Graph-structured data arise ubiquitously in many application domains. A fundamental problem is to quantify their similarities. Graph kernels are often used for this purpose, which decompose graphs into substructures and compare these substructures. However, most of the existing graph kernels do not have the property of scale-adaptivity, i.e., they cannot compare graphs at multiple levels of granularities. Many real-world graphs such as molecules exhibit structure at varying levels of granularities. To tackle this problem, we propose a new graph kernel called Tree++ in this paper. At the heart of Tree++ is a graph kernel called the path-pattern graph kernel. The path-pattern graph kernel first builds a truncated BFS tree rooted at each vertex and then uses paths from the root to every vertex in the truncated BFS tree as features to represent graphs. The path-pattern graph kernel can only capture graph similarity at fine granularities. In order to capture graph similarity at coarse granularities, we incorporate a new concept called super path into it. The super path contains truncated BFS trees rooted at the vertices in a path. Our evaluation on a variety of real-world graphs demonstrates that Tree++ achieves the best classification accuracy compared with previous graph kernels.
In this paper, we consider the nonparametric least square regression in a Reproducing Kernel Hilbert Space (RKHS). We propose a new randomized algorithm that has optimal generalization error bounds with respect to the square loss, closing a long-standing gap between upper and lower bounds. Moreover, we show that our algorithm has faster finite-time and asymptotic rates on problems where the Bayes risk with respect to the square loss is small. We state our results using standard tools from the theory of least square regression in RKHSs, namely, the decay of the eigenvalues of the associated int…
Kernel ridge regression (KRR) has recently attracted renewed interest due to its potential for explaining the transient effects, such as double descent, that emerge during neural network training. In this work, we study how the alignment between the target function and the kernel affects the performance of the KRR. We focus on the truncated KRR (TKRR) which utilizes an additional parameter that controls the spectral truncation of the kernel matrix. We show that for polynomial alignment, there is an \emph{over-aligned} regime, in which TKRR can achieve a faster rate than what is achievable by f…
Kernel ridge regression (KRR) is a well-known and popular nonparametric regression approach with many desirable properties, including minimax rate-optimality in estimating functions that belong to common reproducing kernel Hilbert spaces (RKHS). The approach, however, is computationally intensive for large data sets, due to the need to operate on a dense $n \times n$ kernel matrix, where $n$ is the sample size. Recently, various approximation schemes for solving KRR have been considered, and some analyzed. Some approaches such as Nyström approximation and sketching have been shown to preserve …
Structural interpretability in SVMs with truncated orthogonal polynomial kernels
Dpto. de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid（数学系，科学学院，自治大学马德里）；Instituto de Ciencias Matemáticas (ICMAT), Campus de Cantoblanco UAM（数学科学研究所（ICMAT），UAM坎托布兰科校区）；Dpto. de Biociencias, Facultad de Ciencias Biomédicas y de la Salud, Universidad Europea de Madrid（生物科学系，生物医学与健康科学学院，欧洲大学马德里）；Dpto. de Física y Matemáticas, Facultad de Ciencias, Universidad de Alcalá（物理与数学系，科学学院，阿尔卡拉大学）
In this paper, we propose the distributed tree kernels (DTK) as a novel method to reduce time and space complexity of tree kernels. Using a linear complexity algorithm to compute vectors for trees, we embed feature spaces of tree fragments in low-dimensional spaces where the kernel computation is directly done with dot product. We show that DTKs are faster, correlate with tree kernels, and obtain a statistically similar performance in two natural language processing tasks.
Kernels ensuing from tree ensembles such as random forest (RF) or gradient boosted trees (GBT), when used for kernel learning, have been shown to be competitive to their respective tree ensembles (particularly in higher dimensional scenarios). On the other hand, it has been also shown that performance of the kernel algorithms depends on the degree of the kernel-target alignment. However, the kernel-target alignment for kernel learning based on the tree ensembles has not been investigated and filling this gap is the main goal of our work. Using the eigenanalysis of the kernel matrix, we demonst…
Efficient Test-Time Inference via Deterministic Exploration of Truncated Decoding Trees
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；ETH Zurich（苏黎世联邦理工学院）；AI Center Tübingen（图宾根人工智能中心）；University of Tübingen（图宾根大学）；ELLIS Institute Tübingen（图宾根ELLIS研究所）；IMPRS-IS（图宾根IMPRS-IS）
Tree based ensembles such as Breiman's random forest (RF) and Gradient Boosted Trees (GBT) can be interpreted as implicit kernel generators, where the ensuing proximity matrix represents the data-driven tree ensemble kernel. Kernel perspective on the RF has been used to develop a principled framework for theoretical investigation of its statistical properties. Recently, it has been shown that the kernel interpretation is germane to other tree-based ensembles e.g. GBTs. However, practical utility of the links between kernels and the tree ensembles has not been widely explored and systematically…
The Graduate University for Advanced Studies, SOKENDAI（综合研究大学院大学（SOKENDAI））；National Institute of Informatics（国立信息学研究所）
A soft tree is an actively studied variant of a decision tree that updates splitting rules using the gradient method. Although soft trees can take various architectures, their impact is not theoretically well known. In this paper, we formulate and analyze the Neural Tangent Kernel (NTK) induced by soft tree ensembles for arbitrary tree architectures. This kernel leads to the remarkable finding that only the number of leaves at each depth is relevant for the tree architecture in ensemble learning with an infinite number of trees. In other words, if the number of leaves at each depth is fixed, t…
Graduate School of Information Science and Technology, The University of Tokyo（东京大学信息科学与技术研究生院）
The kernel method is a potential approach to analyzing structured data such as sequences, trees, and graphs; however, unordered trees have not been investigated extensively. Kimura et al. (2011) proposed a kernel function for unordered trees on the basis of their subpaths, which are vertical substructures of trees responsible for hierarchical information in them. Their kernel exhibits practically good performance in terms of accuracy and speed; however, linear-time computation is not guaranteed theoretically, unlike the case of the other unordered tree kernel proposed by Vishwanathan and Smola…
Lucerne University of Applied Sciences and Arts（卢塞恩应用科学与艺术大学）
We introduce a novel boosting algorithm called `KTBoost' which combines kernel boosting and tree boosting. In each boosting iteration, the algorithm adds either a regression tree or reproducing kernel Hilbert space (RKHS) regression function to the ensemble of base learners. Intuitively, the idea is that discontinuous trees and continuous RKHS regression functions complement each other, and that this combination allows for better learning of functions that have parts with varying degrees of regularity such as discontinuities and smooth parts. We empirically show that KTBoost significantly outp…
The Graduate University for Advanced Studies, SOKENDAI（综合研究大学院大学）；National Institute of Informatics（国立信息学研究所）
In practical situations, the tree ensemble is one of the most popular models along with neural networks. A soft tree is a variant of a decision tree. Instead of using a greedy method for searching splitting rules, the soft tree is trained using a gradient method in which the entire splitting operation is formulated in a differentiable form. Although ensembles of such soft trees have been used increasingly in recent years, little theoretical work has been done to understand their behavior. By considering an ensemble of infinite soft trees, this paper introduces and studies the Tree Neural Tange…
The recently proposed "generalized min-max" (GMM) kernel can be efficiently linearized, with direct applications in large-scale statistical learning and fast near neighbor search. The linearized GMM kernel was extensively compared in with linearized radial basis function (RBF) kernel. On a large number of classification tasks, the tuning-free GMM kernel performs (surprisingly) well compared to the best-tuned RBF kernel. Nevertheless, one would naturally expect that the GMM kernel ought to be further improved if we introduce tuning parameters. In this paper, we study three simple constructions …
Baidu Research USA（百度研究院美国分院（或直接保留“Baidu Research USA”））
While tree methods have been popular in practice, researchers and practitioners are also looking for simple algorithms which can reach similar accuracy of trees. In 2010, (Ping Li UAI'10) developed the method of "abc-robust-logitboost" and compared it with other supervised learning methods on datasets used by the deep learning literature. In this study, we propose a series of "tunable GMM kernels" which are simple and perform largely comparably to tree methods on the same datasets. Note that "abc-robust-logitboost" substantially improved the original "GDBT" in that (a) it developed a tree-spli…
In this work, we demonstrate the advantage of the pGMM (``powered generalized min-max'') kernel in the context of (ridge) regression. In recent prior studies, the pGMM kernel has been extensively evaluated for classification tasks, for logistic regression, support vector machines, as well as deep neural networks. In this paper, we provide an experimental study on ridge regression, to compare the pGMM kernel regression with the ordinary ridge linear regression as well as the RBF kernel ridge regression. Perhaps surprisingly, even without a tuning parameter (i.e., $p=1$ for the power parameter o…
One-class Classification (OCC) is an area of machine learning which addresses prediction based on unbalanced datasets. Basically, OCC algorithms achieve training by means of a single class sample, with potentially some additional counter-examples. The current OCC models give satisfaction in terms of performance, but there is an increasing need for the development of interpretable models. In the present work, we propose a one-class model which addresses concerns of both performance and interpretability. Our hybrid OCC method relies on density estimation as part of a tree-based learning algorith…
How can we identify the training examples that contribute most to the prediction of a tree ensemble? In this paper, we introduce TREX, an explanation system that provides instance-attribution explanations for tree ensembles, such as random forests and gradient boosted trees. TREX builds on the representer point framework previously developed for explaining deep neural networks. Since tree ensembles are non-differentiable, we define a kernel that captures the structure of the specific tree ensemble. By using this kernel in kernel logistic regression or a support vector machine, TREX builds a su…
McGill University, Department of Mathematics and Statistics（麦吉尔大学数学与统计学系）；Université Marie et Louis Pasteur, CNRS, LmB (UMR 6623)（玛丽·路易斯·帕斯特大学、法国国家科学研究中心、LmB（UMR 6623））
Imperial College London（帝国理工学院）；BASF SE（巴斯夫股份公司）；Cooperative State University（合作州立大学）
Tree ensembles can be well-suited for black-box optimization tasks such as algorithm tuning and neural architecture search, as they achieve good predictive performance with little or no manual tuning, naturally handle discrete feature spaces, and are relatively insensitive to outliers in the training data. Two well-known challenges in using tree ensembles for black-box optimization are (i) effectively quantifying model uncertainty for exploration and (ii) optimizing over the piece-wise constant acquisition function. To address both points simultaneously, we propose using the kernel interpretat…
A Kernel Fisher Discriminant Analysis-Based Tree Ensemble Classifier: KFDA Forest
School of Industrial Management Engineering, Korea University（韩国大学工业管理工程学院）；School of Mechanical Engineering, Chungnam National University（Chungnam国立大学机械工程学院）
University of Bonn（波恩大学）；Fraunhofer IAIS（弗劳恩霍夫智能分析与信息系统研究所）
The Weisfeiler-Lehman graph kernels are among the most prevalent graph kernels due to their remarkable time complexity and predictive performance. Their key concept is based on an implicit comparison of neighborhood representing trees with respect to equality (i.e., isomorphism). This binary valued comparison is, however, arguably too rigid for defining suitable similarity measures over graphs. To overcome this limitation, we propose a generalization of Weisfeiler-Lehman graph kernels which takes into account the similarity between trees rather than equality. We achieve this using a specifical…
State Key Lab of Software Development Environment（软件开发环境国家重点实验室）；Beihang University（北京航空航天大学）
In deep neural networks, better results can often be obtained by increasing the complexity of previously developed basic models. However, it is unclear whether there is a way to boost performance by decreasing the complexity of such models. Here, based on an optimization method, we investigate the feasibility of improving graph classification performance while simplifying the model learning process. Inspired by progress in structural information assessment, we optimize the given data sample from graphs to encoding trees. In particular, we minimize the structural entropy of the transformed enco…
Tensor network compressibility of convolutional models
Multiverse Computing（多元宇宙计算公司）；Donostia International Physics Center（多诺斯蒂亚国际物理中心）；Institute for Advanced Studies in Basic Sciences (IASBS)（基础科学高级研究所）；Ikerbasque Foundation for Science（伊克尔巴斯克科学基金会）
