Transformers are Expressive, But Are They Expressive Enough for Regression? (2402.15478v3)

Abstract: Transformers have become pivotal in Natural Language Processing, demonstrating remarkable success in applications like Machine Translation and Summarization. Given their widespread adoption, several works have attempted to analyze the expressivity of Transformers. Expressivity of a neural network is the class of functions it can approximate. A neural network is fully expressive if it can act as a universal function approximator. We attempt to analyze the same for Transformers. Contrary to existing claims, our findings reveal that Transformers struggle to reliably approximate smooth functions, relying on piecewise constant approximations with sizable intervals. The central question emerges as: ''Are Transformers truly Universal Function Approximators?'' To address this, we conduct a thorough investigation, providing theoretical insights and supporting evidence through experiments. Theoretically, we prove that Transformer Encoders cannot approximate smooth functions. Experimentally, we complement our theory and show that the full Transformer architecture cannot approximate smooth functions. By shedding light on these challenges, we advocate a refined understanding of Transformers' capabilities. Code Link: https://github.com/swaroop-nath/transformer-expressivity.

• The paper challenges the claim that Transformers are universal function approximators by exposing their limitations in continuous function regression.
• It introduces a resolution factor δ to establish a mathematical link between a function’s derivative and the required complexity of the Transformer model.
• Empirical experiments reveal significant shortcomings in direct continuous function approximation, highlighting the need for new architectural innovations.

Evaluating the Capability of Transformers in Function Approximation

Overview

Transformers have undeniably revolutionized the field of NLP. With their unparalleled ability to model complex dependencies, these architectures have set new benchmarks across a spectrum of NLP applications. However, their ability to approximate continuous functions remains a subject of keen investigation. Recent works postulate Transformers as Universal Function Approximators—entities capable of modeling any function given sufficient parameters and training data. Our examination, however, uncovers limitations in their ability to approximate continuous functions, leading us to scrutinize and experimentally challenge their alleged universal approximation capabilities.

Theoretical Insights

The expressivity of a neural model like the Transformer can be quantitatively analyzed through its effectiveness in function approximation. This involves assessing whether a Transformer can model a wide class of functions, specifically continuous functions, to an acceptable degree of accuracy. Our theoretical analysis reveals a significant challenge: Transformers, as configured in their original or slightly modified forms, encounter difficulties approximating continuous functions directly. This is attributed to the necessity of employing piecewise constant functions for approximation, which inherently introduces errors when modeling highly variable continuous functions. We find that the essence of this limitation lies in the resolution factor δ, a measure dictating the granularity of the piecewise constant approximation. The smaller the value of δ, the finer the approximation, yet the more computationally demanding the model becomes due to the increased layers required for accurate approximation. Our work mathematically formulates this relationship, exposing a direct link between the derivative of the target function and the necessary specifications of the Transformer model—pointing towards an exponential increase in complexity for adequately approximating functions with significant rates of change.

Empirical Validation

To empirically substantiate our theoretical findings, we undertake a series of experiments designed to test the Transformer’s efficacy in continuous function approximation. Through careful experimental design, we contrast the model's performance in approximating continuous functions against its capabilities in modeling piecewise constant functions. The experiments, structured across varying dimensions of the model and the data, consistently highlight a marked discrepancy in performance. Specifically, we observe a considerable failure rate when Transformers are tasked with direct continuous function approximation, in contrast to their relative success with piecewise constant functions. These findings are further illustrated through qualitative analyses, including t-SNE visualizations, which vividly depict the model's struggle in capturing the intricacies of continuous function landscapes.

Implications and Future Directions

Our work prompts a reassessment of the perceived universal function approximation capabilities of Transformers. While their prowess in NLP and related areas is indisputable, our findings urge a refined understanding of their limitations in more general computational tasks. This insight opens avenues for future research dedicated to enhancing Transformers' expressivity, potentially through architectural innovations or hybrid modeling approaches. Expanding upon our foundational work, subsequent investigations could delve into discrete evaluations of Transformer components, aiming to pinpoint and remedy specific sources of the observed limitations in function approximation. Additionally, considering alternative paradigms of function approximation within neural architectures could yield novel insights, potentially guiding the development of more versatile and computationally efficient models.

In conclusion, while Transformers continue to dominate in their ability to model complex dependencies and patterns in data, our exploration reveals significant challenges in their application as universal function approximators for continuous spaces. By bringing to light these limitations and providing a pathway for future explorations, we contribute to the ongoing dialogue on the theoretical and practical boundaries of Transformer models, with the hope of catalyzing advancements that bolster their computational repertoire.