--- title: 'Socratic-Geo: Autonomous Geometric Reasoning' url: https://www.emergentmind.com/topics/socratic-geo-framework type: topic --- # Socratic-Geo: Autonomous Geometric Reasoning Socratic-Geo is a fully autonomous multi-agent framework for synthetic data generation and geometric reasoning, designed to address limitations in multimodal large language models (MLLMs) with respect to geometric problem-solving. By dynamically coupling data synthesis with model learning using a closed-loop interplay among three specialized agents—a Teacher, a Solver, and a Generator—Socratic-Geo achieves state-of-the-art performance in geometric reasoning tasks with orders-of-magnitude greater data efficiency relative to prior methods [2602.03414]. ## 1. System Architecture Socratic-Geo features a closed-loop curriculum engine composed of three distinct but interacting agents: Teacher (T), Solver (S), and Generator (G). The curriculum $\mathcal{C}_t$ is iteratively expanded as follows: $$ \mathcal{C}_{t+1} = \mathcal{C}_t \cup \{(I', q', a') ~|~ \exists (I, q, a^*) \in \mathcal{C}_t, S \text{ fails } k \text{ times on } (I, q), T \text{ invents and validates } (I', q', a') \} $$ - **Solver (S):** Given an image $I$ and question $q$, the Solver attempts solutions via policy $\pi_S$, receiving binary rewards $R \in \{0,1\}$. - **Teacher (T):** Diagnoses failures of S, constructs new programmatic problems through the RePI (Reflective Problem Invention) mechanism, and employs Reflect for solvability and validity checks. Valid triplets $(I', q', a', c')$ are added to the evolving curriculum. - **Generator (G):** Trains on the collection of (instruction, image) pairs accumulated from T, distilling drawing intelligence into a diffusion-based image generation model. ### Interaction Workflow ``` ┌─────────┐ fail ┌─────────┐ valid ┌──────────┐ │ Solver │──────────▶│ Teacher │──────────▶│ Curriculum│ │ (S) │◀──────────│ (T) │◀──────────▶│ (𝒞) │ └─────────┘ update └─────────┘ sample └──────────┘ T → collects → G → trains → G* ``` This closed-loop design ensures that data generation is tightly coupled to the Solver’s learning needs, yielding maximally informative synthetic curricula. ## 2. Teacher Agent: RePI and Reflect ### RePI Pipeline - **Purpose:** Invents new problems based on Solver failures and modifies base Python code accordingly to create new images and textual descriptions. - **Process:** From a failed instance $(I, q, a^*)$ and unsuccessful Solver attempts $\{a_S^{(i)}\}_{i=1}^k$, T: - Diagnoses the underlying failure $C$, - Modifies the base code $c \rightarrow c'$, and - Executes $c'$ to generate $(I', q', a')$. ### Reflect Validation - **Reflect:** T attempts to solve $(I', q')$ under its own policy $\pi_T$, producing $a_T$. If $a_T$ matches $a'$ and satisfies geometric consistency checks, $(I', q', a', c')$ is incorporated into the curriculum. ### Scoring Metrics - $\mathrm{RePI}(c') = 1_{\{\mathrm{Exec}(c') = \text{success} \land \mathrm{GeometryCheck}(I') = \text{true}\}}$ - $\mathrm{Reflect}(I', q', a') = 1_{\{V(q', a_T) = 1 \land \text{GeomConsistent}(I', q')\}}$ ### Pseudocode Excerpt ``` Algorithm TeacherModules(𝒫₀, π_S, π_T, G, T_max): 𝒞 ← seedSet 𝒫₀ 𝒟 ← ∅ for t = 1 … T_max: sample (I, q, a*, c) ∼ 𝒞 a⃗_S ← π_S(I, q) // k attempts if ∑_i 𝟙[a_S^(i) = a*] = 0: C ← Diagnose(π_S, a*) repeat c′ ← π_T.Modify(c, C) (I′, q′, a′) ← Exec(c′) until Exec succeeds (RePI=1) a_T ← π_T(I′, q′) v ← Check(a_T, a′) if v=1: 𝒞 ← 𝒞 ∪ {(I′, q′, a′, c′)} d ← π_T.Trans(q′) 𝒟 ← 𝒟 ∪ {(d, I′)} return 𝒞, π_S*, G* ``` ## 3. Solver Agent: Group Relative Policy Optimization The Solver uses a reinforcement learning objective based on Group Relative Policy Optimization (GRPO). Rather than learning by supervised fine-tuning (SFT), S only receives binary rewards $R_i \in \{0,1\}$ for each attempt. - **Preference Sets:** For each problem $(I, q)$ and $k$ attempts: - If $\sum R_i > 0$, $\mathbb{Z}^+ =$ all correct attempts, $\mathbb{Z}^- =$ incorrect. - If $\sum R_i = 0$, $\mathbb{Z}^+ =$ Teacher-provided reference, $\mathbb{Z}^- =$ all attempts. - **GRPO Loss:** $$ L_{\mathrm{GRPO}}(\theta_S) = -\mathbb{E}_{(I, q) \sim \mathcal{C}_t}\left[ \frac{1}{|\mathbb{Z}|} \sum_{a \in \mathbb{Z}} \frac{1}{|a|} \sum_{t=1}^{|a|} \left[\hat{A}_t(a) + \beta(r_t-1)\right]\log \pi_\theta(a_t \mid I, q, a_{