cs.LGMar 14, 2026

Level Up: Defining and Exploiting Transitional Problems for Curriculum Learning

Authors: Amogh Inamdar, Zhenwei Tang, Ashton Anderson, Richard Zemel

Organizations: Department of Computer Science Columbia University · Department of Computer Science University of Toronto

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Abstract

Curriculum learning--ordering training examples in a sequence to aid machine learning--takes inspiration from human learning, but has not gained widespread acceptance. Static strategies for scoring item difficulty rely on indirect proxy scores of varying quality and produce curricula that are not specific to the learner at hand. Dynamic approaches base difficulty estimates on gradient information, requiring considerable extra computation during training. We introduce a novel method for measuring the difficulty of individual problem instances that is calibrated to a series of models of increasing competence, and identify transitional problems that are consistently easier as model ability increases. Applying this method to diverse model series constructed from sets of models that are readily available on many tasks, we find that training on a curriculum that levels up from easier to harder transitional problems most efficiently improves a model to the next tier of competence. These problems induce a natural progression from easier to harder items, which outperforms other training strategies. By measuring difficulty directly relative to model competence, our method yields interpretable problems, learner-specific curricula, and a principled basis for step-by-step improvement.

Figures & tables

Figure 1: Transitional Problems at a level ii can only and consistently be solved by models at a competence level j≥ij\geq i . We find that with modest budgets, training on an ascending-level curriculum on transitional problems produces models that perform better than those trained with other strategies.
Figure 2
Figure 3
Figure 6: Math: Variation in average question length, answer length, and arithmetic operations for transitional problems vs. all solved problems for a 6-model series obtained from the training checkpoints of Qwen2.5-1.5B-Base on GSM8k.
Figure 5
Figure 9: Chess : Average human rating, number of legal moves, and solution length for transitional problems versus all solved problems at each transitional point. For transitional problems, the human rating and solution length correlate with the competence level, while the number of legal moves does not. None of these measures correlate for the full set of puzzles across levels.
Figure 10: Chess : Each subplot depicts results when a chess model at a particular competence level ii is trained on puzzles from a single level (along the x-axis), and tested on its ability to solve game-position problems at level i+1i+1 (accuracies, y-axis). The vertical line in each indicates our hypothesized level to achieve the best performance. The star denotes the actual level that achieves the best performance, which largely aligns with our hypothesis. Error bars represent std across 10 runs.
Figure 11: Gradient L2L^{2} norm of LevelUp , LevelDown , and IID curricula during fine-tuning on chess positions. Left: per-step gradient norm (faint lines) overlaid with a centered rolling mean (solid lines). Right: smoothed difference from the IID baseline.
Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 12: Relative improvement over IID on chess, evaluated on test sets matched to each difficulty measure. Only transitional point-based measures (left) benefit from Level-Up curriculum ordering, while external difficulty measures (right) show marginal improvement or degradation.
Figure 10
Figure 15: Each subplot depicts results when a chess model at a particular competence level ii is trained on game-positions from a single level (along the x-axis), and tested on its ability to solve puzzles at level i+1i+1 (accuracies, y-axis). The vertical line in each indicates our hypothesized level to achieve the best performance. The star denotes the actual level that achieves the best performance, which largely aligns with our hypothesis. Error bars represent std across 10 runs.
Figure 12
Figure 18: Variation in average question length, answer length, and arithmetic operations for transitional problems vs. all solved problems for a 5-model series obtained from the training checkpoints of Qwen2.5-0.5B on GSM8k.
Figure 19: Variation in human measures of difficulty for problems obtained from a 4-model series obtained from the training checkpoints of Qwen2.5-1.5B on GSM8k. Like for the 6-model series, the transitional problems vary much more sharply across human-interpretable features than the set of all solved problems.
Figure 15
Figure 22: Variation in human measures of difficulty for a 5-model series obtained from the Qwen2.5 model family. Due to the large improvement in performance between levels, the set of all solved problems also shows an increase in difficulty. However, the transitional problems show a much sharper change between levels, indicating better differentiation in learning difficulty.
Figure 17
Figure 25: Variation in human measures of difficulty for a 6-model series obtained from increasing kk -shot adaptation on the Qwen2.5-1.5B model. Even with the reduced differentiation in performance between model levels, the transitional problems exhibit a human-interpretable difficulty increase.
BudgetLearning RateWeight Decay#Train#Test#RunsMax StepsBatch Size
Tiny10−310^{-3}10−610^{-6}10,00010,000109064
Low10−310^{-3}10−610^{-6}10,00010,0001090128
Mid10−310^{-3}10−610^{-6}10,00010,00010180256
Appendix
Table 1: Hyperparameter Settings in Chess Experiments.
Size0.5B-ckpts1.5B-ckpts0.5B-family1.5B-family0.5B-k-shot1.5B-k-shot0.5B-Orca
Tiny88–––––
Low10101010101010
Mid1512–12–––
Appendix
Table 2: Hyperparameter Settings in the Math Experiments.
SettingMethodModelDataset#Source#Transitional#Per level
MathIn-distributionQwen2.5-1.5B-BaseGSM8k6,5002,092675 / 462 / 272 / 289 / 394
MathDistillationQwen2.5-0.5B-BaseGSM8k6,5005,4422,181 / 2,113 / 693 / 255
MathNeo-transitionalQwen2.5-0.5B-BaseOrca-formatted5,000539174 / 139 / 114 / 56 / 56
ChessIn-distributionMaia-2Puzzles∼\sim 5.8M180,00020,000
ChessTransferMaia-2Game Positions∼\sim 300B180,00020,000
Appendix
Table 3: Dataset statistics. #Source is the total pool used to identify transitional problems; #Transitional is the number kept for training (identified for Math, capped for Chess); #Per level lists the per-stage split.
DatasetPhaseModelRuntime
GSM8kCollect checkpointsQwen2.5-1.5B-Base25 min 22 sec
Transitional/GSM8kCurriculum (per method/baseline)Qwen2.5-1.5B-Base8 min 5 sec
Appendix
Table 4: Per-phase runtime for the math experiments.

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