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Dimension-Free Minimax Rates for Learning Pairwise Interactions in Attention-Style Models

large-language-models › model-architecture

📄 Abstract

Abstract: We study the convergence rate of learning pairwise interactions in single-layer attention-style models, where tokens interact through a weight matrix and a non-linear activation function. We prove that the minimax rate is $M^{-\frac{2\beta}{2\beta+1}}$ with $M$ being the sample size, depending only on the smoothness $\beta$ of the activation, and crucially independent of token count, ambient dimension, or rank of the weight matrix. These results highlight a fundamental dimension-free statistical efficiency of attention-style nonlocal models, even when the weight matrix and activation are not separately identifiable and provide a theoretical understanding of the attention mechanism and its training.

Authors (4)

Shai Zucker

Xiong Wang

Fei Lu

Inbar Seroussi

Submitted

October 13, 2025

arXiv Category

stat.ML

arXiv PDF

Key Contributions

This paper proves a dimension-free minimax rate for learning pairwise interactions in single-layer attention-style models. The rate depends only on the sample size and activation smoothness, not on token count or ambient dimension, highlighting the fundamental statistical efficiency of attention mechanisms. It provides theoretical understanding of why attention models can be effective.

Business Value

Provides theoretical justification for the effectiveness of attention mechanisms, guiding the design and training of more efficient and powerful deep learning models, particularly in areas like NLP and potentially beyond.

Paper Metadata

Innovation Type

Theoretical Advancement

Deployment Feasibility

High (theoretical foundation)

Limitations Addressed

Addresses the lack of theoretical understanding regarding the statistical efficiency and convergence rates of attention-style models, particularly concerning their ability to learn interactions without being hindered by high dimensionality or large numbers of tokens.

Performance Gains

Theoretical proof of a dimension-free minimax rate, demonstrating superior statistical efficiency.

Technical Tags

attention modelspairwise interactionsminimax ratesdimension-free learningstatistical efficiencynonlocal modelsweight matrixactivation functionconvergence ratestheoretical analysis

Research Topics

Machine Learning TheoryDeep Learning TheoryStatistical LearningNeural Network Architectures

Methods & Architectures

Minimax rate analysisTheoretical proofsConvergence analysis Attention-style modelsSingle-layer attention

Applications & Tasks

Machine Learning Theory Natural Language Processing (implied by attention) Learning interactionsStatistical estimationConvergence analysis Learning pairwise interactions in attention modelsAnalyzing convergence rates

Related Fields

Machine Learning TheoryDeep LearningNatural Language ProcessingStatistical Theory

Keywords

attentionpairwise interactionsminimax ratedimension-freestatistical efficiencynonlocal modelsconvergencelearning theoryneural networkstransformersweight matrixactivation function

Academic Context

#Machine Learning Theory#Deep Learning Theory#Statistical Learning#Neural Network Architectures

Commercial Potential

Competitive Edge

Provides fundamental theoretical insights into the efficiency of attention mechanisms, explaining their success and guiding future architectural designs.

Resource Requirements

Compute Needs

Theoretical analysis, no direct implementation requirements mentioned.

Data Requirements

Theoretical analysis, not tied to specific datasets.

Deployment Constraints

Theoretical results provide understanding, not direct deployment guidance.

Scalability

Highlights inherent scalability benefits (dimension-free rates) of attention models.

Production Readiness

Maturity Level

Theoretical Research

Patent Potential

Low (primarily theoretical)

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