Nonparametric Modern Hopfield Models

TL;DR

This paper introduces a nonparametric interpretation of modern Hopfield models, leading to efficient sparse variants with theoretical guarantees and practical validation in memory retrieval and learning tasks.

Contribution

It provides a nonparametric perspective on modern Hopfield models, enabling the development of efficient sparse variants with strong theoretical properties.

Findings

Sparse modern Hopfield models have sub-quadratic complexity.

The models retain key properties like connection to transformer attention.

Empirical validation shows effectiveness in retrieval and learning tasks.

Abstract

We present a nonparametric interpretation for deep learning compatible modern Hopfield models and utilize this new perspective to debut efficient variants. 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. Interestingly, 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 \textit{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. 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 our framework in both synthetic and realistic settings for memory retrieval and learning tasks.