EurusPRM-Stage2

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Introduction

EurusPRM-Stage2 is trained using Implicit PRM, which obtains free process rewards at no additional cost but just needs to simply train an ORM on the cheaper response-level labels. During inference, implicit process rewards are obtained by forward passing and calculating the log-likelihood ratio on each step.

The key ingredient of Implicit PRM is the reward representation, as demonstrated below:

✨

Proposition: Consider an ORM where the reward is parameterized by the log-likelihood ratio of two causal LMs, i.e.

$$r_{\varphi }(\mathbf{y}):=\beta \log \frac{{\pi }_{\varphi }(\mathbf{y})}{{\pi }_{\text{ref}}(\mathbf{y})}\mathrm{.}r\_\backslash phi(\backslash mathbf\{y\})\; :=\; \backslash beta\; \backslash log\; \backslash frac\{\backslash pi\_\backslash phi(\backslash mathbf\{y\})\}\{\backslash pi\_\backslash text\{ref\}(\backslash mathbf\{y\})\}.$$

Define

is the exponential average of $r_{\theta }r\_\backslash theta$ at step $tt$.

Hence, $q_{\theta }^{t}q\_\backslash theta^t$represents an exact expectation of outcome reward $r_{\theta }r\_\backslash theta$ at step $tt$, i.e., the Q value.

The proposition indicates that when modeling

$$r_{\varphi }(\mathbf{y}):=\beta \log \frac{{\pi }_{\varphi }(\mathbf{y})}{{\pi }_{\text{ref}}(\mathbf{y})}r\_\backslash phi(\backslash mathbf\{y\})\; :=\; \backslash beta\; \backslash log\; \backslash frac\{\backslash pi\_\backslash phi(\backslash mathbf\{y\})\}\{\backslash pi\_\backslash text\{ref\}(\backslash mathbf\{y\})\}$$

to train an ORM with the standard pipeline, where $\beta \backslash beta$ is a hyperparameter, $\varphi \backslash phi$ can implicitly learn a Q function. Hence, process reward $r_{\varphi }^{t}r\_\backslash phi^t$ can be obtained by:

Therefore, we can indeed obtain PRMs simply by collecting response-level data and training an ORM, without any burden of annotating step labels.

The proposition is agnostic to specific choices of the training objective of ORMs. It can be instantiated with different objectives as vanilla ORM training, with the only difference being substituting the $r_{\varphi }\left(\mathbf{y}\right)r\_\backslash phi\; \backslash left(\; \backslash mathbf\{y\}\; \backslash right)$ with $\beta \log \frac{{\pi }_{\varphi }(\mathbf{y})}{{\pi }_{\text{ref}}(\mathbf{y})}\backslash beta\; \backslash log\; \backslash frac\{\backslash pi\_\backslash phi(\backslash mathbf\{y\})\}\{\backslash pi\_\backslash text\{ref\}(\backslash mathbf\{y\})\}$.

For example, DPO already meets our assumption and serves as a strong variant, while in this work, we instantiate our implicit PRM with cross entropy (CE) loss due to memory efficiency:

$${\mathcal{L}}_{CE}=l\cdot \log \sigma (\beta \log \frac{{\pi }_{\varphi }(\mathbf{y})}{{\pi }_{\text{ref}}(\mathbf{y})})+(1-l)\cdot \log [1-\sigma (\beta \log \frac{{\pi }_{\varphi }(\mathbf{y})}{{\pi }_{\text{ref}}(\mathbf{y})})]\backslash small\; \backslash mathcal\{L\}\_\{CE\}\; =\; l\; \backslash cdot\; \backslash log\; \backslash sigma\; \backslash left(\; \backslash beta\; \backslash log\; \backslash frac\{\backslash pi\_\backslash phi(\backslash mathbf\{y\})\}\{\backslash pi\_\backslash text\{ref\}(\backslash mathbf\{y\})\}\; \backslash right)\; +\; (1\; -\; l)\; \backslash cdot\; \backslash log\; \backslash left[\; 1\; -\; \backslash sigma\; \backslash left(\; \backslash beta\; \backslash log\; \backslash frac\{\backslash pi\_\backslash phi(\backslash mathbf\{y\})\}\{\backslash pi\_\backslash text\{ref\}(\backslash mathbf\{y\})\}\; \backslash right)\; \backslash right]$$

We started the second-stage training on top of EurusPRM-Stage1 with fine-grained step-level labels. To obtain step-level labels, we employed Llama-3.1-70B-Inst and Qwen2.5-72B-Inst to insert nuance errors into correct solutions. We also mixed response-level data in this stage. The model was continually trained with $L_{CE}L\_\{CE\}$ with a learning rate of 5e-7 and a batch-size of 64.

Usage

We show an example leveraging EurusPRM-Stage2 below:

import torch
from transformers import AutoTokenizer,AutoModelForCausalLM
coef=0.001
d = {'query':'Convert the point $(0,3)$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$',
     'answer':[
"Step 1: To convert the point (0,3) from rectangular coordinates to polar coordinates, we need to find the radius (r) and the angle theta (\u03b8).",
            "Step 1: Find the radius (r). The radius is the distance from the origin (0,0) to the point (0,3). Since the x-coordinate is 0, the distance is simply the absolute value of the y-coordinate. So, r = |3| = 3.",
            "Step 2: Find the angle theta (\u03b8). The angle theta is measured counterclockwise from the positive x-axis. Since the point (0,3) lies on the positive y-axis, the angle theta is 90 degrees or \u03c0\/2 radians.",
            "Step 3: Write the polar coordinates. The polar coordinates are (r, \u03b8), where r > 0 and 0 \u2264 \u03b8 < 2\u03c0. In this case, r = 3 and \u03b8 = \u03c0\/2.\n\nTherefore, the polar coordinates of the point (0,3) are (3, \u03c0\/2).\n\n\n\\boxed{(3,\\frac{\\pi}{2})}"
     ]
     }
model = AutoModelForCausalLM.from_pretrained('PRIME-RL/EurusPRM-Stage2')
tokenizer = AutoTokenizer.from_pretrained('PRIME-RL/EurusPRM-Stage2')
ref_model = AutoModelForCausalLM.from_pretrained('Qwen/Qwen2.5-Math-7B-Instruct')
input_ids = tokenizer.apply_chat_template([
    {"role": "user", "content": d["query"]},
    {"role": "assistant", "content": "\n\n".join(d["answer"])},
], tokenize=True, add_generation_prompt=False,return_tensors='pt')
attention_mask = input_ids!=tokenizer.pad_token_id
step_last_tokens = []
for step_num in range(0, len(d["answer"])+1):
    conv = tokenizer.apply_chat_template([
        {"role":"user", "content":d["query"]},
        {"role":"assistant", "content":"\n\n".join(d["answer"][:step_num])},
    ], tokenize=False, add_generation_prompt=False)
    conv = conv.strip()
    if step_num!=0 and step_num!=len(d['answer']):
        conv+='\n\n'
    currect_ids = tokenizer.encode(conv,add_special_tokens=False)
    step_last_tokens.append(len(currect_ids) - 2)


inputs = {'input_ids':input_ids,'attention_mask':attention_mask,'labels':input_ids}
label_mask = torch.tensor([[0]*step_last_tokens[0]+[1]*(input_ids.shape[-1]-step_last_tokens[0])])
step_last_tokens = torch.tensor([step_last_tokens])

def get_logps(model,inputs):
    logits = model(input_ids=inputs['input_ids'], attention_mask=inputs['attention_mask']).logits
    labels = inputs['labels'][:, 1:].clone().long()
    logits = logits[:, :-1, :]
    labels[labels == -100] = 0
    per_token_logps = torch.gather(logits.log_softmax(-1), dim=2, index=labels.unsqueeze(2)).squeeze(2)
    return per_token_logps

with torch.no_grad():
    per_token_logps = get_logps(model, inputs)
    ref_per_token_logps = get_logps(ref_model,inputs)

raw_reward = per_token_logps - ref_per_token_logps
beta_reward = coef * raw_reward * label_mask[:,1:]
beta_reward = beta_reward.cumsum(-1)
beta_reward = beta_reward.gather(dim=-1, index=step_last_tokens[:,1:])
print(beta_reward)

Evaluation

Evaluation Base Model

We adopt Eurus-2-7B-SFT, Qwen2.5-7B-Instruct and Llama-3.1-70B-Instruct as generation models to evaluate the performance of our implicit PRM. For all models, we set the sampling temperature as 0.5, p of the top-p sampling as 1.

Best-of-N Sampling

We use Best-of-64 as our evaluation metric. The weighting methods are different for several PRMs below.

• For Skywork-o1-Open-PRM-Qwen-2.5-7B, we use simple average reward across all steps.
• For EurusPRM-Stage 1, we use the minimum reward across all steps.
• For EurusPRM-Stage 2, we use the accumulative rewards.

Eurus-2-7B-SFT

Method	Reward Model	MATH	AMC	AIME_2024	OlympiadBench	Minerva Math	Avg
Greedy Pass @ 1	N/A	65.1	30.1	3.3	29.8	32.7	32.2
Majority Voting @ 64	N/A	65.6	53.0	13.3	39.1	22.4	38.7
Best-of-64	Skywork-o1-Open-PRM-Qwen-2.5-7B	47.2	45.8	10.0	32.3	16.2	30.3
	EurusPRM-Stage 1	44.6	41.0	6.7	32.9	17.3	28.5
	EurusPRM-Stage 2	47.2	43.4	13.3	33.8	19.2	31.4
Weighted Best-of-64	Skywork-o1-Open-PRM-Qwen-2.5-7B	64.6	55.4	13.3	41.3	23.2	39.6
	EurusPRM-Stage 1	66.0	54.2	13.3	39.6	29.0	40.4
	EurusPRM-Stage 2	66.0	54.2	13.3	39.7	29.0	40.4

Llama-3.1-70B-Instruct

Method	Reward Model	MATH	AMC	AIME 2024	OlympiadBench	Minerva Math	Avg
Greedy Pass @ 1	N/A	64.6	30.1	16.7	31.9	35.3	35.7
Majority Voting @ 64	N/A	80.2	53.0	26.7	40.4	38.6	47.8
Best-of-N @ 64	Skywork-o1-Open-PRM-Qwen-2.5-7B	77.8	56.6	23.3	39.0	31.6	45.7
	EurusPRM-Stage 1	77.8	44.6	26.7	35.3	41.5	45.2
	EurusPRM-Stage 2	80.6	59.0	20.0	37.6	44.9	48.4
Weighted Best-of-64	Skywork-o1-Open-PRM-Qwen-2.5-7B	81.2	56.6	23.3	42.4	38.2	48.3
	EurusPRM-Stage 1	80.4	53.0	26.7	40.9	46.7	49.5
	EurusPRM-Stage 2	80.4	53.0	26.7	41.0	46.3	49.5

Qwen2.5-7B-Instruct

Method	Reward Model	MATH	AMC	AIME 2024	OlympiadBench	Minerva Math	Avg
Greedy Pass @ 1	N/A	73.3	47.0	13.3	39.4	35.3	41.7
Majority Voting @ 64	N/A	82.0	53.0	16.7	43.0	36.4	46.2
Best-of-N @ 64	Skywork-o1-Open-PRM-Qwen-2.5-7B	85.2	60.2	20.0	44.7	32.7	48.6
	EurusPRM-Stage 1	81.8	47.0	16.7	40.1	41.5	45.4
	EurusPRM-Stage 2	86.0	59.0	16.7	41.4	41.5	48.9
Weighted Best-of-64	Skywork-o1-Open-PRM-Qwen-2.5-7B	83.6	55.4	13.3	43.7	36.8	46.6
	EurusPRM-Stage 1	82.6	53.0	16.7	42.7	45.2	48.0
	EurusPRM-Stage 2	84.8	53.0	16.7	43.2	45.6	48.7

Citation

@misc{cui2024process,
  title={Process Reinforcement through Implicit Rewards},
  author={Ganqu Cui and Lifan Yuan and Zefan Wang and Hanbin Wang and Wendi Li and Bingxiang He and Yuchen Fan and Tianyu Yu and Qixin Xu and Weize Chen and Jiarui Yuan and Huayu Chen and Kaiyan Zhang and Xingtai Lv and Shuo Wang and Yuan Yao and Hao Peng and Yu Cheng and Zhiyuan Liu and Maosong Sun and Bowen Zhou and Ning Ding},
  year={2025}
}

@article{yuan2024implicitprm,
  title={Free Process Rewards without Process Labels},
  author={Lifan Yuan and Wendi Li and Huayu Chen and Ganqu Cui and Ning Ding and Kaiyan Zhang and Bowen Zhou and Zhiyuan Liu and Hao Peng},
  journal={arXiv preprint arXiv:2412.01981},
  year={2024}
}