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[2602.15503] Approximation Theory for Lipschitz Continuous Transformers
Summary
This paper explores the approximation theory for Lipschitz continuous Transformers, establishing a theoretical foundation for their stability and robustness in safety-sensitive applications.
Why It Matters
With the increasing deployment of Transformers in critical areas, ensuring their stability and robustness is essential. This research provides a theoretical framework for designing Transformers that maintain Lipschitz continuity, which is crucial for reliable performance in various applications.
Key Takeaways
- Introduces a class of gradient-descent-type Transformers that are Lipschitz continuous by design.
- Proves a universal approximation theorem for Lipschitz-constrained function spaces.
- Adopts a measure-theoretic approach to interpret Transformers as operators on probability measures.
- Ensures stability without sacrificing the expressivity of the model.
- Provides a rigorous foundation for robust Transformer architectures in safety-critical settings.
Computer Science > Machine Learning arXiv:2602.15503 (cs) [Submitted on 17 Feb 2026] Title:Approximation Theory for Lipschitz Continuous Transformers Authors:Takashi Furuya, Davide Murari, Carola-Bibiane Schönlieb View a PDF of the paper titled Approximation Theory for Lipschitz Continuous Transformers, by Takashi Furuya and 2 other authors View PDF HTML (experimental) Abstract:Stability and robustness are critical for deploying Transformers in safety-sensitive settings. A principled way to enforce such behavior is to constrain the model's Lipschitz constant. However, approximation-theoretic guarantees for architectures that explicitly preserve Lipschitz continuity have yet to be established. In this work, we bridge this gap by introducing a class of gradient-descent-type in-context Transformers that are Lipschitz-continuous by construction. We realize both MLP and attention blocks as explicit Euler steps of negative gradient flows, ensuring inherent stability without sacrificing expressivity. We prove a universal approximation theorem for this class within a Lipschitz-constrained function space. Crucially, our analysis adopts a measure-theoretic formalism, interpreting Transformers as operators on probability measures, to yield approximation guarantees independent of token count. These results provide a rigorous theoretical foundation for the design of robust, Lipschitz continuous Transformer architectures. Subjects: Machine Learning (cs.LG); Machine Learning (stat.ML) Ci...
