C2-Evo: Co-evolving Multimodal Reasoning Framework

• C2-Evo is a fully automatic, closed-loop framework that co-evolves multimodal training data and models, improving mathematical reasoning through adaptive, curriculum-aligned evolution.
• The framework employs nested evolutionary loops for cross-modal data augmentation and error-adaptive model updates, ensuring systematic escalation of task complexity.
• Empirical results show significant performance gains on vision-language benchmarks with minimal data, outperforming static methods and closing the gap with larger models.

C2-Evo is a fully automatic, closed-loop, self-improving framework designed for co-evolving multimodal training data and model capabilities in mathematical reasoning with multimodal LLMs (MLLMs). It systematically addresses the data-model alignment problem by tightly coupling cross-modal data evolution with adaptive model training—outperforming static or separately-evolved approaches on a range of vision-language mathematical reasoning benchmarks (Chen et al., 22 Jul 2025).

1. Architecture and Workflow

C2-Evo maintains two interactive, nested evolutionary loops: a Cross-Modal Data Evolution Loop and a Data-Model Evolution Loop. The workflow begins with a seed dataset $\mathcal{D}_1$ and a base model $\pi_1$. Each outer iteration $t$ comprises (1) data evolution—systematic generation of multimodal problems that escalate in visual and textual complexity—and (2) selective model updating—alternating supervised fine-tuning (SFT) and reinforcement learning (RL) on adaptively filtered samples.

Algorithmic Overview:

• Cross-modal Data Evolution: For each triplet $(I_t, Q_t, A_t)$, GPT-4o generates actions and reasoning, yielding auxiliary construction code $C_t$ executed to obtain $I_{t+1}$. The diagram is formally described (Doubao), used as context for DeepSeek-R1 to generate $m \in [4,10]$ sub-problems, fused into a challenging composed question $Q_{t+1}$. Answers $A_{t+1}$ are validated by repetition and agreement.
• Data-Model Evolution: Newly generated samples are evaluated with the current model $\pi_t$ (32 rollouts), identifying a training subset $\pi_1$0 with error rate $\pi_1$1. The model is first fine-tuned on these, then refined with RL using Group Relative Policy Optimization (GRPO).
• Iteration: Over $\pi_1$2 iterations, the dataset $\pi_1$3 grows in complexity and coverage, while the model $\pi_1$4 improves in step with sample difficulty.

The process is summarized below.

Stage	Key Operations	Notation
Data evolution	Auxiliary constructions, diagram augmentation, sub-problem composition	$\pi_1$5, $\pi_1$6, $\pi_1$7
Error-adaptive selection	Evaluate error rate over 32 model rollouts, select $\pi_1$8	$\pi_1$9, $t$0
Alternating SFT and RL	Cross-entropy on traces ($t$1), GRPO RL with accuracy and format rewards	$t$2, $t$3

2. Cross-Modal Data Evolution

This loop orchestrates the joint generation and augmentation of both visual (diagrammatic) and textual (problem statement) complexity:

• Diagram Augmentation: OCR and diagram parsing extract geometry; GPT-4o proposes auxiliary lines (e.g., parallel, perpendicular, connecting), emits Python/Jupyter code, and executes to yield $t$4 with increased geometric intricacy.
• Structured Problem Construction: Formal description $t$5 (from Doubao) and guiding principles $t$6 inform DeepSeek-R1 to create 4–10 sub-problems conditioned on geometric constraints, theorems, and backward-reasoning hints. These are composed into a single question $t$7, with explicit alignment between sub-problems and visual elements.
• Iterative Refinement: For each $t$8, three independent answer traces are produced; only samples with perfect agreement and valid reasoning survive. Reasoning-trace length $t$9 serves as a complexity metric ($(I_t, Q_t, A_t)$0), which grows monotonically through iterations—from approximately 50 tokens in round 1 to 120 in round 3.

Problem complexity is thus systematically increased across both modalities.

3. Data-Model Evolution and Adaptive Selection

C2-Evo’s outer loop adaptively selects samples that are optimally difficult for the current state of the model, mitigating both underfitting and catastrophic failure:

• Error Rate Evaluation: For each generated sample $(I_t, Q_t, A_t)$1, $(I_t, Q_t, A_t)$2 is rolled out 32 times. The error rate, $(I_t, Q_t, A_t)$3, measures difficulty.
• Selective Training Set Formation: The subset $(I_t, Q_t, A_t)$4 is chosen; easier ($(I_t, Q_t, A_t)$5) and overly hard ($(I_t, Q_t, A_t)$6) samples are excluded, synchronizing data difficulty to model capability.
• Alternation of Learning Paradigms: After each selection, SFT and RL (see below) are applied sequentially. This ensures rapid adaptation to new reasoning patterns and avoids overfitting to only the most recent distribution.

This joint evolution ensures the data-model curriculum is always aligned with the model’s blind spots.

4. Learning Objectives: SFT and RL with GRPO

C2-Evo alternates between standard supervised fine-tuning and RL via Group Relative Policy Optimization:

• SFT Objective: The model parameters are updated to minimize the negative log-likelihood of the gold reasoning trace $(I_t, Q_t, A_t)$7 given image $(I_t, Q_t, A_t)$8 and question $(I_t, Q_t, A_t)$9 over $C_t$0:

$C_t$1

• GRPO RL Objective: RL is cast with a trajectory $C_t$2, with two reward components:
  • $C_t$3 if answer $C_t$4 matches the ground truth, $C_t$5 otherwise
  • $C_t$6 if reasoning trace matches the prescribed format (“> …<answer>…</answer>”), $C_t$7 otherwise
  • Composite reward: $C_t$8 (empirical $C_t$9)
  • GRPO maximizes:

$I_{t+1}$0

with $I_{t+1}$1; initial iteration uses 32 rollouts, reducing to 8 in later rounds.

• Curriculum Schedule: Each iteration begins with SFT (to ensure trace format), followed by 1–2 GRPO RL steps (to optimize accuracy and generalization). Empirical ablation shows that RL is more effective when initialized from the SFT-refreshed model than from the original pretrained parameters.

5. Datasets, Task Spectrum, and Complexity Control

C2-Evo is anchored on the Geometry3k dataset (3,002 problems: 2,101 train/300 val/601 test). Each base item is a diagram, a natural-language prompt, and a formal annotation. At every iteration, C2-Evo augments the dataset with 500–800 additional multimodal samples (total $I_{t+1}$21,500 new at iteration 3).

Task Types:

• Pure geometric reasoning (e.g., angles, chords, parallelism, perpendicularity)
• Geometry-driven algebraic computations (e.g., Pythagorean theorem, coordinate geometry)

Complexity Controls:

• Visual: number and sophistication of auxiliary lines inserted
• Textual: number of sub-problems composed into each multimodal task
• Selection: error-rate filtering enforces a “just-hard-enough” training set per model state

6. Benchmarks and Empirical Performance

C2-Evo’s performance is evaluated on a suite of benchmarks comprising both previously curated and self-augmented multimodal reasoning challenges:

Model	Geo-Sub	Geo-Sub-Aux	MathVista-ALL	MathVerse
Qwen2-VL-7B (baseline)	40.4	40.4	60.0	30.2
C²-Evo iter 1	45.5	48.7	62.3	33.6
C²-Evo iter 2	46.9	50.9	62.7	34.8
C²-Evo iter 3 (final)	50.9	54.9	63.4	34.9
Δ (base → iter 3)	+10.5	+14.5	+3.4	+4.7

Geo-Sub (274-item subset) and Geo-Sub-Aux (with auxiliary lines) directly measure geometric reasoning and diagram comprehension. MathVista (GEO, ARI, GPS sub-tasks, combined) and MathVerse (full visual math problems) test broad visual-mathematical generalization (Chen et al., 22 Jul 2025).

With the Qwen2-VL-7B backbone, C2-Evo achieves +14.5 points on Geo-Sub-Aux and +10.5 on Geo-Sub over baseline. For a 2B-parameter model, accuracy rises from 28.0 to 38.2 (Geo-Sub) and from 29.1 to 42.2 (Geo-Sub-Aux) at final iteration. Notably, C2-Evo achieves these gains with less than 1% of the data used by certain closed-source systems, while approaching performance parity with GPT-4o on many sub-tasks. Qualitative analyses indicate that C2-Evo’s solutions feature more structured, detailed reasoning with correct employment of geometric constructs.

7. Framework Significance and Implications

C2-Evo offers an instantiation of tightly coupled, curriculum-aligned multimodal data/model co-evolution, enabling scalable, automatic improvement in reasoning-intensive multimodal domains. Its closed-loop structure prevents divergence between task difficulty and model capability, a limitation in prior self-improving schemes that evolve data or models in isolation (Chen et al., 22 Jul 2025). The fine-grained complexity control and adaptive training sample selection contribute to rapid and efficient model improvement with minimal dataset scale.

A plausible implication is that frameworks employing coupled data/model co-evolution with adaptive sample difficulty can accelerate progress in domains (such as mathematical reasoning or science diagrams) where data scaling is bottlenecked by annotation costs and task complexity.