Curriculum Learning as Transport: Understanding Curricula with Wasserstein Geodesics

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 compare easy-to-hard and hard-to-easy orderings, matched-exposure static baselines, endpoint smoothness, and traversal speed under fixed training budgets. We find that curriculum effects are strongly context-dependent: no single strategy dominates across tasks, difficulty axes, and budgets. Instead, curricula mainly shift where a fixed training budget is used most efficiently across difficulty levels. 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.