# On the Emergence of Implicit Curriculum in RLVR Learning Dynamics

arXiv: https://arxiv.org/abs/2602.14872 · HTML: https://mlsanity.com/paper/2602.14872/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- First posted on arXiv 16 Feb 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Built on by (most cited first, 10 of 10)

- [EvoTrainer: Co-Evolving LLM Policies and Training Harnesses for Autonomous Agentic Reinforcement Learning](https://mlsanity.com/paper/2606.03108/) (2026)
- [Kernelized Advantage Estimation: From Nonparametric Statistics to LLM Reasoning](https://mlsanity.com/paper/2604.28005/) (2026)
- [You Only Need Minimal RLVR Training: Extrapolating LLMs via Rank-1 Trajectories](https://mlsanity.com/paper/2605.21468/) (2026)
- [Question Begets Question: Self-Evolving Curriculum for Reinforcement Fine-Tuning on Competition Mathematics](https://mlsanity.com/paper/2608.01522/) (2026)
- [When Errors Can Be Beneficial: A Categorization of Imperfect Rewards for Policy Gradient](https://mlsanity.com/paper/2604.25872/) (2026)
- [BASIS: Batchwise Advantage Estimation from Single-Rollout Information Sharing for LLM Reasoning](https://mlsanity.com/paper/2605.27293/) (2026)
- [Agentic Transformers Provably Learn to Search via Reinforcement Learning](https://mlsanity.com/paper/2606.00183/) (2026)
- [CurveRL: Principled Distribution-Aware Context Reweighting for LLM Reasoning](https://mlsanity.com/paper/2605.24331/) (2026)
- [What Pretraining and Midtraining Make Learnable from Rewards?](https://mlsanity.com/paper/2609.38446/) (2026)
- [Verification with Transfer: Exact Information Frontiers and Their Price in Calls](https://mlsanity.com/paper/2610.12211/) (2026)

## Abstract (arXiv)

Reinforcement learning with verifiable rewards (RLVR) has been a main driver of recent breakthroughs in large reasoning models. Yet it remains a mystery how rewards based solely on final outcomes can help overcome the long-horizon barrier to extended reasoning. To understand this, we develop a theory of the training dynamics of RLVR for transformers on compositional reasoning tasks. Our theory shows that mixed-difficulty training naturally induces an implicit curriculum: without any explicit schedule, easier problems become learnable first and shape the frontier for harder ones, creating a learning progression from easy to hard during optimization. The effectiveness of this curriculum is governed by the smoothness of the difficulty spectrum. When the spectrum is smooth, training dynamics enter a well-behaved relay regime, in which persistent gradient signals on easier problems make slightly harder ones tractable and keep training at the edge of competence. When the spectrum contains abrupt discontinuities, training undergoes grokking-type phase transitions with prolonged plateaus before progress recurs. As a technical contribution, our analysis develops and adapts techniques from Fourier analysis on finite groups to our setting. We validate the predicted mechanisms empirically via controlled synthetic experiments and real-model RLVR runs.