# Curriculum Learning as Transport: Understanding Curricula with Wasserstein Geodesics

arXiv: https://arxiv.org/abs/2609.09099 · HTML: https://mlsanity.com/paper/2609.09099/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 36 references that are in the map.
- First posted on arXiv 8 Sep 2026.

## Builds on (most cited first, 25 of 25)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Did Aristotle Use a Laptop? A Question Answering Benchmark with Implicit Reasoning Strategies](https://mlsanity.com/paper/2101.02235/) (2021)
- [An Empirical Study of Example Forgetting during Deep Neural Network Learning](https://mlsanity.com/paper/1812.05159/) (2018)
- [Training Region-based Object Detectors with Online Hard Example Mining](https://mlsanity.com/paper/1604.03540/) (2016)
- [Dataset Cartography: Mapping and Diagnosing Datasets with Training Dynamics](https://mlsanity.com/paper/2009.10795/) (2020)
- [DoReMi: Optimizing Data Mixtures Speeds Up Language Model Pretraining](https://mlsanity.com/paper/2305.10429/) (2023)
- [On The Power of Curriculum Learning in Training Deep Networks](https://mlsanity.com/paper/1904.03626/) (2019)
- [Automated Curriculum Learning for Neural Networks](https://mlsanity.com/paper/1704.03003/) (2017)
- [Teacher-Student Curriculum Learning](https://mlsanity.com/paper/1707.00183/) (2017)
- [Competence-based Curriculum Learning for Neural Machine Translation](https://mlsanity.com/paper/1903.09848/) (2019)
- [RegMix: Data Mixture as Regression for Language Model Pre-training](https://mlsanity.com/paper/2407.01492/) (2024)
- [When Do Curricula Work?](https://mlsanity.com/paper/2012.03107/) (2020)
- [DoGE: Domain Reweighting with Generalization Estimation](https://mlsanity.com/paper/2310.15393/) (2023)
- [SynLogic: Synthesizing Verifiable Reasoning Data at Scale for Learning Logical Reasoning and Beyond](https://mlsanity.com/paper/2505.19641/) (2025)
- [The Unreasonable Effectiveness of Easy Training Data for Hard Tasks](https://mlsanity.com/paper/2401.06751/) (2024)
- [Curriculum Reinforcement Learning using Optimal Transport via Gradual Domain Adaptation](https://mlsanity.com/paper/2210.10195/) (2022)
- [What Makes a Good Curriculum? Disentangling the Effects of Data Ordering on LLM Mathematical Reasoning](https://mlsanity.com/paper/2510.19099/) (2025)
- [A Psychology-based Unified Dynamic Framework for Curriculum Learning](https://mlsanity.com/paper/2408.05326/) (2024)
- [Optimizing Pre-Training Data Mixtures with Mixtures of Data Expert Models](https://mlsanity.com/paper/2502.15950/) (2025)
- [Topic Over Source: The Key to Effective Data Mixing for Language Models Pre-training](https://mlsanity.com/paper/2502.16802/) (2025)
- [R&B: Domain Regrouping and Data Mixture Balancing for Efficient Foundation Model Training](https://mlsanity.com/paper/2505.00358/) (2025)

## Abstract (arXiv)

Curriculum learning is governed by several coupled design choices---how difficulty is defined, how examples are ordered, how much exposure each level receives, and how quickly training moves across levels---making it hard to isolate what actually helps. We present Wasserstein curriculum paths, a simple transport-based framework that decouples these factors by representing curricula as trajectories of training distributions over discrete difficulty levels. Across a calibrated synthetic suite with 12 tasks and 33 difficulty axes, we use this framework to isolate the effects of ordering, matched exposure, endpoint smoothness, and pacing under fixed training budgets. We find that curriculum effects are strongly context-dependent: no single strategy dominates across tasks, difficulty axes, and budgets, and curricula mainly change where a fixed budget is spent most effectively. Within this framework, easy-to-hard ordering improves hard-level performance relative to exposure-matched static sampling, showing that the benefit is not explained by cumulative exposure alone. We further show that endpoint smoothness and pacing substantially affect where along the difficulty spectrum a curriculum is effective. Finally, we show that the same transport view naturally supports extensions to learned pacing through geometry and to structured difficulty spaces beyond one-dimensional orderings.