Random Access to Advice Strings and Collapsing Results

We propose a model of computation where a Turing machine is given random access to an advice string. With random access, an advice string of exponential length becomes meaningful for polynomially bounded complexity classes. We compare the power of complexity classes under this model. It gives a more stringent notion than the usual model of computation with relativization. Under this model of random access, we prove that there exist advice strings such that the polynomial-time hierarchy PH and parity polynomial-time ⊕P all collapse to P. Our main proof technique uses the decision tree lower bounds for constant depth circuits [Y], [C], [Ha], and the algebraic machinery of Razborov [R] and Smolensky [S].     

Omnivariate decision trees.

Univariate decision trees at each decision node consider the value of only one feature leading to axis-aligned splits. In a linear multivariate decision tree, each decision node divides the input space into two with a hyperplane. In a nonlinear multivariate tree, a multilayer perceptron at each node divides the input space arbitrarily, at the expense of increased complexity and higher risk of overfitting. We propose omnivariate trees where the decision node may be univariate, linear, or nonlinear depending on the outcome of comparative statistical tests on accuracy thus matching automatically the complexity of the node with the subproblem defined by the data reaching that node. Such an architecture frees the designer from choosing the appropriate node type, doing model selection automatically at each node. Our simulation results indicate that such a decision tree induction method generalizes better than trees with the same types of nodes everywhere and induces small trees.     

Robust Polynomials and Quantum Algorithms

We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main results are: *For every n-bit Boolean function f there is an n-variate polynomial p of degree O(n) that robustly approximates it, in the sense that p(x) remains close to f(x) if we slightly vary each of the n inputs of the polynomial. *There is an O(n)-query quantum algorithm that robustly recovers n noisy input bits. Hence every n-bit function can be quantum computed with O(n) queries in the presence of noise. This contrasts with the classical model of Feige et al., where functions such as parity need Θ(n log n) queries. We give several extensions and applications of these results.     

Model selection in omnivariate decision trees using Structural Risk Minimization

As opposed to trees that use a single type of decision node, an omnivariate decision tree contains nodes of different types. We propose to use Structural Risk Minimization (SRM) to choose between node types in omnivariate decision tree construction to match the complexity of a node to the complexity of the data reaching that node. In order to apply SRM for model selection, one needs the VC-dimension of the candidate models. In this paper, we first derive the VC-dimension of the univariate model, and estimate the VC-dimension of all three models (univariate, linear multivariate or quadratic multivariate) experimentally. Second, we compare SRM with other model selection techniques including Akaike’s Information Criterion (AIC), Bayesian Information Criterion (BIC) and cross-validation (CV) on standard datasets from the UCI and Delve repositories. We see that SRM induces omnivariate trees that have a small percentage of multivariate nodes close to the root and they generalize more or at least as accurately as those constructed using other model selection techniques.     

Classifiability-based omnivariate decision trees

Top-down induction of decision trees is a simple and powerful method of pattern classification. In a decision tree, each node partitions the available patterns into two or more sets. New nodes are created to handle each of the resulting partitions and the process continues. A node is considered terminal if it satisfies some stopping criteria (for example, purity, i.e., all patterns at the node are from a single class). Decision trees may be univariate, linear multivariate, or nonlinear multivariate depending on whether a single attribute, a linear function of all the attributes, or a nonlinear function of all the attributes is used for the partitioning at each node of the decision tree. Though nonlinear multivariate decision trees are the most powerful, they are more susceptible to the risks of overfitting. In this paper, we propose to perform model selection at each decision node to build omnivariate decision trees. The model selection is done using a novel classifiability measure that captures the possible sources of misclassification with relative ease and is able to accurately reflect the complexity of the subproblem at each node. The proposed approach is fast and does not suffer from as high a computational burden as that incurred by typical model selection algorithms. Empirical results over 26 data sets indicate that our approach is faster and achieves better classification accuracy compared to statistical model select algorithms.     

决策树技术及其当前研究方向

介绍了决策树归纳技术及其发展过程,阐述了目前最流行的决策树技术的基本方法及简化决策树的主要方法。同时介绍了决策树技术面临的挑战,目前在与其他技术结合,寻找新的构造方法,简化方法,训练与检验数据的大小及特性与其本身特性的关系,不确定环境下决策,时间复杂度与准确性之间的关系,软件实现等方面的研究,以及它在工程上的应用,展望了它的未来发展前景。     

一种模型决策森林算法

随机森林(RF)具有抗噪能力强,预测准确率高,能够处理高维数据等优点,因此在机器学习领域得到了广泛的应用。模型决策树(MDT)是一种加速的决策树算法,虽然能够提高决策树算法的训练效率,但是随着非纯伪叶结点规模的增大,模型决策树的精度也在下降。针对上述问题,提出了一种模型决策森林算法(MDF)以提高模型决策树的分类精度。MDF算法将MDT作为基分类器,利用随机森林的思想,生成多棵模型决策树。算法首先通过旋转矩阵得到不同的样本子集,然后在这些样本子集上训练出多棵不同的模型决策树,再将这些树通过投票的方式进行集成,最后根据得到的模型决策森林给出分类结果。在标准数据集上的实验结果表明,提出的模型决策森林在分类精度上明显优于模型决策树算法,并且MDF在树的数量较少时也能取到不错的精度,避免了因树的数量增加时间复杂度增高的问题。     

Case study of inaccuracies in the granulation of decision trees

Cybernetics studies information process in the context of interaction with physical systems. Because such information is sometimes vague and exhibits complex interactions; it can only be discerned using approximate representations. Machine learning provides solutions that create approximate models of information and decision trees are one of its main components. However, decision trees are susceptible to information overload and can get overly complex when a large amount of data is inputted in them. Granulation of decision tree remedies this problem by providing the essential structure of the decision tree, which can decrease its utility. To evaluate the relationship that exists between granulation and decision tree complexity, data uncertainty and prediction accuracy, the deficiencies obtained by nursing homes during annual inspections were taken as a case study. Using rough sets, three forms of granulation were performed: (1) attribute grouping, (2) removing insignificant attributes and (3) removing uncertain records. Attribute grouping significantly reduces tree complexity without having any strong effect upon data consistency and accuracy. On the other hand, removing insignificant features decrease data consistency and tree complexity, while increasing the error in prediction. Finally, decrease in the uncertainty of the dataset results in an increase in accuracy and has no impact on tree complexity.     

Creating Decision Trees from Rules using RBDT‐1

Most of the methods that generate decision trees for a specific problem use the examples of data instances in the decision tree–generation process. This article proposes a method called RBDT‐1—rule‐based decision tree—for learning a decision tree from a set of decision rules that cover the data instances rather than from the data instances themselves. The goal is to create on demand a short and accurate decision tree from a stable or dynamically changing set of rules. The rules could be generated by an expert, by an inductive rule learning program that induces decision rules from the examples of decision instances such as AQ‐type rule induction programs, or extracted from a tree generated by another method, such as the ID3 or C4.5. In terms of tree complexity (number of nodes and leaves in the decision tree), RBDT‐1 compares favorably with AQDT‐1 and AQDT‐2, which are methods that create decision trees from rules. RBDT‐1 also compares favorably with ID3 while it is as effective as C4.5 where both (ID3 and C4.5) are well‐known methods that generate decision trees from data examples. Experiments show that the classification accuracies of the decision trees produced by all methods under comparison are indistinguishable.     

Evolving Fuzzy Min–Max Neural Network Based Decision Trees for Data Stream Classification

Learning from data streams is a challenging task which demands a learning algorithm with several high quality features. In addition to space complexity and speed requirements needed for processing the huge volume of data which arrives at high speed, the learning algorithm must have a good balance between stability and plasticity. This paper presents a new approach to induce incremental decision trees on streaming data. In this approach, the internal nodes contain trainable split tests. In contrast with traditional decision trees in which a single attribute is selected as the split test, each internal node of the proposed approach contains a trainable function based on multiple attributes, which not only provides the flexibility needed in the stream context, but also improves stability. Based on this approach, we propose evolving fuzzy min–max decision tree (EFMMDT) learning algorithm in which each internal node of the decision tree contains an evolving fuzzy min–max neural network. EFMMDT splits the instance space non-linearly based on multiple attributes which results in much smaller and shallower decision trees. The extensive experiments reveal that the proposed algorithm achieves much better precision in comparison with the state-of-the-art decision tree learning algorithms on the benchmark data streams, especially in the presence of concept drift.     

Regular Languages of Thin Trees

An infinite tree is called thin if it contains only countably many infinite branches. Thin trees can be seen as intermediate structures between infinite words and infinite trees. In this work we investigate properties of regular languages of thin trees. Our main tool is an algebra suitable for thin trees. Using this framework we characterize various classes of regular languages: commutative, open in the standard topology, and definable in weak MSO logic among all trees. We also show that in various meanings thin trees are not as rich as all infinite trees. In particular we observe a collapse of the parity index to the level (1, 3) and a collapse of the topological complexity to co-analytic sets. Moreover, a gap property is shown: a regular language of thin trees is either weak MSO-definable among all trees or co-analytic-complete.     

用粗糙集理论建立决策树的一种方法

在数据挖掘中，分期是一个很重要的问题，有很多流行的分类器可以创建决策树木产生类模型。本文介绍了通过信息增益或熵的比较来构造一棵决策树的数桩挖掘算法思想，给出了用粗糙集理论构造决策树的一种方法，并用曲面造型方面的实例说明了决策树的生成过程。通过与ID3方法的比较，该种方法可以降低决策树的复杂性，优化决策树的结构，能挖掘较好的规则信息。     

The random subspace method for constructing decision forests

Much of previous attention on decision trees focuses on the splitting criteria and optimization of tree sizes. The dilemma between overfitting and achieving maximum accuracy is seldom resolved. A method to construct a decision tree based classifier is proposed that maintains highest accuracy on training data and improves on generalization accuracy as it grows in complexity. The classifier consists of multiple trees constructed systematically by pseudorandomly selecting subsets of components of the feature vector, that is, trees constructed in randomly chosen subspaces. The subspace method is compared to single-tree classifiers and other forest construction methods by experiments on publicly available datasets, where the method's superiority is demonstrated. We also discuss independence between trees in a forest and relate that to the combined classification accuracy     

Decision tree approximations of Boolean functions

Decision trees are popular representations of Boolean functions. We show that, given an alternative representation of a Boolean function f, say as a read-once branching program, one can find a decision tree T which approximates f to any desired amount of accuracy. Moreover, the size of the decision tree is at most that of the smallest decision tree which can represent f and this construction can be obtained in quasi-polynomial time. We also extend this result to the case where one has access only to a source of random evaluations of the Boolean function f instead of a complete representation. In this case, we show that a similar approximation can be obtained with any specified amount of confidence (as opposed to the absolute certainty of the former case.) This latter result implies proper PAC-learnability of decision trees under the uniform distribution without using membership queries.     

Fisher’s decision tree

Univariate decision trees are classifiers currently used in many data mining applications. This classifier discovers partitions in the input space via hyperplanes that are orthogonal to the axes of attributes, producing a model that can be understood by human experts. One disadvantage of univariate decision trees is that they produce complex and inaccurate models when decision boundaries are not orthogonal to axes. In this paper we introduce the Fisher’s Tree, it is a classifier that takes advantage of dimensionality reduction of Fisher’s linear discriminant and uses the decomposition strategy of decision trees, to come up with an oblique decision tree. Our proposal generates an artificial attribute that is used to split the data in a recursive way.The Fisher’s decision tree induces oblique trees whose accuracy, size, number of leaves and training time are competitive with respect to other decision trees reported in the literature. We use more than ten public available data sets to demonstrate the effectiveness of our method.     

A metric normalization of tree edit distance

Traditional normalized tree edit distances do not satisfy the triangle inequality. We present a metric normalization method for tree edit distance, which results in a new normalized tree edit distance fulfilling the triangle inequality, under the condition that the weight function is a metric over the set of elementary edit operations with all costs of insertions/deletions having the same weight. We prove that the new distance, in the range [0, 1], is a genuine metric as a simple function of the sizes of two ordered labeled trees and the tree edit distance between them, which can be directly computed through tree edit distance with the same complexity. Based on an efficient algorithm to represent digits as ordered labeled trees, we show that the normalized tree edit metric can provide slightly better results than other existing methods in handwritten digit recognition experiments using the approximating and eliminating search algorithm (AESA) algorithm.     

Decision forest: Twenty years of research

A decision tree is a predictive model that recursively partitions the covariate’s space into subspaces such that each subspace constitutes a basis for a different prediction function. Decision trees can be used for various learning tasks including classification, regression and survival analysis. Due to their unique benefits, decision trees have become one of the most powerful and popular approaches in data science. Decision forest aims to improve the predictive performance of a single decision tree by training multiple trees and combining their predictions. This paper provides an introduction to the subject by explaining how a decision forest can be created and when it is most valuable. In addition, we are reviewing some popular methods for generating the forest, fusion the individual trees’ outputs and thinning large decision forests.     

Noise-Tolerant Occam Algorithms and Their Applications to Learning Decision Trees

In the distribution-independent model of concept learning of Valiant, Angluin and Laird have introduced a formal model of noise process, called classification noise process, to study how to compensate for randomly introduced errors, or noise, in classifying the example data. In this article, we investigate the problem of designing efficient learning algorithms in the presence of classification noise. First, we develop a technique of building efficient robust learning algorithms, called noise-tolerant Occam algorithms, and show that using them, one can construct a polynomial-time algorithm for learning a class of Boolean functions in the presence of classification noise. Next, as an instance of such problems of learning in the presence of classification noise, we focus on the learning problem of Boolean functions represented by decision trees. We present a noise-tolerant Occam algorithm for k-DL (the class of decision lists with conjunctive clauses of size at most k at each decision introduced by Rivest) and hence conclude that k-DL is polynomially learnable in the presence of classification noise. Further, we extend the noise-tolerant Occam algorithm for k-DL to one for r-DT (the class of decision trees of rank at most r introduced by Ehrenfeucht and Haussler) and conclude that r-DT is polynomially learnable in the presence of classification noise.     

Functional Trees

In the context of classification problems, algorithms that generate multivariate trees are able to explore multiple representation languages by using decision tests based on a combination of attributes. In the regression setting, model trees algorithms explore multiple representation languages but using linear models at leaf nodes. In this work we study the effects of using combinations of attributes at decision nodes, leaf nodes, or both nodes and leaves in regression and classification tree learning. In order to study the use of functional nodes at different places and for different types of modeling, we introduce a simple unifying framework for multivariate tree learning. This framework combines a univariate decision tree with a linear function by means of constructive induction. Decision trees derived from the framework are able to use decision nodes with multivariate tests, and leaf nodes that make predictions using linear functions. Multivariate decision nodes are built when growing the tree, while functional leaves are built when pruning the tree. We experimentally evaluate a univariate tree, a multivariate tree using linear combinations at inner and leaf nodes, and two simplified versions restricting linear combinations to inner nodes and leaves. The experimental evaluation shows that all functional trees variants exhibit similar performance, with advantages in different datasets. In this study there is a marginal advantage of the full model. These results lead us to study the role of functional leaves and nodes. We use the bias-variance decomposition of the error, cluster analysis, and learning curves as tools for analysis. We observe that in the datasets under study and for classification and regression, the use of multivariate decision nodes has more impact in the bias component of the error, while the use of multivariate decision leaves has more impact in the variance component.