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Add to EdgeJerry Yao-Chieh Hu
Nonparametric Modern Hopfield Models
Apr 05, 2024
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 \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 -- 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.
Uniform Memory Retrieval with Larger Capacity for Modern Hopfield Models
Apr 04, 2024We propose a two-stage memory retrieval dynamics for modern Hopfield models, termed $\mathtt{U\text{-}Hop}$, with enhanced memory capacity. Our key contribution is a learnable feature map $\Phi$ which transforms the Hopfield energy function into a kernel space. This transformation ensures convergence between the local minima of energy and the fixed points of retrieval dynamics within the kernel space. Consequently, the kernel norm induced by $\Phi$ serves as a novel similarity measure. It utilizes the stored memory patterns as learning data to enhance memory capacity across all modern Hopfield models. Specifically, we accomplish this by constructing a separation loss $\mathcal{L}_\Phi$ that separates the local minima of kernelized energy by separating stored memory patterns in kernel space. Methodologically, $\mathtt{U\text{-}Hop}$ memory retrieval process consists of: \textbf{(Stage~I.)} minimizing separation loss for a more uniformed memory (local minimum) distribution, followed by \textbf{(Stage~II.)} standard Hopfield energy minimization for memory retrieval. This results in a significant reduction of possible meta-stable states in the Hopfield energy function, thus enhancing memory capacity by preventing memory confusion. Empirically, with real-world datasets, we demonstrate that $\mathtt{U\text{-}Hop}$ outperforms all existing modern Hopfield models and SOTA similarity measures, achieving substantial improvements in both associative memory retrieval and deep learning tasks.
Outlier-Efficient Hopfield Layers for Large Transformer-Based Models
Apr 04, 2024We introduce an Outlier-Efficient Modern Hopfield Model (termed $\mathtt{OutEffHop}$) and use it to address the outlier-induced challenge of quantizing gigantic transformer-based models. Our main contribution is a novel associative memory model facilitating \textit{outlier-efficient} associative memory retrievals. Interestingly, this memory model manifests a model-based interpretation of an outlier-efficient attention mechanism ($\text{Softmax}_1$): it is an approximation of the memory retrieval process of $\mathtt{OutEffHop}$. Methodologically, this allows us to debut novel outlier-efficient Hopfield layers a powerful attention alternative with superior post-quantization performance. Theoretically, the Outlier-Efficient Modern Hopfield Model retains and improves the desirable properties of the standard modern Hopfield models, including fixed point convergence and exponential storage capacity. Empirically, we demonstrate the proposed model's efficacy across large-scale transformer-based and Hopfield-based models (including BERT, OPT, ViT and STanHop-Net), benchmarking against state-of-the-art methods including $\mathtt{Clipped\_Softmax}$ and $\mathtt{Gated\_Attention}$. Notably, $\mathtt{OutEffHop}$ achieves on average $\sim$22+\% reductions in both average kurtosis and maximum infinity norm of model outputs accross 4 models.
BiSHop: Bi-Directional Cellular Learning for Tabular Data with Generalized Sparse Modern Hopfield Model
Apr 04, 2024We introduce the \textbf{B}i-Directional \textbf{S}parse \textbf{Hop}field Network (\textbf{BiSHop}), a novel end-to-end framework for deep tabular learning. BiSHop handles the two major challenges of deep tabular learning: non-rotationally invariant data structure and feature sparsity in tabular data. Our key motivation comes from the recent established connection between associative memory and attention mechanisms. Consequently, BiSHop uses a dual-component approach, sequentially processing data both column-wise and row-wise through two interconnected directional learning modules. Computationally, these modules house layers of generalized sparse modern Hopfield layers, a sparse extension of the modern Hopfield model with adaptable sparsity. Methodologically, BiSHop facilitates multi-scale representation learning, capturing both intra-feature and inter-feature interactions, with adaptive sparsity at each scale. Empirically, through experiments on diverse real-world datasets, we demonstrate that BiSHop surpasses current SOTA methods with significantly less HPO runs, marking it a robust solution for deep tabular learning.
On Computational Limits of Modern Hopfield Models: A Fine-Grained Complexity Analysis
Feb 14, 2024We investigate the computational limits of the memory retrieval dynamics of modern Hopfield models from the fine-grained complexity analysis. Our key contribution is the characterization of a phase transition behavior in the efficiency of all possible modern Hopfield models based on the norm of patterns. Specifically, we establish an upper bound criterion for the norm of input query patterns and memory patterns. Only below this criterion, sub-quadratic (efficient) variants of the modern Hopfield model exist, assuming the Strong Exponential Time Hypothesis (SETH). To showcase our theory, we provide a formal example of efficient constructions of modern Hopfield models using low-rank approximation when the efficient criterion holds. This includes a derivation of a lower bound on the computational time, scaling linearly with $\Max\{$# of stored memory patterns, length of input query sequence$\}$. In addition, we prove its memory retrieval error bound and exponential memory capacity.
STanHop: Sparse Tandem Hopfield Model for Memory-Enhanced Time Series Prediction
Dec 28, 2023We present STanHop-Net (Sparse Tandem Hopfield Network) for multivariate time series prediction with memory-enhanced capabilities. At the heart of our approach is STanHop, a novel Hopfield-based neural network block, which sparsely learns and stores both temporal and cross-series representations in a data-dependent fashion. In essence, STanHop sequentially learn temporal representation and cross-series representation using two tandem sparse Hopfield layers. In addition, StanHop incorporates two additional external memory modules: a Plug-and-Play module and a Tune-and-Play module for train-less and task-aware memory-enhancements, respectively. They allow StanHop-Net to swiftly respond to certain sudden events. Methodologically, we construct the StanHop-Net by stacking STanHop blocks in a hierarchical fashion, enabling multi-resolution feature extraction with resolution-specific sparsity. Theoretically, we introduce a sparse extension of the modern Hopfield model (Generalized Sparse Modern Hopfield Model) and show that it endows a tighter memory retrieval error compared to the dense counterpart without sacrificing memory capacity. Empirically, we validate the efficacy of our framework on both synthetic and real-world settings.
Beyond PID Controllers: PPO with Neuralized PID Policy for Proton Beam Intensity Control in Mu2e
Dec 28, 2023We introduce a novel Proximal Policy Optimization (PPO) algorithm aimed at addressing the challenge of maintaining a uniform proton beam intensity delivery in the Muon to Electron Conversion Experiment (Mu2e) at Fermi National Accelerator Laboratory (Fermilab). Our primary objective is to regulate the spill process to ensure a consistent intensity profile, with the ultimate goal of creating an automated controller capable of providing real-time feedback and calibration of the Spill Regulation System (SRS) parameters on a millisecond timescale. We treat the Mu2e accelerator system as a Markov Decision Process suitable for Reinforcement Learning (RL), utilizing PPO to reduce bias and enhance training stability. A key innovation in our approach is the integration of a neuralized Proportional-Integral-Derivative (PID) controller into the policy function, resulting in a significant improvement in the Spill Duty Factor (SDF) by 13.6%, surpassing the performance of the current PID controller baseline by an additional 1.6%. This paper presents the preliminary offline results based on a differentiable simulator of the Mu2e accelerator. It paves the groundwork for real-time implementations and applications, representing a crucial step towards automated proton beam intensity control for the Mu2e experiment.
On Sparse Modern Hopfield Model
Sep 22, 2023
We introduce the sparse modern Hopfield model as a sparse extension of the modern Hopfield model. Like its dense counterpart, the sparse modern Hopfield model equips a memory-retrieval dynamics whose one-step approximation corresponds to the sparse attention mechanism. Theoretically, our key contribution is a principled derivation of a closed-form sparse Hopfield energy using the convex conjugate of the sparse entropic regularizer. Building upon this, we derive the sparse memory retrieval dynamics from the sparse energy function and show its one-step approximation is equivalent to the sparse-structured attention. Importantly, we provide a sparsity-dependent memory retrieval error bound which is provably tighter than its dense analog. The conditions for the benefits of sparsity to arise are therefore identified and discussed. In addition, we show that the sparse modern Hopfield model maintains the robust theoretical properties of its dense counterpart, including rapid fixed point convergence and exponential memory capacity. Empirically, we use both synthetic and real-world datasets to demonstrate that the sparse Hopfield model outperforms its dense counterpart in many situations.
Feature Programming for Multivariate Time Series Prediction
Jun 09, 2023
We introduce the concept of programmable feature engineering for time series modeling and propose a feature programming framework. This framework generates large amounts of predictive features for noisy multivariate time series while allowing users to incorporate their inductive bias with minimal effort. The key motivation of our framework is to view any multivariate time series as a cumulative sum of fine-grained trajectory increments, with each increment governed by a novel spin-gas dynamical Ising model. This fine-grained perspective motivates the development of a parsimonious set of operators that summarize multivariate time series in an abstract fashion, serving as the foundation for large-scale automated feature engineering. Numerically, we validate the efficacy of our method on several synthetic and real-world noisy time series datasets.