📐 FineMath

What is it?

📐 FineMath consists of 34B tokens (FineMath-3+) and 54B tokens (FineMath-3+ with InfiMM-WebMath-3+) of mathematical educational content filtered from CommonCrawl. To curate this dataset, we trained a mathematical content classifier using annotations generated by LLama-3.1-70B-Instruct. We used the classifier to retain only the most educational mathematics content, focusing on clear explanations and step-by-step problem solving rather than advanced academic papers.

The Dataset Curation section details the process for creating the dataset.

What is being released?

The dataset is released in two versions:

• FineMath-3+: 34B tokens, 21.4M documents containing mathematical reasoning and problem solving, formatted with Markdown and LaTeX.
• FineMath-4+ (a subset of FineMath-3+): 9.6B tokens, 6.7M documents of higher quality with detailed explanations. Models trained on this dataset perform better on GSM8k and MATH.

We also release a filtered English text-only portion of the InfiMM-WebMath-40B dataset, classified using the same approach as FineMath:

• InfiMM-WebMath-3+: 20.5B tokens, 13.9M documents.
• InfiMM-WebMath-4+ (a subset of InfiMM-WebMath-3+): 8.5B tokens, 6.3M documents.

How to load the dataset

Use one of the available configs: finemath-3plus, finemath-4plus, infiwebmath-3plus, or infiwebmath-4plus.

from datasets import load_dataset

# Load the high-quality subset
data = load_dataset("HuggingFaceTB/finemath", "finemath-4plus", split="train", num_proc=8)

# Or load the larger subset
data = load_dataset("HuggingFaceTB/finemath", "finemath-3plus", split="train", num_proc=8)

Dataset curation

Recent language models like DeepSeekMath and MathStral have demonstrated strong mathematical capabilities, trained on specialized datasets that aren't publicly available. We developed a pipeline to identify and extract high-quality mathematical content from CommonCrawl, with several iterations of refinement to improve quality.

Phase 1: Initial content extraction and classification

We began by re-extracting pages from CommonCrawl WARCs using URLs from the FineWeb dataset, collecting both the latest and largest versions of each page to capture the evolution of pages across the years. Unlike FineWeb which uses Trafilatura, we employed Resiliparse for text extraction as it better preserves forum discussions and QA answers that often contain crucial reasoning steps and solutions.

For initial quality assessment, we used Llama-3.1-70B-Instruct to generate annotations on a 3-point scale:

1. Contains general mathematical content
2. Shows logical reasoning in mathematical context
3. Contains clear step-by-step solutions at appropriate level

A multilingual-e5-small-based classifier finetuned on these annotations was used to score the initial corpus. However, this first version performed below the OpenWebMath baseline, leading to several important refinements.

Phase 2: Recalling more candidate pages

Analysis revealed that FineWeb's C4 filter removes pages containing '{' characters, inadvertently filtering out content with LaTeX notation. To address this and expand coverage, we:

1. Identified promising website domains by selecting those where at least 10% of pages received a classifier score ≥ 2
2. Added URLs from OpenWebMath and InfiMM-WebMath datasets
3. Recovered URLs of pages filtered by FineWeb's '{' rule from its rejection logs
4. Re-extracted all content from scratch using the OpenWebMath pipeline, which properly handles mathematical notation across various HTML markup formats and standardizes them to LaTeX

Phase 3: Refined quality assessment

The expanded corpus underwent a more fine-grained quality evaluation:

Once again, we used LLama-3.1-70B-Instruct to score a sample of newly extracted pages on a 5-point scale (full prompt available in here): We finetuned a new classifier on these annotations and scored the entire corpus. After leaving only pages with a score of 3 or higher, and deduplicating the samples using simple single-band MinHash-LSH, we obtained FineMath-3+ with 34B tokens.

The same classifier was applied to InfiMM-WebMath's text content, focusing more on reasoning rather than advanced mathematics.

Both datasets were additionally filtered using FineWeb's language classification pipeline to remove non-English content.

Decontamination

Following Qwen2.5-Math's approach, we removed samples with 13-gram overlaps against test sets from GSM8k, MATH, MMLU and ARC. Decontamination logs are available at HuggingFaceTB/finemath_contamination_report.

Results and Performance

Our evaluations show several key findings:

1. FineMath-3+ outperforms the base InfiWebMath on GSM8k and MATH benchmarks
2. FineMath-4+ demonstrates superior performance compared to both FineMath-3+ and InfiWebMath-4+ on GSM8k and MATH
3. Combining the datasets (50% FineMath-3+ with 50% InfiWebMath-3+) yields approximately 50B tokens while matching the performance of FineMath-3+
4. Deduplicating the pages repeated between FineMath and InfiWebMath reduces performance compared to a non-deduplicated combination

Dataset Schema

{
    'url': string,               # Source page URL
    'fetch_time': int64,         # Crawler timestamp
    'content_mime_type': string, # MIME type
    'warc_filename': string,     # Common Crawl WARC source file
    'warc_record_offset': int32, # WARC record offset, in bytes
    'warc_record_length': int32, # WARC record size, in bytes
    'text': string,             # Page content
    'token_count': int32,       # Number of Llama tokens
    'char_count': int32,        # Character count
    'metadata': string,         # Additional OpenWebMath metadata
    'score': float64,           # Raw quality score
    'int_score': int64,         # Integer quality score
    'crawl': string,            # Common Crawl crawl identifier
    'snapshot_type': string,    # Whether the page is the latest or the largest for this URL
    'language': string,         # Document language
    'language_score': float64   # LangID probability
}

Considerations for Using the Data

Social Impact of Dataset

With the release of this dataset, we aim to make high-quality mathematical educational content more accessible to the machine learning community. While multiple language models have demonstrated strong mathematical capabilities, the datasets used to train these capabilities are often not publicly available. By releasing FineMath, we hope to:

• Make the dataset creation process more transparent
• Reduce the barrier to entry for training models with strong mathematical capabilities
• Provide a benchmark for mathematical content quality filtering

Discussion of Biases

The dataset may have certain inherent biases:

• Focus on English language content
• Emphasis on popular educational approaches to mathematics
• Bias towards certain types of mathematical notation and formatting

Other Known Limitations

• The dataset is limited to English language content
• The filtering criteria may not capture advanced mathematical content (e.g. advanced research subjects)
• Some mathematical notation (e.g. image-based) may not be preserved
• Long-form content may have varying quality even within high-scoring documents

Licensing Information

The dataset is released under the Open Data Commons Attribution License (ODC-By) v1.0 license. The use of this dataset is also subject to CommonCrawl's Terms of Use.

Future work

There are several avenues for future work:

• Expand language coverage beyond English
• Improve mathematical notation extraction and preservation
• Develop more sophisticated quality metrics
• Create specialized subsets for different educational levels

Citation Information

@misc{lozhkov2024finemath,  
    author       = { Lozhkov, Anton and Ben Allal, Loubna and Bakouch, Elie and von Werra, Leandro and Wolf, Thomas },  
    title        = { FineMath: the Finest Collection of Mathematical Content }, 
    year         = 2024,  
    url          = { https://huggingface.co/datasets/HuggingFaceTB/finemath },  
    doi          = { 10.57967/hf/3847 },
    publisher    = { Hugging Face }
}