Prompting a Pretrained Transformer Can Be a Universal Approximator
(2402.14753)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 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, the attention mechanism is uniquely suited for universal approximation with prefix-tuning a single attention head being sufficient to approximate any continuous function. 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.
Overview
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This paper theoretically evaluates whether prompting and prefix-tuning can turn pretrained transformers into universal approximators for any sequence-to-sequence function.
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It demonstrates that a single attention head, with proper prefix-tuning, can approximate any smooth continuous function on a hypersphere, highlighting the universality of transformer models.
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The study suggests that while transformer depth requirement scales linearly with sequence length for approximation, the prefix or prompt length required increases with the function's complexity and the desired accuracy.
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The findings have practical implications for the design and application of transformer models, suggesting routes for optimizing transformers for efficient prefix-tuning and prompting.
Exploring the Depths of Prefix-Tuning: A Theoretical Perspective on Transformer Universality
Introduction
Recent advancements in LLMs and the generative AI domain have significantly benefitted from the efficient fine-tuning of transformer models. While practices such as prompting, prefix-tuning, and soft prompting have become commonplace, their theoretical underpinnings, particularly pertaining to their efficacy in universally approximating sequence-to-sequence functions, remain largely unexplored. This investigation aims to theoretically analyze whether prompting and prefix-tuning can serve as universal approximators, thereby potentially modifying the output of a pretrained transformer to arbitrary precision.
Theoretical Framework
To understand the capabilities of transformers in accommodating various sequence-to-sequence transformations through prefixing, we consider a formal evaluation through the lenses of universal approximation. Leveraging the theoretical groundwork laid by traditional neural network approximation theorems, we scrutinize the attention mechanism’s ability to approximate continuous functions on hyperspheres. Particularly, we show that with prefix-tuning, a single attention head can effectively approximate any smooth continuous function, underlining the transformer architecture’s inherent universality. Furthermore, we derive bounds on the necessary prefix length to achieve a desired approximation error, enhancing our comprehension of the transformer’s approximation capacities.
Results and Observations
Our analysis reveals several noteworthy outcomes:
- A single attention head, when properly prefixed, suffices for approximating any continuous function on a hypersphere, revealing the inherent universality of the attention mechanism.
- The depth of the transformer required for approximation scales linearly with the sequence length, independent of the desired accuracy, presenting an efficient scheme contrary to existing beliefs on necessitating depth adjustments for accuracy improvements.
- We unfurl that the prompt or prefix length scales unfavorably with the target function’s complexity and the desired approximation error. This suggests limitations in the practical applicability of prefix-tuning and prompting, especially for complex functions or higher desired accuracies.
Practical Implications
The unraveled theoretical insights offer profound implications for both the design and application of transformer models:
- This work suggests a pathway to ensuring that a pretrained model has the intrinsic capability to function as a token-wise universal approximator by incorporating specific attention heads during training.
- The findings may guide the development of novel transformer architectures optimized for efficient prefix-tuning and prompting, aiming at enhanced model adaptability with minimal training overhead.
- Understanding the bounds of prefix length for desired accuracy levels aids in estimating the practicability and computational feasibility of prompting strategies for given tasks.
Future Directions
Despite its theoretical rigor, our examination adheres to a specific class of predefined transformer models, potentially divergent from those trained on real-world datasets. This opens avenues for future work to investigate the approximation capabilities of realistically pretrained transformers with prefix-tuning. Moreover, probing into the inverse bounds and exploring the practical limits of prefix-tuning and prompting in real-world applications constitute essential steps forward.
Conclusion
The elucidation of the theoretical aspects of prefix-tuning and prompting as universal approximators not only adds a significant chapter to the understanding of transformers but also paves the path for designing more robust, adaptable, and efficient AI systems. The theoretical milestones achieved beckon further empirical and theoretical exploration, promising an exciting trajectory for future research in the realm of transformer models.