What Makes a Good Curriculum? Disentangling the Effects of Data Ordering on LLM Mathematical Reasoning

TL;DR

We address how data ordering in curriculum learning influences LLM mathematical reasoning under fixed budgets and offline training. Our metric-driven framework decomposes difficulty into $\text{Problem Difficulty}$, $\text{Model Surprisal}$, $\text{Confidence Margin}$, $\text{Predictive Uncertainty}$, and $\text{Decision Variability}$, and evaluates forward and reverse curricula across three base models on multiple benchmarks. We find that no curriculum dominates universally; the optimal direction depends on model capacity and task complexity, with curricula reshaping both optimization dynamics and representations, and strategies prioritizing decision-uncertain samples offering additional gains. These findings provide concrete guidance for curriculum design and show that how data is presented can be as important as what data is presented in shaping learning outcomes.

Abstract

Curriculum learning (CL) - ordering training data from easy to hard - has become a popular strategy for improving reasoning in large language models (LLMs). Yet prior work employs disparate difficulty metrics and training setups, leaving open fundamental questions: When does curriculum help? Which direction - forward or reverse - is better? And does the answer depend on what we measure? We address these questions through a unified offline evaluation framework that decomposes curriculum difficulty into five complementary dimensions: Problem Difficulty, Model Surprisal, Confidence Margin, Predictive Uncertainty, and Decision Variability. Through controlled post-training experiments on mathematical reasoning benchmarks with Llama3.1-8B, Mistral-7B, and Gemma3-4B, we find that (i) no curriculum strategy dominates universally - the relative effectiveness of forward versus reverse CL depends jointly on model capability and task complexity; (ii) even within a single metric, samples at different difficulty levels produce distinct gains depending on task demands; and (iii) task-aligned curricula focus on shaping the model's final representations and generalization, whereas inner-state curricula modulate internal states such as confidence and uncertainty. Our findings challenge the notion of a universal curriculum strategy and offer actionable guidance across model and task regimes, with some metrics indicating that prioritizing decision-uncertain samples can further enhance learning outcomes.

Figures (4)

• Figure 1: Overview of the metric-driven curriculum framework. Left: From the training set, we estimate difficulty metrics, construct curricula, and train models under controlled settings. Right (top): Five metric families spanning problem-side and model-side perspectives. Right (bottom): Curriculum schedules, including Original (random order), FCL (easy$\rightarrow$hard), RCL (hard$\rightarrow$easy), and SGC (single-tier focus). Each bar denotes a training instance, with height reflecting its metric value.
• Figure 2: Results of metric-tiered curricula on Llama3-8B for four benchmarks. For each metric, the training set is split into three equal-size tiers (Low, Medium, High), each comprising one third of the data and shuffled; models are trained on one tier at a time. Bars show accuracy (%).
• Figure 3: Internal-state metrics across different curriculum strategies. We present TLP, SLP, TLE, SLE, and LG (from left to right, defined in Section \ref{['sec:metric-definition']}) for Llama3-8B trained under Forward CL (FCL), Reverse CL (RCL), and the shuffled baseline (SHUF) across five benchmarks.
• Figure 4: Training loss dynamics over 100 training steps under different curriculum learning strategies. We compare FCL, RCL, and the shuffled baseline (SHUF) across five metric dimensions (from left to right): TLP, TLE, LG, ACC, and VACC, showing how loss evolves over training under different CL strategies.