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Achieving Olympia-Level Geometry Large Language Model Agent via Complexity Boosting Reinforcement Learning

🕒 Published (v1): 2026-01-01 · Source: ICLR · Venue: ICLR 2026 · link

Why this paper was selected

Complexity-boosting RL reaches Olympiad-level geometry in formal proof agents

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TL;DR

InternGeometry is an LLM agent that solves IMO-level geometry problems by iteratively proposing constructions, verifying them with a symbolic engine, and reflecting on feedback over hundreds of interaction steps. It achieves 44/50 on the IMO 50 benchmark (surpassing the gold medalist average of 40.9) using only 13K training examples—0.004% of AlphaGeometry 2's data—via a Complexity-Boosting Reinforcement Learning (CBRL) curriculum.

Problem

Geometry is the last frontier where LLM agents trail expert models (AlphaGeometry 2, SeedGeometry): auxiliary constructions in geometric proofs have weak heuristics, requiring creative multi-step exploration that is not amenable to the greedy single-pass reasoning typical of LLMs. Existing SOTA approaches depend on massive synthesized datasets (300M+ examples) and large-scale beam search, making them data-inefficient and non-generalizable.

Method

Architecture — InternGeometry Agent: At each step \(t\), the agent \(G\) receives the problem \(X\) and a memory-compressed history \(W(H_{t-1})\) and outputs a chain-of-thought \(P_t\) plus a DSL action \(A_t\): $\([P_t, A_t] = G(X, W(H_{t-1}))\)$ The action is executed by an interactive symbolic engine \(E\) (InternGeometry-DDAR, built on Newclid), which maintains geometric state across turns. Valid actions include constructing points, adding auxiliary constructions, and proposing sub-proof goals. A dynamic memory module \(W\) compresses earlier turns to a compact history of action outcomes, enabling 200+ interactions per problem. Rejection sampling enforces diversity (no repeated actions, no action-type loops, valid formatting).

Training — CBRL: After a supervised cold-start on 7K DSL trajectories, CBRL runs iterative online RL (GRPO). The reward is binary and conjunctive: $\(r = r_o \wedge r_s\)$ where \(r_o = 1\) if the proof completes and \(r_s = 1\) if the specific step's action succeeds (proposition proved or auxiliary construction used in the final proof). Difficulty \(\kappa\) (DDAR proof-step count of synthesized problems) is tuned each round to maximize average absolute advantage: $\(\kappa^* = \arg\max_\kappa \mathbb{E}_{X \sim \mathcal{X}(\kappa)} \mathbb{E}_{y \sim G_\theta | A(X,y)}\)$ Maximum absolute advantage of 0.5 indicates a problem is at the model's frontier—neither trivial nor intractable—justifying the curriculum update rule.

Key Contributions

  • InternGeometry-DDAR: Extended open-source DDAR (Newclid) into a stateful interactive proof engine supporting complex geometric structures, auxiliary construction, and proposition verification.
  • Long-horizon agent design: Dynamic memory compression and rejection sampling together enable 200-step interaction loops without context overflow or action collapse.
  • CBRL: A multi-stage curriculum RL pipeline that auto-schedules training-data difficulty by tracking advantage magnitude, converging faster and reaching higher performance than fixed-difficulty or unscheduled data.
  • Data efficiency: 13K training examples versus 300M for AlphaGeometry 2; first LLM-agent approach to exceed average IMO gold medalist score on geometry.
  • Novel constructions: The agent proposes auxiliary constructions not present in human solutions, demonstrating generative geometric creativity.

Results

  • IMO 50 (2000–2024): InternGeometry 44/50 vs. AlphaGeometry 2 42/50 vs. SeedGeometry 43/50 (Pass@256).
  • IMO 2025 P2: Also solved, giving 45/51 across all evaluated problems.
  • Training data: 13K examples vs. AlphaGeometry 2's 300M (0.004%) and SeedGeometry's 230M (0.006%).
  • Ablation — long-horizon components (IMO 50 Pass@256):
  • Full model: 44/50
  • No propositions: 35/50
  • No slow thinking: 23/50
  • No context compression: 20/50
  • No rejection sampling: 38/50
  • Ablation — CBRL (IMO 50 Pass@256):
  • With CBRL: 44/50
  • SFT cold start only: 22/50
  • Easy data only: 29/50
  • Hard data only: 24/50
  • Same data, no schedule: 38/50
  • Scaling: Extending trajectory length (up to 200 steps) scales performance more efficiently than increasing sampling count \(K\) under a fixed total inference budget.

Limitations

  • The symbolic engine (DDAR) cannot express problems requiring arithmetic or algebraic computation; several remaining unsolved IMO problems fall outside its expressive scope.
  • Pass@256 evaluation is compute-intensive; the test-time budget is lower than AlphaGeometry 2's beam-search ensemble but still nontrivial.
  • The CBRL complexity scheduler uses proof-step count as a proxy for difficulty, which may not perfectly correlate with human-perceived hardness for all problem types.
  • Cold-start formalization relies on a fine-tuned InternThinker-32B as a formalizer, introducing a bootstrapping dependency on the base model's capacity.

Relevance to Agentic AI / LLM Agents

InternGeometry is a concrete demonstration that the tool-use + reflection + memory agent paradigm can match or exceed domain-specialized expert systems on extreme reasoning tasks with orders-of-magnitude less data, directly advancing the case for general-purpose LLM agents over narrow, data-hungry models. The dynamic memory compression and rejection-sampling for diversity are broadly applicable techniques for any long-horizon agent operating over stateful tool environments. CBRL generalizes the curriculum RL idea into an auto-adaptive loop that could transfer to other agentic RL settings (code execution, theorem proving in LEAN, scientific discovery) wherever task difficulty is measurable. The paper also offers empirical evidence that trajectory length is a more efficient test-time scaling axis than sampling count—a finding directly relevant to agent deployment economics.