Nonparametric Modern Hopfield Models

2024-04-05

arXiv.org

AI Summary

Modern Hopfield networks suffer from theoretical and practical gaps in efficient memory modeling, particularly regarding scalability, parameterization, and integration with deep learning architectures. Method: We propose a nonparametric framework that formulates memory storage and retrieval as query-memory pair-wise nonparametric regression, naturally enabling end-to-end deep learning integration. We introduce the first sparse-structured modern Hopfield model, achieving subquadratic time complexity without an explicit energy function—thereby inheriting Transformer-style attention, fixed-point convergence guarantees, and exponential memory capacity. Efficiency is systematically unified across variants via linear/random masking, Top-K selection, and positive random feature expansion. Results: Rigorous theoretical analysis, alongside experiments on synthetic and real-world tasks, validates both theoretical soundness and empirical effectiveness. Our work establishes the first principled foundation and practical methodology for sparse, nonparametric modern Hopfield networks.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Mixture of Experts (MoE)
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We present a nonparametric construction for deep learning compatible modern Hopfield models and utilize this framework to debut an efficient variant. Our key contribution stems from interpreting the memory storage and retrieval processes in modern Hopfield models as a nonparametric regression problem subject to a set of query-memory pairs. Crucially, our framework not only recovers the known results from the original dense modern Hopfield model but also fills the void in the literature regarding efficient modern Hopfield models, by introducing extit{sparse-structured} modern Hopfield models with sub-quadratic complexity. We establish that this sparse model inherits the appealing theoretical properties of its dense analogue -- connection with transformer attention, fixed point convergence and exponential memory capacity -- even without knowing details of the Hopfield energy function. Additionally, we showcase the versatility of our framework by constructing a family of modern Hopfield models as extensions, including linear, random masked, top-$K$ and positive random feature modern Hopfield models. Empirically, we validate the efficacy of our framework in both synthetic and realistic settings.

Problem

• Develop nonparametric deep learning compatible Hopfield models
• Introduce sparse-structured Hopfield models with sub-quadratic complexity
• Extend framework to construct diverse modern Hopfield model variants

Innovation

• Nonparametric regression for Hopfield models
• Sparse-structured models with sub-quadratic complexity
• Family of Hopfield models including linear variants