We conduct a systematic study of the approximation properties of Transformer for sequence modeling with long, sparse and complicated memory. We investigate the mechanisms through which different components of Transformer, such as the dot-product self-attention, positional encoding and feed-forward layer, affect its expressive power, and we study their combined effects through establishing explicit approximation rates. Our study reveals the roles of critical parameters in the Transformer, such as the number of layers and the number of attention heads. These theoretical insights are validated experimentally and offer natural suggestions for alternative architectures.
Recently, Transformer networks have redefined the state of the art in many NLP tasks. However, these models suffer from quadratic computational cost in the input sequence length $n$ to compute pairwise attention in each layer. This has prompted recent research into sparse Transformers that sparsify the connections in the attention layers. While empirically promising for long sequences, fundamental questions remain unanswered: Can sparse Transformers approximate any arbitrary sequence-to-sequence function, similar to their dense counterparts? How does the sparsity pattern and the sparsity level…
Indian Institute of Technology, Hyderabad（印度理工学院海得拉巴分校）；RWTH Aachen（亚琛工业大学）；Zerotha Research（泽罗萨研究院）；Cerence GmbH（赛轮思公司）；The University of Tokyo（东京大学）；IBM Research（IBM研究院）
The recent works proposing transformer-based models for graphs have proven the inadequacy of Vanilla Transformer for graph representation learning. To understand this inadequacy, there is a need to investigate if spectral analysis of the transformer will reveal insights into its expressive power. Similar studies already established that spectral analysis of Graph neural networks (GNNs) provides extra perspectives on their expressiveness. In this work, we systematically study and establish the link between the spatial and spectral domain in the realm of the transformer. We further provide a the…
On the Expressive Power of Floating-Point Transformers
Department of Artificial Intelligence, Korea University（韩国大学人工智能系）；Department of Mathematics and Statistics, Sejong University（世宗大学数学与统计学系）；Department of Mathematical Sciences, Gwangju Institute of Science and Technology（全州科学技术研究院数学科学系）
University of Antwerp（安特卫普大学）；Hasselt University（哈塞尔特大学）
We investigate the power of message-passing neural networks (MPNNs) in their capacity to transform the numerical features stored in the nodes of their input graphs. Our focus is on global expressive power, uniformly over all input graphs, or over graphs of bounded degree with features from a bounded domain. Accordingly, we introduce the notion of a global feature map transformer (GFMT). As a yardstick for expressiveness, we use a basic language for GFMTs, which we call MPLang. Every MPNN can be expressed in MPLang, and our results clarify to which extent the converse inclusion holds. We consid…
Vision-Transformers are widely used in various vision tasks. Meanwhile, there is another line of works starting with the MLP-mixer trying to achieve similar performance using mlp-based architectures. Interestingly, until now those mlp-based architectures have not been adapted for NLP tasks. Additionally, until now, mlp-based architectures have failed to achieve state-of-the-art performance in vision tasks. In this paper, we analyze the expressive power of mlp-based architectures in modeling dependencies between multiple different inputs simultaneously, and show an exponential gap between the a…
Efficient Transformers have been developed for long sequence modeling, due to their subquadratic memory and time complexity. Sparse Transformer is a popular approach to improving the efficiency of Transformers by restricting self-attention to locations specified by the predefined sparse patterns. However, leveraging sparsity may sacrifice expressiveness compared to full-attention, when important token correlations are multiple hops away. To combine advantages of both the efficiency of sparse transformer and the expressiveness of full-attention Transformer, we propose \textit{Diffuser}, a new s…
ETH Zürich（苏黎世联邦理工学院）；University of Pittsburgh（匹兹堡大学）；ICSI（国际计算机科学研究所）；UC Berkeley（加州大学伯克利分校）
We propose a novel method called Long Expressive Memory (LEM) for learning long-term sequential dependencies. LEM is gradient-based, it can efficiently process sequential tasks with very long-term dependencies, and it is sufficiently expressive to be able to learn complicated input-output maps. To derive LEM, we consider a system of multiscale ordinary differential equations, as well as a suitable time-discretization of this system. For LEM, we derive rigorous bounds to show the mitigation of the exploding and vanishing gradients problem, a well-known challenge for gradient-based recurrent seq…
Theoretical Constraints on the Expressive Power of $\mathsf{RoPE}$-based Tensor Attention Transformers
The University of Hong Kong（香港大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；The Simons Institute for the Theory of Computing at UC Berkeley（加州大学伯克利分校西蒙斯理论计算研究所）；Anhui University（安徽大学）
Normalizing flows provide a general mechanism for defining expressive probability distributions, only requiring the specification of a (usually simple) base distribution and a series of bijective transformations. There has been much recent work on normalizing flows, ranging from improving their expressive power to expanding their application. We believe the field has now matured and is in need of a unified perspective. In this review, we attempt to provide such a perspective by describing flows through the lens of probabilistic modeling and inference. We place special emphasis on the fundament…
Despite that going deep has proven successful in many neural architectures, the existing graph transformers are relatively shallow. In this work, we explore whether more layers are beneficial to graph transformers, and find that current graph transformers suffer from the bottleneck of improving performance by increasing depth. Our further analysis reveals the reason is that deep graph transformers are limited by the vanishing capacity of global attention, restricting the graph transformer from focusing on the critical substructure and obtaining expressive features. To this end, we propose a no…
Generative flows are promising tractable models for density modeling that define probabilistic distributions with invertible transformations. However, tractability imposes architectural constraints on generative flows, making them less expressive than other types of generative models. In this work, we study a previously overlooked constraint that all the intermediate representations must have the same dimensionality with the original data due to invertibility, limiting the width of the network. We tackle this constraint by augmenting the data with some extra dimensions and jointly learning a g…
Normalizing flows have received a great deal of recent attention as they allow flexible generative modeling as well as easy likelihood computation. While a wide variety of flow models have been proposed, there is little formal understanding of the representation power of these models. In this work, we study some basic normalizing flows and rigorously establish bounds on their expressive power. Our results indicate that while these flows are highly expressive in one dimension, in higher dimensions their representation power may be limited, especially when the flows have moderate depth.
KAIST（韩国科学技术院）；LG AI Research（LG人工智能研究院）；University of Illinois Chicago（伊利诺伊大学芝加哥分校）
We show that standard Transformers without graph-specific modifications can lead to promising results in graph learning both in theory and practice. Given a graph, we simply treat all nodes and edges as independent tokens, augment them with token embeddings, and feed them to a Transformer. With an appropriate choice of token embeddings, we prove that this approach is theoretically at least as expressive as an invariant graph network (2-IGN) composed of equivariant linear layers, which is already more expressive than all message-passing Graph Neural Networks (GNN). When trained on a large-scale…
Sequence models assign probabilities to variable-length sequences such as natural language texts. The ability of sequence models to capture temporal dependence can be characterized by the temporal scaling of correlation and mutual information. In this paper, we study the mutual information of recurrent neural networks (RNNs) including long short-term memories and self-attention networks such as Transformers. Through a combination of theoretical study of linear RNNs and empirical study of nonlinear RNNs, we find their mutual information decays exponentially in temporal distance. On the other ha…
Atmospheric Space and Technology Research Associates (ASTRA)（大气空间与技术研究中心 (ASTRA)）
We develop multiple Deep Learning (DL) models that advance the state-of-the-art predictions of the global auroral particle precipitation. We use observations from low Earth orbiting spacecraft of the electron energy flux to develop a model that improves global nowcasts (predictions at the time of observation) of the accelerated particles. Multiple Machine Learning (ML) modeling approaches are compared, including a novel multi-task model, models with tail- and distribution-based loss functions, and a spatio-temporally sparse 2D-convolutional model. We detail the data preparation process as well…
Despite the widespread adoption of Transformer models for NLP tasks, the expressive power of these models is not well-understood. In this paper, we establish that Transformer models are universal approximators of continuous permutation equivariant sequence-to-sequence functions with compact support, which is quite surprising given the amount of shared parameters in these models. Furthermore, using positional encodings, we circumvent the restriction of permutation equivariance, and show that Transformer models can universally approximate arbitrary continuous sequence-to-sequence functions on a …
MIT（麻省理工学院）；Broad Institute of MIT and Harvard（麻省理工学院和哈佛大学博德研究所）；Massachusetts General Hospital（麻省总医院）；Massachusetts General Hospital Research Institute（麻省总医院研究所）；Harvard Medical School（哈佛医学院）；Harvard-MIT Division of Health Sciences and Technology（哈佛-麻省理工卫生科学与技术部）
Supervised machine learning applications in health care are often limited due to a scarcity of labeled training data. To mitigate this effect of small sample size, we introduce a pre-training approach, Patient Contrastive Learning of Representations (PCLR), which creates latent representations of ECGs from a large number of unlabeled examples. The resulting representations are expressive, performant, and practical across a wide spectrum of clinical tasks. We develop PCLR using a large health care system with over 3.2 million 12-lead ECGs, and demonstrate substantial improvements across multipl…
The goals of this paper are two-fold. The first goal is to serve as an expository tutorial on the working of deep learning models which emphasizes geometrical intuition about the reasons for success of deep learning. The second goal is to complement the current results on the expressive power of deep learning models and their loss surfaces with novel insights and results. In particular, we describe how deep neural networks carve out manifolds especially when the multiplication neurons are introduced. Multiplication is used in dot products and the attention mechanism and it is employed in capsu…
Google Research（谷歌研究）；TReNDS Center, Georgia State University（TReNDS中心，佐治亚州立大学）；Maynooth University（迈诺特大学）；Georgia State University/Georgia Institute of Technology/Emory University Center for Translational Research in Neuroimaging and Data Science (TReNDS Center)（佐治亚州立大学/佐治亚理工学院/埃默里大学神经影像与数据科学转化研究中心（TReNDS中心））；Department of Computer Science, Maynooth University（迈诺特大学计算机科学系）
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）
The identification of nonlinear dynamics from observations is essential for the alignment of the theoretical ideas and experimental data. The last, in turn, is often corrupted by the side effects and noise of different natures, so probabilistic approaches could give a more general picture of the process. At the same time, high-dimensional probabilities modeling is a challenging and data-intensive task. In this paper, we establish a parallel between the dynamical systems modeling and language modeling tasks. We propose a transformer-based model that incorporates geometrical properties of the da…
Transformers are being used extensively across several sequence modeling tasks. Significant research effort has been devoted to experimentally probe the inner workings of Transformers. However, our conceptual and theoretical understanding of their power and inherent limitations is still nascent. In particular, the roles of various components in Transformers such as positional encodings, attention heads, residual connections, and feedforward networks, are not clear. In this paper, we take a step towards answering these questions. We analyze the computational power as captured by Turing-complete…
Max-Planck-Institut für Quantenoptik（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心（MCQST））；Dahlem Center for Complex Quantum Systems, Freie Universität Berlin（柏林自由大学达勒姆复杂量子系统中心）；Freie Universität Berlin（柏林自由大学）
Tensor-network techniques have enjoyed outstanding success in physics, and have recently attracted attention in machine learning, both as a tool for the formulation of new learning algorithms and for enhancing the mathematical understanding of existing methods. Inspired by these developments, and the natural correspondence between tensor networks and probabilistic graphical models, we provide a rigorous analysis of the expressive power of various tensor-network factorizations of discrete multivariate probability distributions. These factorizations include non-negative tensor-trains/MPS, which …
Existing analyses of the expressive capacity of Transformer models have required excessively deep layers for data memorization, leading to a discrepancy with the Transformers actually used in practice. This is primarily due to the interpretation of the softmax function as an approximation of the hardmax function. By clarifying the connection between the softmax function and the Boltzmann operator, we prove that a single layer of self-attention with low-rank weight matrices possesses the capability to perfectly capture the context of an entire input sequence. As a consequence, we show that one-…
Transformers have recently taken the center stage in language modeling after LSTM's were considered the dominant model architecture for a long time. In this project, we investigate the performance of the Transformer architectures-BERT and Transformer-XL for the language modeling task. We use a sub-word model setting with the Finnish language and compare it to the previous State of the art (SOTA) LSTM model. BERT achieves a pseudo-perplexity score of 14.5, which is the first such measure achieved as far as we know. Transformer-XL improves upon the perplexity score to 73.58 which is 27\% better …
