Prompt Space Optimizing Few-shot Reasoning Success with Large Language Models

The main weakness of the method is its sensitivity to the number of principal components to include, $k$. Figure 5 shows that accuracy varies hugely (by 20 to 30 percent) if we don't choose exactly the right value for $k$. Comparatively the method achieves modest gains (1 to 3 percent) compared to previous work. The accuracy drops happen even if we change $k$ by just 1 or 2. The authors state that they cannot automatically determine the optimal parameter value for a dataset.

In the Introduction, the authors state that previous methods have "lack of guidance on finding optimal prompts" in contrast the the presented method. I don't think this is true as previous methods (including Auto-CoT) are very similar; the most significant difference is the way the questions to include in the prompt are selected, which is using PCA in the presented method.

Connected to this, some claims the paper makes are too extensive in my opinion, like the presented method being "an innovative mathematical solution" which "aims to develop a deep understanding on how to design CoT prompting".

The algorithm itself is presented as if it would be a novel algorithm inspired by PCA, but I think it's actually PCA without making the data zero mean. I found some of the explanations (e.g., linear independence, some parts of PCA, the proof for using SVD to compute PCA) superfluous.

A weakness of the evaluation is that the authors used gpt-35-turbo instead of the widely used davinci model. This makes the results very different from previous work (e.g., the CoT paper reported 46.9% accuracy on GSM8K and 68.9% on SVAMP with GPT 175B compared to 75.8% and 80.2% in the present paper). Even though gpt-35-turbo may outperform davinci, it would be good to report those results too for the sake of comparison to previous work.

Some smaller issues:

• I think that in-context learning is not a prompt engineering technique as stated in the abstract.
• In the Introduction, the authors state that CoT prompting demonstrates that the reasoning ability of LLMs perfectly matches the scaling laws. I believe this should be rephrased as these scaling laws are not mathematically exact to be perfectly matched.
• In the first paragraph of 4.2, I think some text is missing from "average of 3.2% in Table 2".

1. The PROMPT SPACE exhibits certain instabilities. The choice of encoding model and hyperparameter settings can profoundly impact the model's performance. I hope the authors can provide a more thorough analysis in this regard. Specific concerns are detailed in the questions section.
2. Both the method in this paper and Auto-CoT aim to identify the most representative samples. While experimental results suggest that PROMPT SPACE is superior, I believe the paper should provide a clearer analysis of why matrix decomposition in the PROMPT SPACE is more effective, compared to clustering.

The paper misses strong example selection baselines. The paper mainly extends the Auto-CoT in the example selection part, and compares it to Auto-CoT (where examples are selected from clusters). There are many more advanced (e.g., learning-based methods) example selection strategy like [1][2][3][4][5][6]

[1] Learning To Retrieve Prompts for In-Context Learning. EMNLP 2022.

[2] Dynamic Prompt Learning via Policy Gradient for Semi-structured Mathematical Reasoning. ICLR 2023.

[3] Complexity-Based Prompting for Multi-Step Reasoning. ICLR 2022.

[4] Compositional Exemplars for In-context Learning. ICML 2023

[5] complementary explanations for effective in-context learning. ACL Findings 23.

[6] Automatic Prompt Augmentation and Selection with Chain-of-Thought from Labeled Data. Arxiv 23

The paper claims the proposed approach can choose an optimal number of examples. From section 3, I don’t see a very clear and formal description of how the optimal number of examples are selected. Is it achieved using a validation set (which seems to be suggested on 4.4)? Also, as in 4.4, it suggests the question space of AQUA (a more diverse and complex dataset) is even smaller than quite simple formulaic datasets like SingleEQ, which seems to invalidate the claim of the paper on that the prompt space method selects questions that represent the space; intuitively, the AQUA should not be less diverse than the more synthetic and narrow datasets like SingleEQ.

The presentation of the method/experiment is very unclear, many of the details are omitted, making it hard to tell if the proposed method is indeed superior compared to existing methods. In addition, many of the design choices (e.g., the question set for each task, number of basis questions $k$, choice of the embedding model) are not well justified and seem to be determined based on the best test set performance (which could mean info-leaking on the test set).

• For doing SVD over a question set: end of page 4, step 1 "Embedding questions", the authors said there are $m$ questions in a task, which they further used to embed and construct a matrix to select the top k basis questions from. Where do those $m$ questions come from? The training set or the test set of each task? and what is $m$ for each task? Note both zero-shot CoT and few-shot CoT are completely unsupervised and do not require a training dataset.

• Optimal number of basis questions: In the beginning of section 4, the authors mentioned " 3. Prompt Space can determine the optimal number of basis questions for significantly improving the performance of LLMs on each dataset. ". However, in section 4.4, the authors said "there remains a challenge that we cannot automatically determine the optimal number of basis questions for each dataset". Can the optimal number of basis questions be automatically determined or not? Based on Figure 5, there is a big discrepancy on the performance depending on the number of basis questions, how is the final performance determined in Table 1 and 2? Is it based on which k gives the best test performance?

• Which embedding models to use: in Section 4.3, different embedding models give quite different results, for example, if choosing a different embedding model, the proposed method's performance could be worse compared to the baselines. Why choosing the MiniLLM:384 as the embedding model? Is it again based on the test set performance?