DeepSeekMath: Open LLM for Math Reasoning

Updated 3 August 2025

DeepSeekMath is a family of open-source LLMs specialized in advanced mathematical reasoning and problem solving.
It employs a dedicated corpus of 120B math tokens and a GRPO-based reinforcement learning strategy to optimize multi-step logic and symbolic computations.
Benchmark results indicate high chain-of-thought accuracy, achieving up to 60.9% with self-consistency, signaling its competitive performance in formal math tasks.

DeepSeekMath is a family of open-source LLMs specifically engineered for mathematical reasoning and problem solving. Developed by continuing the pretraining of code-intensive transformer models using a dedicated, high-quality corpus of mathematical texts, DeepSeekMath pushes the boundaries of open LLMs in both general quantitative problem solving and formal mathematics. The approach integrates innovations in data curation, reinforcement learning, and evaluation methodology to address the unique structural, symbolic, and deductive demands of mathematics.

1. Model Design and Training Paradigm

DeepSeekMath’s architecture originates from the DeepSeek-Coder-Base-v1.5 7B—a decoder-only transformer first optimized for code-based reasoning. The model is then further pretrained on 120B math-related tokens, explicitly curated for mathematical richness. This continued pretraining strategy allows DeepSeekMath to build upon the logical and structural patterns acquired during code pretraining, specializing its network for mathematical domain transfer (Shao et al., 5 Feb 2024).

The cornerstone of its performance is the data selection pipeline that mines 120B tokens from Common Crawl, employing fastText-based classification, deduplication, and iterative bootstrapping over math-dense domains (notably including mathoverflow.net and OpenWebMath) to ensure both breadth and mathematical depth. This corpus encompasses a multilingual, multi-format mathematical landscape, exposing the model to a spectrum of problem types and formalism.

2. Reinforcement Learning via Group Relative Policy Optimization (GRPO)

DeepSeekMath’s mathematical reasoning capabilities are augmented by reinforcement learning through the Group Relative Policy Optimization (GRPO) algorithm. Unlike canonical Proximal Policy Optimization (PPO), GRPO dispenses with a separate value function (critic). Instead, for each question, the model samples a group of outputs, computes relative rewards, and normalizes these within the group to estimate an advantage. The critical gradient coefficient for each token, given as

$\text{GC}_{\text{GRPO}}(q, o, t, \pi_{\text{ref}}) = \hat{A}_{i, t} + \beta \cdot \left( \frac{\pi_{\text{ref}}(o_{i, t} | q, o_{i, <t})}{\pi_\theta(o_{i, t} | q, o_{i, <t})} - 1 \right)$

incorporates both token-level relative reward and regularization with respect to a reference policy. Here, $\hat{A}_{i, t}$ is the group-based, normalized token advantage; $\beta$ modulates the KL penalty; and $\pi_{\text{ref}}$ is a static reference model.

This scheme optimizes memory and computational demands, as the value baseline is computed from intra-group statistics. It further prioritizes correct or more mathematically rigorous reasoning in heterogeneous output groups, sharpening chain-of-thought performance (Shao et al., 5 Feb 2024, Vojnovic et al., 25 Feb 2025).

3. Benchmark Results and Performance Analysis

DeepSeekMath achieves a top-1 accuracy of 51.7% on the MATH competition benchmark—placing it close to closed-source models like Gemini-Ultra and GPT-4 while using significantly fewer parameters (7B vs models with much larger capacity). When evaluated using self-consistency over 64 samples (majority voting among multiple outputs), accuracy rises to 60.9%. This indicates that DeepSeekMath’s output distribution reliably clusters around correct reasoning trajectories, provided mild aggregation/voting (Shao et al., 5 Feb 2024).

On step-by-step chain-of-thought settings (e.g., GSM8K, code-intensive program-of-thought questions), DeepSeekMath surpasses most contemporary open models, especially in problems demanding multi-stage reasoning and explicit calculation.

4. Data Curation and Pipeline

The DeepSeekMath corpus construction hinges on iterative bootstrapping:

Iteration Step	Methodology	Outcome
1. Seed Corpus	Positive: OpenWebMath, Negative: random CC	Initial fastText classifier training
2. Classify Common Crawl	HTML scraping, classifier scoring	Recall math-rich HTML documents
3. Quality Control	Deduplication, near-deduplication, ranking	Prune and refine
4. Iterative Expansion	Identify new domains (e.g., mathoverflow)	Augment training set
5. Final Compilation	Combine/curate 120B math tokens	Compose high-quality math reasoning set

This approach ensures the model is exposed to mathematical syntax, notation, and a diversity of problem genres, reducing noise from non-mathematical web sources.

5. Advances in Mathematical Reasoning and Limitations

DeepSeekMath offers strong chain-of-thought capabilities: it can generate detailed, logically connected multi-step solutions to both competition-grade and grade-school problems. Python code synthesis (program-of-thought) is a native capability, enabling direct computational reasoning, which can be leveraged for symbolic algebra, numerical calculation, and answer verification (Shao et al., 5 Feb 2024).

Nevertheless, certain domains remain limited. For example, while the model can synthesize valid step sequences for logic, number theory, and algebraic manipulation, geometric reasoning (which often requires spatial abstraction) and formal theorem proving (requiring Lean or Isabelle statement-level encoding) still trail the best proprietary systems.

6. Integration of Selective LLMing and Precision Gains

Subsequent studies highlight that DeepSeekMath’s training regime, which uniformly trains on all math tokens, may be further optimized for efficiency. The Rho-1 model introduces Selective LLMing (SLM), training only on “useful” tokens with high excess loss compared to a reference model. Rho-1 achieves comparable MATH benchmark results (51.8% at 7B scale) using merely ~3% of DeepSeekMath’s token count. This suggests that integrating SLM—focusing updates only on informative tokens—could dramatically improve training efficiency for DeepSeekMath without performance loss (Lin et al., 11 Apr 2024).

7. Future Research Directions and Open Challenges

DeepSeekMath’s architecture and pipeline open several avenues:

Data efficiency: SLM and related methods may reduce the volume of required pretraining tokens by focusing only on token subsets with high learning potential.
Scaling and curriculum: Dynamic skill adaptation frameworks (e.g., graph-based curriculum learning (Chen et al., 26 Dec 2024)) may further scaffold the introduction of advanced mathematical concepts, mirroring human learning curves.
Verifier-guided search limitations: In multi-step reasoning, verifier-guided search can underperform naive repeated sampling at scale due to verifier misranking; mitigating these scaling flaws remains an open problem (Yu et al., 1 Feb 2025).
Instruction fusion and mistake-driven learning: Techniques such as MathFusion (problem pair fusion (Pei et al., 20 Mar 2025)) and LEMMA (learning from errors (Pan et al., 21 Mar 2025)) demonstrate that exposing and correcting model-generated reasoning mistakes during training enhances step-level reflection and downstream accuracy.

8. Significance and Broader Impact

DeepSeekMath sets a milestone for open, transparent, and high-performance mathematical LLMs. It achieves near-parity in competition-level math reasoning with proprietary giants by leveraging innovations in training data engineering, RL-based instruction tuning, and efficient data utilization. Its design informs best practices for open-source mathematical LLMs and offers a platform for further research in mathematical tool use, formal proof systems, and multi-step logical inference. The methodology demonstrates the competitive potential of open models when paired with meticulous dataset curation, robust RL alignment methods, and efficient utilization of domain structure—paving the way for further advances in automated mathematical reasoning.