---
library_name: transformers
tags:
  - math
  - reasoning
  - reinforcement-learning
  - qwen2
  - mathematics
  - chain-of-thought
license: apache-2.0
language:
  - en
  - zh
base_model: Qwen/Qwen2.5-Math-1.5B-Instruct
pipeline_tag: text-generation
---

# Nexus-1.5B

<p align="center">
  <img src="https://img.shields.io/badge/Base%20Model-Qwen2.5--Math--1.5B--Instruct-orange" />
  <img src="https://img.shields.io/badge/Parameters-1.54B-blue" />
  <img src="https://img.shields.io/badge/Method-LPRO-green" />
  <img src="https://img.shields.io/badge/MATH--500-80.2-red" />
  <img src="https://img.shields.io/badge/GSM8K-85.2-red" />
</p>

**Nexus-1.5B** is a 1.54-billion-parameter mathematical reasoning model developed by [Neuriton](https://www.facebook.com/neuriton), trained via **Length-Penalized Reward Optimization (LPRO)** — a novel reinforcement learning alignment method that improves both accuracy and response conciseness simultaneously.

Built on top of `Qwen2.5-Math-1.5B-Instruct`, Nexus-1.5B achieves **80.2 on MATH-500** and **85.2 on GSM8K** (CoT), surpassing its base model by **+4.4 points** on MATH-500 while reducing average response length by **14%**.

---

## What is LPRO?

Standard GRPO (Group Relative Policy Optimization) suffers from two key problems:

1. **Length bias** — short responses receive disproportionately large gradient signals, implicitly penalizing long correct derivations.
2. **Entropy collapse** — symmetric probability-ratio clipping causes the policy to converge to a narrow set of solution patterns, limiting further improvement.

**LPRO** fixes both with three targeted modifications:

| Component | What it does |
|---|---|
| **Asymmetric clipping** | Decouples the lower and upper clip bounds (`ε_low=0.20`, `ε_high=0.28`) to preserve policy entropy |
| **Token-level normalization** | Replaces per-response weight `1/G` with global weight `1/Σ|oᵢ|` to produce an unbiased gradient estimate |
| **Length-penalized advantage** | Adds a group-standardized length penalty: `Aᵢ = (rᵢ - μᵣ)/(σᵣ + ε) - λ·(Lᵢ - μ_L)/(σ_L + ε)` |

The final objective is:

$$\mathcal{J}_{\text{LPRO}}(\theta) = \mathbb{E}\left[\frac{1}{\sum_{i=1}^{G}|o_i|} \sum_{i=1}^{G}\sum_{t=1}^{|o_i|} \min\!\left(r_{i,t}(\theta)\,\hat{A}_{i,t},\ \text{clip}_{\text{asym}}(r_{i,t}(\theta))\,\hat{A}_{i,t}\right)\right]$$

---

## Model Details

| Property | Value |
|---|---|
| **Base model** | `Qwen/Qwen2.5-Math-1.5B-Instruct` |
| **Parameters** | 1.54B |
| **Architecture** | Transformer Decoder (28 layers, GQA, RoPE, SwiGLU, RMSNorm) |
| **Context length** | 8,192 tokens |
| **Vocabulary size** | 128,256 |
| **Training method** | LPRO (RL fine-tuning, no distillation) |
| **Training data** | 100 difficulty-filtered problems from MATH-500 |
| **Group size G** | 4 |
| **Length penalty λ** | 0.10 |
| **Learning rate** | 1e-6 |
| **PPO epochs/iter** | 4 |

---

## Benchmark Results

### Chain-of-Thought (CoT)

| Model | GSM8K | MATH-500 | MMLU-STEM | CMATH | GaoKao Cloze | GaoKao QA |
|---|---|---|---|---|---|---|
| Qwen2-Math-1.5B-Instruct | 84.2 | 69.4 | 54.9 | 79.6 | 59.7 | 50.7 |
| Qwen2.5-Math-1.5B-Instruct | 84.8 | 75.8 | 57.5 | 83.0 | 65.5 | 54.1 |
| **Nexus-1.5B** | **85.2** | **80.2** | **60.3** | **83.5** | **67.2** | **56.9** |

### Tool-Integrated Reasoning (TIR)

| Model | MATH-500 | Minerva Math | GaoKao 2023 EN | Olympiad Bench | College Math |
|---|---|---|---|---|---|
| Qwen2.5-Math-1.5B-Instruct | 80.0 | 34.0 | 68.0 | 49.0 | 54.0 |
| **Nexus-1.5B** | **84.0** | **40.0** | **74.0** | **56.0** | **57.0** |

### Ablation: Effect of Length Penalty (λ)

| λ | MATH-500 Acc. | Avg. Response Length |
|---|---|---|
| 0.0 (GRPO baseline) | 77.4 | 312 tokens |
| **0.1 (Nexus-1.5B)** | **80.2** | **268 tokens** |
| 0.3 (over-penalized) | 78.0 | 201 tokens |

> **Key insight:** At λ=0.1, accuracy and conciseness improve simultaneously. The length penalty acts as a de-noising regularizer — discouraging redundant steps rather than suppressing genuinely long derivations.

---

## How to Use

```python
from transformers import AutoModelForCausalLM, AutoTokenizer

model_name = "Dat1710/nexus-1.5b"

tokenizer = AutoTokenizer.from_pretrained(model_name)
model = AutoModelForCausalLM.from_pretrained(
    model_name,
    torch_dtype="auto",
    device_map="auto"
)

# Chain-of-Thought prompt
system_prompt = "Please reason step by step, and put your final answer within \\boxed{}."

messages = [
    {"role": "system", "content": system_prompt},
    {"role": "user", "content": "Find all functions f: ℝ⁺ → ℝ⁺ such that for each x ∈ ℝ⁺, there is exactly one y ∈ ℝ⁺ satisfying xf(y) + yf(x) ≤ 2."}
]

text = tokenizer.apply_chat_template(
    messages,
    tokenize=False,
    add_generation_prompt=True
)

model_inputs = tokenizer([text], return_tensors="pt").to(model.device)

generated_ids = model.generate(
    **model_inputs,
    max_new_tokens=2048,
    temperature=0.7,
    do_sample=True,
)

generated_ids = [
    output_ids[len(input_ids):]
    for input_ids, output_ids in zip(model_inputs.input_ids, generated_ids)
]

response = tokenizer.batch_decode(generated_ids, skip_special_tokens=True)[0]
print(response)
```

### Tool-Integrated Reasoning (TIR)

```python
system_prompt = (
    "Please integrate natural language reasoning with programs to solve the problem above, "
    "and put your final answer within \\boxed{}."
)
```

---

## Evaluation Prompt Format

**CoT (8-shot for GSM8K, 4-shot for MATH-500):**
```
<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
{problem}<|im_end|>
<|im_start|>assistant
```

**TIR (zero-shot):**
```
<|im_start|>system
Please integrate natural language reasoning with programs to solve the problem above,
and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
{problem}<|im_end|>
<|im_start|>assistant
```

---

## Training Details

### Data Curation

Training problems are sourced from **MATH-500** and filtered by difficulty using a learnable-zone criterion: a problem is retained if, among 8 sampled solutions from the base model, **between 2 and 5 are correct**. This yields 100 training problems that provide meaningful gradient signal — neither trivially easy nor intractably hard.

### Training Procedure

1. **Group sampling:** For each prompt, sample G=4 responses from the current policy.
2. **Reward computation:** Rule-based binary reward (correctness via symbolic answer matching) + small format bonus (α=0.1) for well-formed `\boxed{}` output.
3. **Advantage computation:** Compute length-penalized group z-score advantages.
4. **Policy update:** Maximize LPRO objective for 4 epochs per iteration.
5. **Iterate:** Set old policy ← new policy and repeat.

### Reward Function

$$r_i = \mathbf{1}[\hat{a}(o_i) = a^*] + 0.1 \cdot \mathbf{1}[\text{format}(o_i)]$$

where $\hat{a}(o_i)$ is the extracted answer from the last `\boxed{}` expression, verified via symbolic equivalence.

---

## Limitations

- **Scale:** Nexus-1.5B operates at 1.54B parameters. Hard olympiad problems (e.g., AIME) remain challenging for models at this scale.
- **Language:** Primarily optimized for English and Chinese mathematical text. Performance on other languages is not evaluated.
- **Domain:** Designed for mathematical reasoning. General language understanding or instruction-following tasks are outside the model's training distribution.
- **TIR dependency:** Tool-integrated reasoning requires a sandboxed Python interpreter at inference time.

---

## Citation

If you use Nexus-1.5B in your research, please cite:

```bibtex
@techreport{neuriton2026nexus,
  title     = {Nexus-1.5B: Length-Penalized Reward Optimization for Robust Mathematical Reasoning},
  author    = {Neuriton Team},
  institution = {Neuriton},
  year      = {2026},
  month     = {Summer},
  note      = {Technical Report}
}
```

---

## Acknowledgements

We thank the Qwen Team at Alibaba Group for open-sourcing the Qwen2.5-Math model family, and the authors of DAPO for the asymmetric clipping insight that is central to LPRO.

---

*Developed by [Neuriton](https://neuriton.ai) · Summer 2026*