Aug 8, 2024

By

ModelBox Team

In the ever-evolving world of AI, mathematical reasoning remains a critical challenge. Today, we’re excited to announcr that ModelBox supports Qwen2-Math inference, the newest member of the Qwen2 open-source family, designed specifically to tackle advanced mathematical problems with unparalleled precision.

## Qwen2-Math-72B-Instruct

### Benchmark Performance

Qwen2-Math-72B-Instruct stands out as our flagship model, outperforming leading AI models like GPT-4o, Claude-3.5-Sonnet, Gemini-1.5-Pro, and Llama-3.1-405B in various math benchmarks, including Math, GSM8K, and MMLU-STEM. Its accuracy and problem-solving abilities are unmatched, particularly in complex, multi-step logical reasoning tasks.

### Foundation and Development

Built on the robust Qwen2-1.5B/7B/72B base, Qwen2-Math underwent rigorous pre-training using a specially curated math-specific corpus. This dataset includes high-quality mathematical texts, books, code, exam questions, and synthetically generated math data, ensuring the model’s exceptional performance in both English and Chinese math benchmarks.

### Instruction Fine-Tuning

We’ve fine-tuned Qwen2-Math-72B using a dedicated reward model, further optimizing it through dense feedback signals and binary correctness indicators. This approach has led to superior results in challenging exams like OlympiadBench, CollegeMath, AIME2024, and AMC2023, showcasing its capability to handle difficult and diverse mathematical problems.

## Case Studies

Qwen2-Math's prowess extends beyond benchmarks. It has successfully solved various competition problems, including those from prestigious math Olympiads like the IMO and Zhautykov Olympiad. These real-world applications highlight its potential to contribute significantly to fields requiring advanced mathematical reasoning.

Take IMO 2022 for example.

Problem

Let 𝑅+R+ denote the set of positive real numbers. Find all functions 𝑓:𝑅+→𝑅+*f*:R+→R+ such that for each 𝑥∈𝑅+*x*∈R+, there is exactly one 𝑦∈𝑅+*y*∈R+ satisfying

𝑥𝑓(𝑦)+𝑦𝑓(𝑥)≤2*xf*(*y*)+*yf*(*x*)≤2

Qwen2-Math-72B-Instruct Solution

To find all functions 𝑓:𝑅+→𝑅+*f*:R+→R+ such that for each 𝑥∈𝑅+*x*∈R+, there is exactly one 𝑦∈𝑅+*y*∈R+ satisfying

𝑥𝑓(𝑦)+𝑦𝑓(𝑥)≤2,*xf*(*y*)+*yf*(*x*)≤2,

we will proceed step by step.

Step 1: Analyze the given inequality Given 𝑥∈𝑅+*x*∈R+, there exists a unique 𝑦∈𝑅+*y*∈R+ such that

𝑥𝑓(𝑦)+𝑦𝑓(𝑥)≤2.*xf*(*y*)+*yf*(*x*)≤2.

We need to explore the implications of this condition.

Step 2: Consider the case 𝑥=𝑦*x*=*y* Substitute 𝑦=𝑥*y*=*x* into the inequality:

𝑥𝑓(𝑥)+𝑥𝑓(𝑥)=2𝑥𝑓(𝑥)≤2.*xf*(*x*)+*xf*(*x*)=2*xf*(*x*)≤2.

This simplifies to:

𝑥𝑓(𝑥)≤1.*xf*(*x*)≤1.

Thus, for all 𝑥∈𝑅+*x*∈R+,

𝑓(𝑥)≤1𝑥.*f*(*x*)≤*x*1.

Step 3: Consider the uniqueness condition For each 𝑥∈𝑅+*x*∈R+, there is exactly one 𝑦∈𝑅+*y*∈R+ such that 𝑥𝑓(𝑦)+𝑦𝑓(𝑥)≤2*xf*(*y*)+*yf*(*x*)≤2. We need to ensure that this 𝑦*y* is unique.

Step 4: Test the function 𝑓(𝑥)=1𝑥*f*(*x*)=*x*1 Consider the function 𝑓(𝑥)=1𝑥*f*(*x*)=*x*1. Substitute 𝑓(𝑥)=1𝑥*f*(*x*)=*x*1 and 𝑓(𝑦)=1𝑦*f*(*y*)=*y*1 into the inequality:

𝑥⋅1𝑦+𝑦⋅1𝑥=𝑥𝑦+𝑦𝑥.*x*⋅*y*1+*y*⋅*x*1=*yx*+*xy*.

We need:

𝑥𝑦+𝑦𝑥≤2.*yx*+*xy*≤2.

Using the AM-GM inequality, we know:

𝑥𝑦+𝑦𝑥≥2.*yx*+*xy*≥2.

Equality holds if and only if 𝑥𝑦=1*yx*=1, which implies 𝑥=𝑦*x*=*y*.

Thus, for 𝑓(𝑥)=1𝑥*f*(*x*)=*x*1, the inequality 𝑥𝑓(𝑦)+𝑦𝑓(𝑥)≤2*xf*(*y*)+*yf*(*x*)≤2 holds if and only if 𝑥=𝑦*x*=*y*. This satisfies the condition that there is exactly one 𝑦*y* for each 𝑥*x*.

Step 5: Verify if other functions can satisfy the condition Suppose there exists another function 𝑔:𝑅+→𝑅+*g*:R+→R+ such that 𝑔(𝑥)≠1𝑥*g*(*x*)=*x*1 for some 𝑥*x*. Then, for some 𝑥*x*, 𝑔(𝑥)<1𝑥*g*(*x*)<*x*1 or 𝑔(𝑥)>1𝑥*g*(*x*)>*x*1.

If 𝑔(𝑥)<1𝑥

*g*(*x*)<*x*1, then 𝑥𝑔(𝑦)+𝑦𝑔(𝑥)*xg*(*y*)+*yg*(*x*) might not satisfy the inequality for all 𝑦*y*.If 𝑔(𝑥)>1𝑥

*g*(*x*)>*x*1, then 𝑥𝑔(𝑦)+𝑦𝑔(𝑥)*xg*(*y*)+*yg*(*x*) might exceed 2 for some 𝑦*y*.

Thus, any function other than 𝑓(𝑥)=1𝑥*f*(*x*)=*x*1 would not satisfy the uniqueness condition.

Conclusion The only function 𝑓:𝑅+→𝑅+*f*:R+→R+ that satisfies the given condition is:

𝑓(𝑥)=1𝑥.*f*(*x*)=*x*1.

### Data Decontamination

To ensure the integrity of our results, we meticulously decontaminated both our pre-training and fine-tuning datasets. We removed overlapping samples with key benchmarks like GSM8K, MATH, and SAT Math, ensuring that Qwen2-Math's performance reflects true generalization rather than memorization.

Qwen2-Math sets a new standard in mathematical AI. Its success in surpassing top-tier models on complex math tasks makes it a valuable tool for academia and industry alike.

## Why Choose ModelBox?

ModelBox provides a comprehensive platform for AI model integration, management, and optimization. By supporting the latest models like Qwen2-Math-72B-Instruct we ensure that our users have access to cutting-edge technology with the following advantages:

Unified API Key: Simplifies the integration of various LLMs, including most mainstream Claude Sonnet 3.5, GPT4o Mini, Mistral Large 2, etc, streamlining the development process.

Prompt Management: Facilitates easier debugging and testing with structured outputs.

Analytics: Allows users to monitor usage and performance, ensuring optimal resource utilization.

Optimization: Enables experimentation and evaluation of different models to find the best fit for specific applications.

ModelBox supports Qwen2-Math and takes a leap together with this phenomenal model.

### Learn more about ModelBox

Official Website: https://www.model.box/

Models: https://app.model.box/models

Medium: https://medium.com/@modelbox

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