arXiv preprint · September 2025

Ladders of Thought

A Self-Evolving Curriculum of Progressively Simplified Reasoning Traces

Minghui Liu Thomas Magelinski Dehao Yuan Qi Yu Furong Huang

University of Maryland, College Park · Capital One

TL;DR

LoT turns each reasoning problem into a ladder of faithful, progressively easier variants—then uses a bandit scheduler to adapt the training mix as the student learns.

+32.1ppAddSub over CoT KD
with OPT-2.7B

+25.2ppStrategyQA over CoT KD
with OPT-1.3B

1–8BStudent models tested
across four model families

3–8%Training-time overhead
from the adaptive scheduler

The challenge

Small models need a gentler climb.

Knowledge distillation gives smaller language models expert traces, but not always the learning path needed to absorb them. The gap between teacher and student can leave brittle shortcuts instead of transferable reasoning.

Overview of LoT: progressive simplification, difficulty bucketing, and multi-armed-bandit sampling

Figure 1

LoT rewrites a problem into successively easier forms, groups examples by minimal reasoning depth, and lets an adaptive scheduler choose the most useful level throughout training.

Progressive rewrites

Replace premises used in one reasoning step with their logically entailed conclusion. Each rewrite preserves the task and answer while reducing the remaining inferential depth.

Step-based difficulty

Estimate difficulty with the minimal number of reasoning steps—not surface length—then organize examples into easy, medium, and hard buckets.

Self-evolving curriculum

Treat each bucket as a bandit arm. Validation gains become rewards, shifting the training distribution toward the levels producing the most learning progress.

Progressive simplification

The same problem, one step closer.

LoT does not decompose a task into unrelated subproblems. It progressively replaces already-resolved premises with intermediate conclusions, keeping the original problem identity intact.

✓ Semantically faithful rewrites
✓ No answer leakage
✓ Interpretable, monotonic difficulty

A notebook and pen word problem rewritten across four progressively easier stages — Each rewrite reduces the number of remaining reasoning steps while preserving solvability.

Results

Better transfer, not just better memorization.

Across math and multi-hop reasoning, the largest gains appear on out-of-distribution tasks that depend on compositional inference.

Arithmetic 8.26 → 40.37

AddSub pass@5 for OPT-2.7B: a 32.11-point gain over standard CoT distillation.

Multi-hop 22.6 → 47.8

StrategyQA pass@5 for OPT-1.3B: a 25.2-point gain over standard CoT distillation.

Larger students 25.69 → 56.88

AddSub pass@5 for Llama 3.1–8B: the gains persist as student scale increases.

GSM8K validation accuracy over training steps for four curriculum strategies — **Faster learning.** Self-evolving sampling reaches high validation accuracy early and remains competitive through training.

Average math accuracy for flat, easy-to-hard, hard-to-easy, and self-evolving curricula — **Best final average.** Self-evolving sampling reaches 42.55% across math benchmarks, ahead of fixed curricula.

Experimental scope

Two domains.
Ten benchmarks.

∑

Math reasoning

Train on GSM8K; evaluate on GSM8K, AddSub, ASDiv, MultiArith, and SVAMP.

∴

Multi-hop reasoning

Train on EntailmentBank; evaluate on EntailmentBank, QASC, OpenBookQA, StrategyQA, and MuSiQue.

Citation

Build on Ladders of Thought.

@misc{liu2026laddersthought,
  title         = {Ladders of Thought: A Self-Evolving Curriculum of Progressively Simplified Reasoning Traces},
  author        = {Minghui Liu and Thomas Magelinski and Dehao Yuan and Qi Yu and Furong Huang},
  year          = {2026},
  eprint        = {2609.25643},
  archivePrefix = {arXiv},
  primaryClass  = {cs.AI},
  url           = {https://arxiv.org/abs/2609.25643}
}