\textit{Attention} computes the dependency between representations, and it encourages the model to focus on the important selective features. Attention-based models, such as Transformer and graph attention network (GAT), are widely utilized for sequential data and graph-structured data. This paper suggests a new interpretation and generalized structure of the attention in Transformer and GAT. For the attention in Transformer and GAT, we derive that the attention is a product of two parts: 1) the RBF kernel to measure the similarity of two instances and 2) the exponential of $L^{2}$ norm to compute the importance of individual instances. From this decomposition, we generalize the attention in three ways. First, we propose implicit kernel attention with an implicit kernel function instead of manual kernel selection. Second, we generalize $L^{2}$ norm as the $L^{p}$ norm. Third, we extend our attention to structured multi-head attention. Our generalized attention shows better performance on classification, translation, and regression tasks.
By using the framework of Determinantal Point Processes (DPPs), some theoretical results concerning the interplay between diversity and regularization can be obtained. In this paper we show that sampling subsets with kDPPs results in implicit regularization in the context of ridgeless Kernel Regression. Furthermore, we leverage the common setup of state-of-the-art DPP algorithms to sample multiple small subsets and use them in an ensemble of ridgeless regressions. Our first empirical results indicate that ensemble of ridgeless regressors can be interesting to use for datasets including redunda…
TU Dortmund University（多特蒙德工业大学）；Washington University in St. Louis（圣路易斯华盛顿大学）；TU Darmstadt（达姆施塔特工业大学）
Non-linear kernel methods can be approximated by fast linear ones using suitable explicit feature maps allowing their application to large scale problems. We investigate how convolution kernels for structured data are composed from base kernels and construct corresponding feature maps. On this basis we propose exact and approximative feature maps for widely used graph kernels based on the kernel trick. We analyze for which kernels and graph properties computation by explicit feature maps is feasible and actually more efficient. In particular, we derive approximative, explicit feature maps for …
Kernel methods have achieved very good performance on large scale regression and classification problems, by using the Nyström method and preconditioning techniques. The Nyström approximation -- based on a subset of landmarks -- gives a low rank approximation of the kernel matrix, and is known to provide a form of implicit regularization. We further elaborate on the impact of sampling diverse landmarks for constructing the Nyström approximation in supervised as well as unsupervised kernel methods. By using Determinantal Point Processes for sampling, we obtain additional theoretical results con…
Kernels are powerful and versatile tools in machine learning and statistics. Although the notion of universal kernels and characteristic kernels has been studied, kernel selection still greatly influences the empirical performance. While learning the kernel in a data driven way has been investigated, in this paper we explore learning the spectral distribution of kernel via implicit generative models parametrized by deep neural networks. We called our method Implicit Kernel Learning (IKL). The proposed framework is simple to train and inference is performed via sampling random Fourier features.…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；ETH Zürich（苏黎世联邦理工学院）
Modern implicit generative models such as generative adversarial networks (GANs) are generally known to suffer from issues such as instability, uninterpretability, and difficulty in assessing their performance. If we see these implicit models as dynamical systems, some of these issues are caused by being unable to control their behavior in a meaningful way during the course of training. In this work, we propose a theoretically grounded method to guide the training trajectories of GANs by augmenting the GAN loss function with a kernel-based regularization term that controls local and global dis…
State University of New York at Buffalo（纽约州立大学布法罗分校）
Manifold learning is a fundamental problem in machine learning with numerous applications. Most of the existing methods directly learn the low-dimensional embedding of the data in some high-dimensional space, and usually lack the flexibility of being directly applicable to down-stream applications. In this paper, we propose the concept of implicit manifold learning, where manifold information is implicitly obtained by learning the associated heat kernel. A heat kernel is the solution of the corresponding heat equation, which describes how "heat" transfers on the manifold, thus containing ample…
Tsinghua University（清华大学）；University of Toronto（多伦多大学）
Recent progress in variational inference has paid much attention to the flexibility of variational posteriors. One promising direction is to use implicit distributions, i.e., distributions without tractable densities as the variational posterior. However, existing methods on implicit posteriors still face challenges of noisy estimation and computational infeasibility when applied to models with high-dimensional latent variables. In this paper, we present a new approach named Kernel Implicit Variational Inference that addresses these challenges. As far as we know, for the first time implicit va…
Synthetic data generation has become a key ingredient for training machine learning procedures, addressing tasks such as data augmentation, analysing privacy-sensitive data, or visualising representative samples. Assessing the quality of such synthetic data generators hence has to be addressed. As (deep) generative models for synthetic data often do not admit explicit probability distributions, classical statistical procedures for assessing model goodness-of-fit may not be applicable. In this paper, we propose a principled procedure to assess the quality of a synthetic data generator. The proc…
Modern implicit generative models such as generative adversarial networks (GANs) are generally known to suffer from issues such as instability, uninterpretability, and difficulty in assessing their performance. If we see these implicit models as dynamical systems, some of these issues are caused by being unable to control their behavior in a meaningful way during the course of training. In this work, we propose a theoretically grounded method to guide the training trajectories of GANs by augmenting the GAN loss function with a kernel-based regularization term that controls local and global dis…
Technion（以色列理工学院）；Microsoft Research（微软研究院）；Toyota Technological Institute at Chicago（芝加哥丰田工业大学）；Princeton University（普林斯顿大学）
We provide a detailed asymptotic study of gradient flow trajectories and their implicit optimization bias when minimizing the exponential loss over "diagonal linear networks". This is the simplest model displaying a transition between "kernel" and non-kernel ("rich" or "active") regimes. We show how the transition is controlled by the relationship between the initialization scale and how accurately we minimize the training loss. Our results indicate that some limit behaviors of gradient descent only kick in at ridiculous training accuracies (well beyond $10^{-100}$). Moreover, the implicit bia…
Georgia Institute of Technology（佐治亚理工学院）；University of Michigan（密歇根大学）
Attention mechanisms take an expectation of a data representation with respect to probability weights. This creates summary statistics that focus on important features. Recently, (Martins et al. 2020, 2021) proposed continuous attention mechanisms, focusing on unimodal attention densities from the exponential and deformed exponential families: the latter has sparse support. (Farinhas et al. 2021) extended this to use Gaussian mixture attention densities, which are a flexible class with dense support. In this paper, we extend this to two general flexible classes: kernel exponential families and…
Transformer is a powerful architecture that achieves superior performance on various sequence learning tasks, including neural machine translation, language understanding, and sequence prediction. At the core of the Transformer is the attention mechanism, which concurrently processes all inputs in the streams. In this paper, we present a new formulation of attention via the lens of the kernel. To be more precise, we realize that the attention can be seen as applying kernel smoother over the inputs with the kernel scores being the similarities between inputs. This new formulation gives us a bet…
In this work, we draw attention to a connection between skill-based models of game outcomes and Gaussian process classification models. The Gaussian process perspective enables a) a principled way of dealing with uncertainty and b) rich models, specified through kernel functions. Using this connection, we tackle the problem of predicting outcomes of football matches between national teams. We develop a player kernel that relates any two football matches through the players lined up on the field. This makes it possible to share knowledge gained from observing matches between clubs (available in…
State University of New York at Buffalo（纽约州立大学布法罗分校）
Learning with kernels is an important concept in machine learning. Standard approaches for kernel methods often use predefined kernels that require careful selection of hyperparameters. To mitigate this burden, we propose in this paper a framework to construct and learn a data-dependent kernel based on random features and implicit spectral distributions that are parameterized by deep neural networks. The constructed network (called KernelNet) can be applied to deep generative modeling in various scenarios, including two popular learning paradigms in deep generative models, MMD-GAN and implicit…
Google Research（谷歌研究院）；IBM T.J. Watson Research Center（IBM T.J. 沃森研究中心）
In order to fully utilize "big data", it is often required to use "big models". Such models tend to grow with the complexity and size of the training data, and do not make strong parametric assumptions upfront on the nature of the underlying statistical dependencies. Kernel methods fit this need well, as they constitute a versatile and principled statistical methodology for solving a wide range of non-parametric modelling problems. However, their high computational costs (in storage and time) pose a significant barrier to their widespread adoption in big data applications. We propose an algori…
University of Cambridge（剑桥大学）；Google Research（谷歌研究院）；Columbia University（哥伦比亚大学）；The Alan Turing Institute（艾伦·图灵研究所）
The problem of efficient approximation of a linear operator induced by the Gaussian or softmax kernel is often addressed using random features (RFs) which yield an unbiased approximation of the operator's result. Such operators emerge in important applications ranging from kernel methods to efficient Transformers. We propose parameterized, positive, non-trigonometric RFs which approximate Gaussian and softmax-kernels. In contrast to traditional RF approximations, parameters of these new methods can be optimized to reduce the variance of the approximation, and the optimum can be expressed in cl…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；University of California, Los Angeles（加州大学洛杉矶分校）
Multi-head attention empowers the recent success of transformers, the state-of-the-art models that have achieved remarkable success in sequence modeling and beyond. These attention mechanisms compute the pairwise dot products between the queries and keys, which results from the use of unnormalized Gaussian kernels with the assumption that the queries follow a mixture of Gaussian distribution. There is no guarantee that this assumption is valid in practice. In response, we first interpret attention in transformers as a nonparametric kernel regression. We then propose the FourierFormer, a new cl…
In machine learning, no data point stands alone. We believe that context is an underappreciated concept in many machine learning methods. We propose Attention-Based Clustering (ABC), a neural architecture based on the attention mechanism, which is designed to learn latent representations that adapt to context within an input set, and which is inherently agnostic to input sizes and number of clusters. By learning a similarity kernel, our method directly combines with any out-of-the-box kernel-based clustering approach. We present competitive results for clustering Omniglot characters and includ…
Chair of Statistical Field Theory, École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院统计场论主席）；Laboratory of Computational Neuroscience, École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院计算神经科学实验室）
Random Feature (RF) models are used as efficient parametric approximations of kernel methods. We investigate, by means of random matrix theory, the connection between Gaussian RF models and Kernel Ridge Regression (KRR). For a Gaussian RF model with $P$ features, $N$ data points, and a ridge $λ$, we show that the average (i.e. expected) RF predictor is close to a KRR predictor with an effective ridge $\tildeλ$. We show that $\tildeλ > λ$ and $\tildeλ \searrow λ$ monotonically as $P$ grows, thus revealing the implicit regularization effect of finite RF sampling. We then compare the risk (i.e. t…
Georgetown University（乔治城大学）；Meng Cheng Family Medicine Clinic（孟诚家庭医学科诊所）；Tri-Service General Hospital（三军总医院）；Fu Jen Catholic University（辅仁大学）
We present an end-to-end, interpretable, deep-learning architecture to learn a graph kernel that predicts the outcome of chronic disease drug prescription. This is achieved through a deep metric learning collaborative with a Support Vector Machine objective using a graphical representation of Electronic Health Records. We formulate the predictive model as a binary graph classification problem with an adaptive learned graph kernel through novel cross-global attention node matching between patient graphs, simultaneously computing on multiple graphs without training pair or triplet generation. Re…
Tsinghua University（清华大学）；University of Toronto（多伦多大学）
Recently there have been increasing interests in learning and inference with implicit distributions (i.e., distributions without tractable densities). To this end, we develop a gradient estimator for implicit distributions based on Stein's identity and a spectral decomposition of kernel operators, where the eigenfunctions are approximated by the Nyström method. Unlike the previous works that only provide estimates at the sample points, our approach directly estimates the gradient function, thus allows for a simple and principled out-of-sample extension. We provide theoretical results on the er…
Recent studies identified that sequential Recommendation is improved by the attention mechanism. By following this development, we propose Relation-Aware Kernelized Self-Attention (RKSA) adopting a self-attention mechanism of the Transformer with augmentation of a probabilistic model. The original self-attention of Transformer is a deterministic measure without relation-awareness. Therefore, we introduce a latent space to the self-attention, and the latent space models the recommendation context from relation as a multivariate skew-normal distribution with a kernelized covariance matrix from c…
Instituto Superior Técnico & Instituto de Telecomunicações, Universidade de Lisboa, Portugal（里斯本大学理工学院及电信研究所，葡萄牙）；Champalimaud Research, Lisbon, Portugal（Champalimaud研究，葡萄牙里斯本）；TransPerfect, Lisbon, Portugal（TransPerfect，葡萄牙里斯本）；Carnegie Mellon University, Pittsburgh, USA（卡内基梅隆大学，美国匹兹堡）
Graph kernels have recently emerged as a promising approach for tackling the graph similarity and learning tasks at the same time. In this paper, we propose a general framework for designing graph kernels. The proposed framework capitalizes on the well-known message passing scheme on graphs. The kernels derived from the framework consist of two components. The first component is a kernel between vertices, while the second component is a kernel between graphs. The main idea behind the proposed framework is that the representations of the vertices are implicitly updated using an iterative proced…
The attention module, which is a crucial component in Transformer, cannot scale efficiently to long sequences due to its quadratic complexity. Many works focus on approximating the dot-then-exponentiate softmax function in the original attention, leading to sub-quadratic or even linear-complexity Transformer architectures. However, we show that these methods cannot be applied to more powerful attention modules that go beyond the dot-then-exponentiate style, e.g., Transformers with relative positional encoding (RPE). Since in many state-of-the-art models, relative positional encoding is used as…
Carnegie Mellon University（卡内基梅隆大学）；Bosch Center for AI（博世人工智能中心）
Neural networks with sinusoidal activations have been proposed as an alternative to networks with traditional activation functions. Despite their promise, particularly for learning implicit models, their training behavior is not yet fully understood, leading to a number of empirical design choices that are not well justified. In this work, we first propose a simplified version of such sinusoidal neural networks, which allows both for easier practical implementation and simpler theoretical analysis. We then analyze the behavior of these networks from the neural tangent kernel perspective and de…
Action recognition greatly benefits motion understanding in video analysis. Recurrent networks such as long short-term memory (LSTM) networks are a popular choice for motion-aware sequence learning tasks. Recently, a convolutional extension of LSTM was proposed, in which input-to-hidden and hidden-to-hidden transitions are modeled through convolution with a single kernel. This implies an unavoidable trade-off between effectiveness and efficiency. Herein, we propose a new enhancement to convolutional LSTM networks that supports accommodation of multiple convolutional kernels and layers. This re…
Turku Centre for Computer Science – TUCS（图尔库计算机科学中心）；Åbo Akademi University（奥博学术大学）；Goethe University（歌德大学）
We introduce an attention-based Bi-LSTM for Chinese implicit discourse relations and demonstrate that modeling argument pairs as a joint sequence can outperform word order-agnostic approaches. Our model benefits from a partial sampling scheme and is conceptually simple, yet achieves state-of-the-art performance on the Chinese Discourse Treebank. We also visualize its attention activity to illustrate the model's ability to selectively focus on the relevant parts of an input sequence.
To date, most state-of-the-art sequence modeling architectures use attention to build generative models for language based tasks. Some of these models use all the available sequence tokens to generate an attention distribution which results in time complexity of $O(n^2)$. Alternatively, they utilize depthwise convolutions with softmax normalized kernels of size $k$ acting as a limited-window self-attention, resulting in time complexity of $O(k{\cdot}n)$. In this paper, we introduce Time-aware Large Kernel (TaLK) Convolutions, a novel adaptive convolution operation that learns to predict the si…
The increasing availability of implicit feedback datasets has raised the interest in developing effective collaborative filtering techniques able to deal asymmetrically with unambiguous positive feedback and ambiguous negative feedback. In this paper, we propose a principled kernel-based collaborative filtering method for top-N item recommendation with implicit feedback. We present an efficient implementation using the linear kernel, and we show how to generalize it to kernels of the dot product family preserving the efficiency. We also investigate on the elements which influence the sparsity …
University of Cambridge（剑桥大学）；MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）
We propose Adaptive Deep Kernel Fitting with Implicit Function Theorem (ADKF-IFT), a novel framework for learning deep kernel Gaussian processes (GPs) by interpolating between meta-learning and conventional deep kernel learning. Our approach employs a bilevel optimization objective where we meta-learn generally useful feature representations across tasks, in the sense that task-specific GP models estimated on top of such features achieve the lowest possible predictive loss on average. We solve the resulting nested optimization problem using the implicit function theorem (IFT). We show that our…
In pursuit of faster computation, Efficient Transformers demonstrate an impressive variety of approaches -- models attaining sub-quadratic attention complexity can utilize a notion of sparsity or a low-rank approximation of inputs to reduce the number of attended keys; other ways to reduce complexity include locality-sensitive hashing, key pooling, additional memory to store information in compacted or hybridization with other architectures, such as CNN. Often based on a strong mathematical basis, kernelized approaches allow for the approximation of attention with linear complexity while retai…
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Transformers are expensive to train due to the quadratic time and space complexity in the self-attention mechanism. On the other hand, although kernel machines suffer from the same computation bottleneck in pairwise dot products, several approximation schemes have been successfully incorporated to considerably reduce their computational cost without sacrificing too much accuracy. In this work, we leverage the computation methods for kernel machines to alleviate the high computational cost and introduce Skyformer, which replaces the softmax structure with a Gaussian kernel to stabilize the mode…
Attention Kernels for Learning Maps Between Heavy-Tailed Measures
California Institute of Technology（加州理工学院）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of California, Berkeley（加州大学伯克利分校）；ICSI（国际计算机科学研究所）
Faculty of Mathematics and Computer Science, Jagiellonian University（雅盖隆大学数学与计算机科学学院）；Ardigen SA（Ardigen SA公司）
Not all supervised learning problems are described by a pair of a fixed-size input tensor and a label. In some cases, especially in medical image analysis, a label corresponds to a bag of instances (e.g. image patches), and to classify such bag, aggregation of information from all of the instances is needed. There have been several attempts to create a model working with a bag of instances, however, they are assuming that there are no dependencies within the bag and the label is connected to at least one instance. In this work, we introduce Self-Attention Attention-based MIL Pooling (SA-AbMILP…
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Amazon Alexa AI（亚马逊Alexa人工智能部门）
The massive amount of trainable parameters in the pre-trained language models (PLMs) makes them hard to be deployed to multiple downstream tasks. To address this issue, parameter-efficient transfer learning methods have been proposed to tune only a few parameters during fine-tuning while freezing the rest. This paper looks at existing methods along this line through the \textit{kernel lens}. Motivated by the connection between self-attention in transformer-based PLMs and kernel learning, we propose \textit{kernel-wise adapters}, namely \textit{Kernel-mix}, that utilize the kernel structure in …
Google（谷歌）；University of Cambridge（剑桥大学）；Alan Turing Institute（艾伦·图灵研究所）；DeepMind（深度思维）
We introduce Performers, Transformer architectures which can estimate regular (softmax) full-rank-attention Transformers with provable accuracy, but using only linear (as opposed to quadratic) space and time complexity, without relying on any priors such as sparsity or low-rankness. To approximate softmax attention-kernels, Performers use a novel Fast Attention Via positive Orthogonal Random features approach (FAVOR+), which may be of independent interest for scalable kernel methods. FAVOR+ can be also used to efficiently model kernelizable attention mechanisms beyond softmax. This representat…
