	4.4	2.2	1.6	0.0	1.2
Qwen2.5-Math-7B-Instruct 🏐	11.1	4.6	2.6	2.2	3.7
Qwen2.5-32B-Instruct ⚽	11.1	7.1	3.4	2.2	3.7
InternLM3-8B-Instruct ⚽	11.1	7.2	4.3	1.0	3.7
Qwen2.5-7B-Instruct ⚽	11.1	8.9	8.1	4.7	7.5
Llama-3.1-70B-Instruct ⚽	15.6	15.0	8.1	3.0	8.0
Qwen2.5-Max ⚽	22.2	15.5	9.9	5.3	9.8
Qwen2.5-72B-Instruct ⚽	13.3	13.7	12.9	7.5	11.7
Qwen2.5-Math-72B-Instruct 🏐	20.0	18.7	16.2	6.7	14.1
Llama-3.3-70B-Instruct ⚽	22.2	25.3	18.2	6.9	16.4
QwQ-32B-Preview 🏀	44.4	41.0	28.6	8.1	24.7
DeepSeek-Distill-Llama-8B 🏀	44.4	53.9	30.4	9.0	28.0
DeepSeek-Distill-Qwen-7B 🏀	44.4	56.3	35.4	17.5	33.8
OpenAI-o1-mini 🏀	60.3	62.2	53.3	15.6	43.1
DeepSeek-Distill-Llama-70B 🏀	62.2	72.9	63.4	32.2	57.6
DeepSeek-Distill-Qwen-32B 🏀	62.2	77.0	66.5	31.3	59.3

AIME2025

LLMs	Greedy	G-Pass@16_0.5	G-Pass@16_0.75	G-Pass@16_1.0	mG-Pass@16
Llama-3.1-8B-Instruct ⚽	0.0	0.0	0.0	0.0	0.0
Llama-3.1-70B-Instruct ⚽	6.7	4.6	0.2	0.0	0.7
InternLM3-8B-Instruct ⚽	13.3	6.7	0.1	0.0	0.8
Qwen2.5-32B-Instruct ⚽	20.0	11.5	0.2	0.0	1.4
Qwen2.5-7B-Instruct ⚽	6.7	9.7	6.2	0.2	4.7
Qwen2.5-72B-Instruct ⚽	20.0	12.2	5.8	0.1	4.9
Llama-3.3-70B-Instruct ⚽	6.7	6.7	6.6	0.5	5.0
Qwen2.5-Math-7B-Instruct 🏐	20.0	8.7	6.7	6.7	6.8
Qwen2.5-Max ⚽	13.3	11.9	6.8	2.9	6.8
Qwen2.5-Math-72B-Instruct 🏐	13.3	13.3	13.3	13.3	13.3
Gemini-2.0-Flash-Exp ⚽	26.7	26.5	21.5	14.0	21.2
QwQ-32B-Preview 🏀	26.7	34.5	32.4	15.6	28.1
OpenAI-o1-mini 🏀	46.7	39.9	32.5	14.0	28.4
DeepSeek-Distill-Llama-8B 🏀	40.0	40.4	21.2	7.9	21.0
DeepSeek-Distill-Qwen-7B 🏀	46.7	46.6	38.3	22.7	36.1
DeepSeek-Distill-Llama-70B 🏀	46.7	52.5	38.6	26.8	37.4
DeepSeek-R1 🏀	66.7	52.6	46.8	24.3	42.5
OpenAI-o3-mini 🏀	53.3	59.0	46.5	29.4	43.6
DeepSeek-Distill-Qwen-32B 🏀	46.7	59.7	50.2	29.5	47.3

🖋Use G-Pass@k in Your Project

You can use the following class in your work, you need to define the parameters of G-Pass@k, such as k, n, and thresholds. Additionally, you must define a function to score each sample pair, which should return a binary (0 or 1) label for each pair of prediction and corresponding gold. The compute method will then return a dictionary containing the metrics for each gold standard value and its corresponding predictions. You can aggregate these metrics across your dataset as needed.

class GPassAtK:
    def __init__(
        self,
        k: Union[int, List[int]],
        n: int = None,
        thresholds: List[float] = [0.0, 0.25, 0.5, 0.75, 1.0],
        sample_scoring_function: Union[Callable[[str, str], float], str] = None,
    ):
        """Computing G-Pass@k from http://arxiv.org/abs/2412.13147

        Args:
            k (int, list): The number of successful attempts to be considered.
            n (int): Number of samples to generate.
            thresholds (list): Thresholds to control successful attempts in k generate.
            sample_scoring_function (callable or str, optional): Function to use to score each sample.
                Either pass the full function (should take a string prediction and a string gold, and return a score between 0 and 1)
                a string (any of `prefix`, `suffix` or `full`) to define the type of exact match that you want, or nothing to defaults to "full".
                    `prefix` checks if the prediction starts with the gold,
                    `suffix` if the prediction ends with the gold,
                    `full` if the prediction and gold are equal
        """
        self.k = as_list(k)
        self.n = n
        self.thresholds = thresholds

        # Managed the logic of the per prediction of sample scoring
        if callable(sample_scoring_function):
            self.score_sample = sample_scoring_function
            self.type_exact_match = None
        else:
            if isinstance(sample_scoring_function, str):
                if sample_scoring_function not in ["prefix", "suffix", "full"]:
                    raise ValueError(
                        f"type_exact_match (used in parametrized_exact_match) must be one of prefix, suffix, or full. Was {sample_scoring_function} instead."
                    )
                self.type_exact_match = sample_scoring_function
            else:
                self.type_exact_match = "full"
            self.score_sample = self.default_sample_scoring

    def compute(self, predictions: List[str], gold: str, **kwargs) -> dict[str, float]:
        """Computes the metric over a list of golds and predictions for one single item with possibly many samples.
        It applies normalisation (if needed) to model prediction and gold, computes their per prediction score,
        then aggregates the scores over the samples using a pass@k.

        Args:
            golds (list[str]): Reference targets
            predictions (list[str]): k predicted strings

        Returns:
            float: Aggregated score over the current sample's items.
        """
        if len(golds) > 1:
            raise Exception("Cannot compute G-Pass@k with several golds")

        if self.n is None:
            self.n = len(predictions)
            logger.warning(
                "n undefined in the G-Pass@k. We assume it's the same as the sample's number of predictions."
            )
        elif len(predictions) < self.n:
            logger.warning(f"Number of predictions is less than {self.n} for G-Pass@k.")

        all_scores = []
        for pred in predictions[: self.n]:
            all_scores.append(self.score_sample(pred, gold))

        return self.g_pass_at_k(all_scores)

    def default_sample_scoring(self, pred: str, gold: str) -> int:
        if self.type_exact_match == "prefix":
            return 1 if pred.startswith(gold) else 0
        if self.type_exact_match == "suffix":
            return 1 if pred.endswith(gold) else 0
        return 1 if gold == pred else 0

    def g_pass_at_k(self, all_scores: list[int]) -> float:
        """Computation of G-Pass@k details from http://arxiv.org/abs/2412.13147"""
        c: int = sum(all_scores)
        n: int = self.n
        ks: int = self.k
        thresholds: List[float] = self.thresholds

        def _compute_g_pass_at_k(n, c, k, m):
            if m > min(c, k) or k > n or c < 0 or n <= 0 or m < 0:
                return 0.0
            return hypergeom.sf(m - 1, n, c, k)

        def compute_g_pass_at_k(n, c, k, t):
            m = max(int(np.ceil(k * t)), 1)
            return _compute_g_pass_at_k(n, c, k, m)

        def compute_mg_pass_at_k(n, c, k):
            low, high = int(np.ceil(k * 0.5)), k

            mg_pass_at_k = 0.0
            for i in range(low + 1, high + 1):
                mg_pass_at_k += _compute_g_pass_at_k(n, c, k, i)
            mg_pass_at_k = 2 * mg_pass_at_k / k

            return mg_pass_at_k

        metrics = {}
        for k in ks:
            for t in thresholds:
                metrics[f"G-Pass@{k}_{t}"] = compute_g_pass_at_k(n, c, k, t)
            metrics[f"mG-Pass@{k}"] = compute_mg_pass_at_k(n, c, k)

        return metrics

    @property
    def all_metrics(self):
        ks: int = self.k
        thresholds: List[float] = self.thresholds

        metrics = []
        for k in ks:
            for t in thresholds:
                metrics.append(f"G-Pass@{k}_{t}")
            metrics.append(f"mG-Pass@{k}")

        return metrics

🖋Use G-Pass@k in OpenCompass

OpenCompass is a toolkit for evaluating the performance of large language models (LLMs). To use GPassK in OpenCompass, you can follow the steps below:

1. Prepare Environment

Follow these steps to ensure your environment is ready:

# Clone the main repository
git clone https://github.com/open-compass/GPassK.git
cd GPassK/opencompass

# Create and activate a conda environment with specific Python and PyTorch versions
conda create -n livemathbench-eval python=3.10 pytorch torchvision torchaudio pytorch-cuda -c nvidia -c pytorch -y
conda activate livemathbench-eval

# Install additional required packages
pip install loguru

# Clone and install OpenCompass for extended functionality
git clone https://github.com/open-compass/opencompass.git
cd opencompass
pip install -e .

2. Prepare Dataset

LiveMathBench dataset can be obtained from HuggingFace. First, you should be granted to access the dataset from the following link: huggingface. Then, refer to security-tokens to set up your HF tokens.

3. Deploy Judge Models

We leverage Qwen2.5-72B-Instruct as the judge model for judging the correctness of generated answers. We recommend to deploy services using deployment tools such as vllm or lmdeploy for invocation by different evaluation tasks.

Below is an example configuration for deploying the judge model using lmdeploy:

lmdeploy serve api_server Qwen/Qwen2.5-72B-Instruct --server-port 8000 \
    --tp 4 \ # at least 4 A100 or equivalent GPUs are required
    --cache-max-entry-count 0.9 \
    --log-level INFO

After setting up the judge model, define the URLs in the eval_urls and eval_model_name within opencompass_config_templates/*.py. Adjust other parameters such as k， temperatures, llm_infos, and other params according to your needs.

Note

Note that omitting eval_urls will default to an internal rule-based judge, which might only apply to datasets with numerical answers

Tip

💡Now you can use the LiveMath-Judge for judging, which greatly reduces deploy and inference costs.

4. Evaluation

To begin the evaluation, first generate the necessary configuration files by running the following script:

cd opencompass
python dump_opencompass_configs.py --config_template_file {config_templates/nono1.py|config_templates/o1.py|config_templates/close.py}

Upon execution, verify the generated configuration files located in `opencompass_configs/:

.
├── deepseek-math-7b-rl_t0-3_p0-8_k50_rp1-0_rs42_l8192@LiveMathBench-v202412-k4_8_16-r3.py
├── deepseek-math-7b-rl_t0-5_p0-8_k50_rp1-0_rs42_l8192@LiveMathBench-v202412-k4_8_16-r3.py
├── deepseek-math-7b-rl_t0-7_p0-8_k50_rp1-0_rs42_l8192@LiveMathBench-v202412-k4_8_16-r3.py
├── deepseek-math-7b-rl_t1-0_p0-8_k50_rp1-0_rs42_l8192@LiveMathBench-v202412-k4_8_16-r3.py

These files follow a naming convention that reflects the model settings and dataset used:

[MODEL_ABBR]_t[TEMPERATUE]_p[TOP_P]_k[TOP_K]_rp[REPETITION_PENALTY]_l[MAX_OUT_LEN]@[DATASET_ABBR]_k[LIST_OF_K]_r[REPLICATION].py

With the configurations prepared, initiate the evaluation process with the commands below:

cd GPassK
conda activate livemathbench-eval
python opencompass/run.py {path/to/config_file} \
      -w ./opencompass_outputs/ \
      --dump-eval-details \

Refer to the OpenCompass documentation for additional arguments that may enhance your evaluation experience.

🖋Use G-Pass@k in Ligheval

Lighteval is your all-in-one toolkit for evaluating LLMs across multiple backends—whether it's transformers, tgi, vllm, or nanotron—with ease.

1. Prepare Environment

Follow these steps to ensure your environment is ready:

# Clone the main repository
git clone https://github.com/open-compass/GPassK.git
cd GPassK/lighteval

# Create and activate a conda environment with specific Python and PyTorch versions
conda create -n lighteval-eval python=3.10 pytorch torchvision torchaudio pytorch-cuda -c nvidia -c pytorch -y
conda activate lighteval-eval

# Clone and install OpenCompass for extended functionality
git clone https://github.com/huggingface/lighteval
cd lighteval
pip install -e .

# Install additional required packages
pip install opencompass vllm

2. Prepare Dataset

3. Deploy Judge Models

Below is an example configuration for deploying the judge model using lmdeploy:

lmdeploy serve api_server Qwen/Qwen2.5-72B-Instruct --server-port 8000 \
    --tp 4 \ # at least 4 A100 or equivalent GPUs are required
    --cache-max-entry-count 0.9 \
    --log-level INFO

After setting up the judge model, define the URLs in the eval_urls and eval_model within lighteval/configs/eval_cfg.yaml. Adjust other parameters such as k， n, model_name_or_path, and other params according to your needs.

4. Evaluation

To begin the evaluation, running the following script:

cd lighteval
python lighteval_run.py

📄 Citation and Tech Report

If you use G-Pass@k in your research, please cite the following paper:

@article{liu2024your,
  title={Are Your LLMs Capable of Stable Reasoning?},
  author={Liu, Junnan and Liu, Hongwei and Xiao, Linchen and Wang, Ziyi and Liu, Kuikun and Gao, Songyang and Zhang, Wenwei and Zhang, Songyang and Chen, Kai},
  journal={arXiv preprint arXiv:2412.13147},
  year={2024}
}