The Two-Hump Problem: Bridging the Difficulty Gap in Mathematical Reinforcement Learning

Applying machine learning techniques to solving long-standing mathematical conjectures can be particularly challenging due to their extreme reward sparsity. As an illustrative example, we consider Kalai's algebraic Hirsch conjecture and recast the construction of its counterexamples as a sparse-reward reinforcement learning problem on graphs. We propose a constrained options-based HRL framework with an equivariant graph neural network policy, which allows us to learn useful temporal abstractions for this task. We evaluate our approach over a wide range of degrees and demonstrate that it consistently outperforms classical RL algorithms as well as greedy search. By exploiting the hierarchical structure of the problem, we effectively provide a first-of-its-kind application of HRL to a problem in commutative algebra.

Reinforcement learning (RL) has demonstrated potential for enhancing reasoning in large language models (LLMs). However, effective RL training, which requires medium-difficulty training samples, faces two fundamental challenges: Effective Data Scarcity and Dynamic Difficulty Shifts, where medium-difficulty samples are scarce and become trivial as models improve. Existing methods mitigate this scarcity to some extent by generating training samples. However, these approaches suffer from anchor-free generation, ignoring co-evolution, and difficulty mismatch. To address these issues, we propose D$^2$Evo, a Dual Difficulty-aware self-Evolution RL framework. In each iteration, our method mines medium-difficulty anchors based on the current Solver's capability, trains the Questioner to generate diverse questions at appropriate difficulty levels, and jointly optimizes both components to enable progressive reasoning gains. Extensive experiments demonstrate that D$^2$Evo outperforms existing methods on mathematical reasoning benchmarks with fewer than 2K real mathematical samples, and exhibits strong generalization on general reasoning benchmarks.

Reinforcement Learning with Verifiable Reward (RLVR) is empirically shown to notably enhance the reasoning performance of large language models (LLMs), particularly in mathematics and programming. However, the mechanistic role of Sample Difficulty in RLVR remains poorly understood. In this paper, we investigate RLVR through the lens of difficulty-wise and one-sample analysis. We find that sample difficulty has a non-monotonic effect on RLVR: easy and medium-difficulty problems yield the strongest and most stable reasoning improvements, whereas overly hard problems often provide weak learning signals, induce degenerate behaviors such as answer repetition or skipping necessary computation, and can ultimately degrade the model's pre-existing capabilities. Beyond the obverse of response, we further analyze the model's internal feature dynamics using Temporal Sparse Autoencoders (T-SAE). Easy problems mainly reinforce direct-answer and basic-computation features while suppressing deliberative-reasoning features; hard problems activate reasoning-related features but become useful only when successful trajectories are sampled; medium-difficulty problems provide a more balanced signal, strengthening both computation and multi-step reasoning features. Motivated by these findings, we propose difficulty-adaptive strategies for hard-sample utilization, using backward-reasoning reformulation and T-SAE-guided training signals to improve reward density and credit assignment during RLVR. Overall, our results identify sample difficulty as a key factor governing both the optimization dynamics and representation evolution of RLVR.