Transformers are Universal In-context Learners (2408.01367v2)

Abstract: Transformers are deep architectures that define "in-context mappings" which enable predicting new tokens based on a given set of tokens (such as a prompt in NLP applications or a set of patches for a vision transformer). In this work, we study in particular the ability of these architectures to handle an arbitrarily large number of context tokens. To mathematically, uniformly address their expressivity, we consider the case that the mappings are conditioned on a context represented by a probability distribution of tokens which becomes discrete for a finite number of these. The relevant notion of smoothness then corresponds to continuity in terms of the Wasserstein distance between these contexts. We demonstrate that deep transformers are universal and can approximate continuous in-context mappings to arbitrary precision, uniformly over compact token domains. A key aspect of our results, compared to existing findings, is that for a fixed precision, a single transformer can operate on an arbitrary (even infinite) number of tokens. Additionally, it operates with a fixed embedding dimension of tokens (this dimension does not increase with precision) and a fixed number of heads (proportional to the dimension). The use of MLPs between multi-head attention layers is also explicitly controlled. We consider both unmasked attentions (as used for the vision transformer) and masked causal attentions (as used for NLP and time series applications). We tackle the causal setting leveraging a space-time lifting to analyze causal attention as a mapping over probability distributions of tokens.

Overview of "Transformers are Universal In-context Learners"

The paper "Transformers are Universal In-context Learners" by Takashi Furuya, Maarten V. de Hoop, and Gabriel Peyré, presents a formal and rigorous analysis of transformer models, specifically focusing on their capacity to act as universal in-context learners. The research aims to theoretically substantiate the expressive power of transformers, especially in handling an arbitrarily large number of context tokens and approximating in-context mappings with arbitrary precision.

Transformer Architecture and In-context Learning

Transformers, since their introduction by Vaswani et al., have significantly impacted NLP and computer vision through their self-attention mechanism. These models leverage contexts of variable length, making them highly effective for tasks requiring considerable contextual understanding. The paper focuses on providing a mathematical formalism to model these in-context mappings with the support of probability distributions over tokens. This approach allows the examination of their expressivity over potentially infinite context lengths while maintaining fixed token embedding dimensions and a controlled number of heads.

Main Contributions

1. Measure-theoretic Framework:

The authors propose viewing transformers as operators over probability distributions of tokens, which they refer to as "in-context mappings." This perspective converts the typical set of tokens into a continuous representation, facilitating the analysis of transformers' expressivity. By utilizing the Wasserstein distance for measuring smoothness and continuity, the paper sets a foundation for understanding transformers' behavior in a more generalized manner that transcends the finite token sets commonly used in practice.

2. Universality Theorem:

The core contribution is the proof that deep transformers are universal approximators of continuous in-context mappings. Specifically, for any continuous in-context function over compact token domains, there exists a transformer architecture that can approximate this function to any desired precision. This result applies uniformly over any number, including an infinite number of tokens, without requiring an increase in the embedding dimension or the number of heads proportional to the precision demanded.

3. Decomposition and Approximation Techniques:

The proof strategy employs a decomposition into simpler attention heads and affine transformations, then demonstrates that any such in-context mapping can be approximated by these components. Through a recursive composition of these mappings, implemented with fixed dimensions and fixed heads, the authors show how a deep transformer achieves the desired approximation.

Implications and Future Directions

Practical Implications:

The theoretical findings provide a robust foundation for the design and implementation of transformer models in actual applications. With the assurance that transformers can approximate any continuous in-context mapping, practitioners can be confident in applying these architectures to diverse and complex tasks in NLP, vision, and beyond. Moreover, the results suggest that transformers do not need arbitrarily large dimensions to achieve high precision, which has significant implications for the efficient deployment of these models in resource-constrained environments.

Theoretical Implications:

The formalization of transformers via measure-theoretic in-context mappings promotes a deeper understanding of their fundamental capabilities. This opens avenues for further research into the quantitative aspects of this approximation, especially in terms of convergence rates and the impact of token distribution smoothness. The current paper also sets the stage for exploring how these theoretical insights might be extended to other architectures and learning paradigms.

Future Work:

Further research might focus on the quantitative properties of these approximations, utilizing metrics like Wasserstein distances to derive more practical bounds on the depth and width of required architectures. Additionally, extending the framework to handle masked attention mechanisms would align the theoretical models closer with those used in practice, especially in autoregressive tasks common in NLP. Investigating the balance between the number of heads and dimensions could yield more efficient architectures, improving the trade-offs between model complexity and performance.

Conclusion

This paper provides a comprehensive and rigorous exploration of the expressive power of transformers, showcasing their ability to act as universal in-context learners. By embedding transformer operations in a measure-theoretic framework and proving their universality, the research significantly advances both theoretical understanding and practical application of these models. With promising directions for future investigation, this work lays a solid groundwork for the continued evolution and optimization of transformer-based architectures.