Prompting a Pretrained Transformer Can Be a Universal Approximator

Aleksandar Petrov, Philip Torr, Adel Bibi
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:40523-40550, 2024.

Abstract

Despite the widespread adoption of prompting, prompt tuning and prefix-tuning of transformer models, our theoretical understanding of these fine-tuning methods remains limited. A key question is whether one can arbitrarily modify the behavior of a pretrained model by prompting or prefix-tuning it. Formally, whether prompting and prefix-tuning a pretrained model can universally approximate sequence-to-sequence functions. This paper answers in the affirmative and demonstrates that much smaller pretrained models than previously thought can be universal approximators when prefixed. In fact, prefix-tuning a single attention head is sufficient to approximate any continuous function making the attention mechanism uniquely suited for universal approximation. Moreover, any sequence-to-sequence function can be approximated by prefixing a transformer with depth linear in the sequence length. Beyond these density-type results, we also offer Jackson-type bounds on the length of the prefix needed to approximate a function to a desired precision.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-petrov24a, title = {Prompting a Pretrained Transformer Can Be a Universal Approximator}, author = {Petrov, Aleksandar and Torr, Philip and Bibi, Adel}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {40523--40550}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/petrov24a/petrov24a.pdf}, url = {https://proceedings.mlr.press/v235/petrov24a.html}, abstract = {Despite the widespread adoption of prompting, prompt tuning and prefix-tuning of transformer models, our theoretical understanding of these fine-tuning methods remains limited. A key question is whether one can arbitrarily modify the behavior of a pretrained model by prompting or prefix-tuning it. Formally, whether prompting and prefix-tuning a pretrained model can universally approximate sequence-to-sequence functions. This paper answers in the affirmative and demonstrates that much smaller pretrained models than previously thought can be universal approximators when prefixed. In fact, prefix-tuning a single attention head is sufficient to approximate any continuous function making the attention mechanism uniquely suited for universal approximation. Moreover, any sequence-to-sequence function can be approximated by prefixing a transformer with depth linear in the sequence length. Beyond these density-type results, we also offer Jackson-type bounds on the length of the prefix needed to approximate a function to a desired precision.} }
Endnote
%0 Conference Paper %T Prompting a Pretrained Transformer Can Be a Universal Approximator %A Aleksandar Petrov %A Philip Torr %A Adel Bibi %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-petrov24a %I PMLR %P 40523--40550 %U https://proceedings.mlr.press/v235/petrov24a.html %V 235 %X Despite the widespread adoption of prompting, prompt tuning and prefix-tuning of transformer models, our theoretical understanding of these fine-tuning methods remains limited. A key question is whether one can arbitrarily modify the behavior of a pretrained model by prompting or prefix-tuning it. Formally, whether prompting and prefix-tuning a pretrained model can universally approximate sequence-to-sequence functions. This paper answers in the affirmative and demonstrates that much smaller pretrained models than previously thought can be universal approximators when prefixed. In fact, prefix-tuning a single attention head is sufficient to approximate any continuous function making the attention mechanism uniquely suited for universal approximation. Moreover, any sequence-to-sequence function can be approximated by prefixing a transformer with depth linear in the sequence length. Beyond these density-type results, we also offer Jackson-type bounds on the length of the prefix needed to approximate a function to a desired precision.
APA
Petrov, A., Torr, P. & Bibi, A.. (2024). Prompting a Pretrained Transformer Can Be a Universal Approximator. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:40523-40550 Available from https://proceedings.mlr.press/v235/petrov24a.html.

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