How can "weak teacher models" such as average human annotators or existing AI systems, effectively supervise LLMs to improve performance on hard reasoning tasks, especially those that challenge and requires expertise or daily practice from the teacher models? In this paper, we seek for empirical answers to this question by investigating various data-driven strategies that offer supervision data at different quality levels upon tasks of varying complexity. Two intuitive strategies emerge for teacher models to provide supervision during alignment training: 1) using lower-quality supervision from complete tasks that match the difficulty of the target reasoning tasks, and 2) leveraging higher-quality supervision from easier subtasks that are less challenging. Interestingly, we find that even when the outcome error rate for hard task supervision is high (e.g., 90\%), training on such data can outperform perfectly correct supervision on easier subtasks on multiple hard math benchmarks. We further identify a more critical factor influencing training performance: step-wise error rates, which indicate the severity of errors in solutions. Specifically, training on hard task supervision with the same outcome error rates but disparate step-wise error rates can lead to a 30\% accuracy gap on MATH benchmark. Our results also reveal that supplementing hard task supervision with the corresponding subtask supervision can yield notable performance improvements than simply combining rephrased hard full task supervision, suggesting new avenues for data augmentation. Data and code are released at \url{https://github.com/hexuan21/Weak-to-Strong}.

In real-world scenarios, labeled samples for dialogue summarization are usually limited (i.e., few-shot) due to high annotation costs for high-quality dialogue summaries. To efficiently learn from few-shot samples, previous works have utilized massive annotated data from other downstream tasks and then performed prompt transfer in prompt tuning so as to enable cross-task knowledge transfer. However, existing general-purpose prompt transfer techniques lack consideration for dialogue-specific information. In this paper, we focus on improving the prompt transfer from dialogue state tracking to dialogue summarization and propose Skeleton-Assisted Prompt Transfer (SAPT), which leverages skeleton generation as extra supervision that functions as a medium connecting the distinct source and target task and resulting in the model's better consumption of dialogue state information. To automatically extract dialogue skeletons as supervised training data for skeleton generation, we design a novel approach with perturbation-based probes requiring neither annotation effort nor domain knowledge. Training the model on such skeletons can also help preserve model capability during prompt transfer. Our method significantly outperforms existing baselines. In-depth analyses demonstrate the effectiveness of our method in facilitating cross-task knowledge transfer in few-shot dialogue summarization.