Universality and Limitations of Prompt Tuning

Summary

This paper examines the theoretical capacity and limitations of prompt tuning. The authors have demonstrated the possibility of constructing a large transformer model that is sufficient for prompt-tuning to exhibit universal approximation over a Lipschitz function space. However, they have also shown the limitations of prompt tuning compared to model fine-tuning by constructing sequence-to-sequence datasets that cannot be learned by prompt-tuning with a given single-layer transformer, even if the soft-prompt embedding size approaches infinity. Additionally, compared with LoRA, prompt-tuning may require more prompt tokens for exact memorization.

Strengths

- Originality - This paper has pretty good originality. It first theoretically studies the theoretical capacity and limitations of prompt tuning. - Clarity - The structures are clear and conclusions are emphasized. - Significance - The problem studied in this paper is important

Weaknesses

- The authors have conducted experiments to support the proposed limitation. However, additional experimental verifications may be necessary to reinforce the theoretical claims and demonstrate the impact on a wider range of tasks and datasets. - The authors have created sequence-to-sequence datasets that are simple in structure but cannot be memorized by prompt-tuning for a given transformer model. Therefore, they argue that prompt-tuning has limited expressiveness compared to model fine-tuning. However, is memorization typically necessary in real-world applications? Does memorization lead to overfitting? As I understand it, such a phenomenon could also be viewed as a potential advantage in preventing overfitting and achieving generalization ability in the process of downstream adaptation. - The authors have provided quite a few mathematical proofs and concluded several insights. I wonder whether there are any guidance for practical prompt-tuning applications. For example, they provide a lower bound on the required number of prompt tokens for exact memorization and point out that such lower bound can be higher than LoRA’s. I would like to know that on which scenarios this could happen and it is recommended to user LoRA rather than prompt-tuning. - The authors need to correct some minor typos. - line 30, typo: the or a - line 503, line 508, are they the same reference?

Questions

Please respond to questions in the "Weaknesses" section.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

2 fair

Limitations

some assumptions are given and not sure whether they are applied in real-world applications

Summary

This paper wants to provide the insights on "when and how to perform prompt tuning to adapt a pretrained transformer to downstream tasks", by theoretically quantifying the universality (i.e., universal approximators) and limitations (i.e., representation capacity) of prompt tuning, in complementing a lot of work focusing on empirically improve soft prompt tuning, such as better initialization or hyper-parameter tuning. Crucially, they show prompt tuning with theoretically very strong transformer models can approximate any seq2seq functions in the set of Lipschitz functions, and they theoretically and empirically show some limitations on weaker transformers (constructed seq2seq datasets as low bounds) by their relatively lower memorization ability.

Strengths

1. This work shows the universality of prompts to the Lipschitz function space, and the limitations of prompt tuning at their memorization ability. The proofs are interesting and well-guaranteed. 2. The mathematical derivations do show your deliberate thinking and rigorous reasoning. 3. The paper writing is almost well-structured.

Weaknesses

1. Just for curious, you define F_L by directly posting the L-lipschitz functions. However, indeed, this causes some unavoidable confusions to readers, as what are "L-lipschitz functions"? Can you provide any context? Especially for some highly theoretical papers? 2. Some writings make me confusing, even though the general structure looks good. 3. Just one question: Is the first section of universality looks very similar to your cited ICLR work on Transformer's universal approximators? I am happy to raise scores if you can make this more clear to me, such that this highly theoretical paper can obtain a more rigorous consideration for its ratings.

Questions

See weaknesses.

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

2 fair

Contribution

2 fair

Limitations

See weaknesses.

Summary

This paper focuses on understanding the role and theoretical underpinnings of soft-prompt tuning in transformer-based architectures, which is a technique used to adapt pretrained language models for new tasks. Despite the empirical effectiveness of this method, there's a lack of theoretical understanding of how it differs from tuning the model weights. The paper takes significant steps in several areas: 1. **Universal Approximation Analysis:** The researchers prove a universality result that guarantees the existence of a transformer with a prompt that can approximate any sequence-to-sequence function within the set of Lipschitz functions, which are a class of functions that have a particular property of bounded rate of change. 2. **Limitations of Limited-Depth Transformers:** The researchers construct datasets that cannot be memorized by a prompt of any length for a single-layer transformer, demonstrating the inherent limitations of prompt-tuning in this setting. The paper extends the analysis to multi-layer transformer settings, providing conditions under which transformers can only learn from invertible functions. This shows additional limitations of the transformer model. 3. **Lower Bound on Tunable Prompt Parameters:** They establish a lower bound on the required number of tunable prompt parameters and compare this with the parameters needed for a low-rank update (LoRA). This helps to illuminate the resource trade-offs between these two methods. 4. **Empirical Results:** The researchers also provide empirical results that back up their theoretical findings, adding credibility to their conclusions.

Strengths

One of the strengths of the paper is its theoretical grounding. It provides a theoretical understanding of the difference between 'tuning parameters before the input' and 'the tuning of model weights'. The paper presents a strong universality result, stating the existence of a strong transformer with a prompt that can approximate any sequence-to-sequence function in the set of Lipschitz functions. In addition, the paper presents a detailed analysis of the limitations of prompt-tuning, especially for limited-depth transformers. The authors create specific datasets that cannot be memorized by any prompt of any length for a single encoder layer. This analysis provides useful insights into the constraints of prompt-tuning. The paper is generally easy-to-follow in idea. Although I do not check the theoretical correctness of these proofs.

Weaknesses

While this paper takes a commendable effort in delving into the theoretical underpinnings of prompt-tuning in transformer-based architectures, it has several significant shortcomings. Firstly, the focus on sequence-to-sequence encoder models poses a substantial gap in the relevance of these findings to practical, real-world application. The universality of sequence-to-sequence transformers cannot be immediately generalized to decoder models, which are commonly used in practice and can handle an arbitrary number of output tokens. Therefore, the study needs to address how its theoretical results can be extended to these types of models. Without this, the scope of the paper's relevance becomes significantly narrow and its conclusions may not be directly applicable to most use-cases. Secondly, the reliance on the universality results from Yun et al.'s work ("Are transformers universal approximators of sequence-to-sequence functions?") reduces the novelty of the paper. While it is valid to build upon previous work, the authors have not demonstrated a significant extension or new application of these universality findings. It would be beneficial for the paper to provide more novel insights or extend the known results to a more general setting beyond what is already known from the referenced work. Lastly, the exploration of limitations in prompt-tuning is an interesting aspect of the paper. However, the limited breadth of the experiments conducted undermines the impact of these findings. The limitation proofs revolve around specific, constructed single-layer transformers, which do not cover the wide array of transformer architectures used in practice. The implications of these limitations would be far more convincing if the authors could generalize the findings to multi-layer transformers or a wider variety of architectures. To improve upon these shortcomings, the authors might consider broadening their analysis to include decoder models, providing more novel extensions to the universality results, and expanding the scope of their limitation analysis to apply to a broader range of transformer architectures.

Questions

1. How to extend the conclusion to decoder models like GPTs that can generate new content? 2. Can you provide some specific examples (with detailed values) for the limitation analysis? E.g. assume the dimension $d = 1$, then x and y are all just numbers. Then I can have a better understanding of why some functions are not learnable via prompt tuning. I'm not familiar with theoretical analysis, and skipped the proofs. I will leave the judgement of proofs to my peer reviewers and area chairs. In addition, several typos should be fixed: "We use the following notations throughout the paper. A bold lower case character, eg x,": eg --> e.g. "If yes, can we construct the a transformer for this universality result?": the a --> a

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

Not fully addressed. The authors do not mention the limitation in encoder-only models.

Summary

This paper presents theoretical analysis of prompt tuning by 1) formally define the attention and prompting operations, 2) deriving the universality of prompt tuning with a strong transformer model, 3) further discussing the limitation of prompt tuning under the case of single and multi-layered transformer models. Explorative experiments on standard models and benchmark datasets demonstrate the correctness of the derived theorems.

Strengths

- The paper is well written with clearly defined definitions and theorem presentations. - Exploring prompt tuning from a theoretical perspective is very interesting and important, since better understanding the limitations of prompts is the key to improve prompting techniques. - The theorems seems correct and solid.

Weaknesses

- The experimental part is a bit weak compared with the theoretical part. - It will be nice to have some further discussions on how to leverage the current limitations of prompt tuning to design better prompting-based models.

Questions

Please see weaknesses part.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

No limitations or potential negative societal impacts discussed in the main context.

Summary

In this work, the authors embark on exploring the capabilities of prompt tuning in the continuous regime, contrasting it with fine-tuning, as an initial endeavor toward theoretical comprehension. The authors prove by construction that prompt tuning admits universal approximation within the space of Lipschitz functions. Additionally, the authors identified inherent limitations of prompt tuning on single-layer transformers by constructing theoretically difficult datasets for prompt tuning. These limitations are then extended to multi-layer settings under a specific prompt-norm restriction.

Strengths

1. The authors characterize the universal nature of prompt tuning by explicitly constructing a transformer network. 2. The authors provide a construction-based argument for sequence-to-sequence datasets that cannot be learned by prompt tuning with a given single-layer transformer. 3. The authors provide the lower bound on the required number of parameters for prompt tuning to memorize any sequence-to-sequence functions. 4. The authors provide a sufficient condition for multi-layer transformers, under which datasets with shared output tokens cannot be learned with prompt tuning. Overall, this paper is well written. The hypothesis of these theories is acceptable and the theoretical analysis is solid.

Weaknesses

Please add more experiments to support the theoretical result.

Questions

N/A

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

4 excellent

Limitations

the authors adequately addressed the limitations in last page.

Reviewer-author discussion

Dear Reviewers, Please take a moment to read the authors' responses. Your insights and feedback are crucial in ensuring a comprehensive evaluation. Thanks. AC