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Yayın Calculating the VC-dimension of decision trees(IEEE, 2009) Aslan, Özlem; Yıldız, Olcay Taner; Alpaydın, Ahmet İbrahim EthemWe propose an exhaustive search algorithm that calculates the VC-dimension of univariate decision trees with binary features. The VC-dimension of the univariate decision tree with binary features depends on (i) the VC-dimension values of the left and right subtrees, (ii) the number of inputs, and (iii) the number of nodes in the tree. From a training set of example trees whose VC-dimensions are calculated by exhaustive search, we fit a general regressor to estimate the VC-dimension of any binary tree. These VC-dimension estimates are then used to get VC-generalization bounds for complexity control using SRM in decision trees, i.e., pruning. Our simulation results shows that SRM-pruning using the estimated VC-dimensions finds trees that are as accurate as those pruned using cross-validation.Yayın Soft decision trees(IEEE, 2012) İrsoy, Ozan; Yıldız, Olcay Taner; Alpaydın, Ahmet İbrahim EthemWe discuss a novel decision tree architecture with soft decisions at the internal nodes where we choose both children with probabilities given by a sigmoid gating function. Our algorithm is incremental where new nodes are added when needed and parameters are learned using gradient-descent. We visualize the soft tree fit on a toy data set and then compare it with the canonical, hard decision tree over ten regression and classification data sets. Our proposed model has significantly higher accuracy using fewer nodes.Yayın Budding trees(IEEE Computer Soc, 2014-08-24) İrsoy, Ozan; Yıldız, Olcay Taner; Alpaydın, Ahmet İbrahim EthemWe propose a new decision tree model, named the budding tree, where a node can be both a leaf and an internal decision node. Each bud node starts as a leaf node, can then grow children, but then later on, if necessary, its children can be pruned. This contrasts with traditional tree construction algorithms that only grows the tree during the training phase, and prunes it in a separate pruning phase. We use a soft tree architecture and show that the tree and its parameters can be trained using gradient-descent. Our experimental results on regression, binary classification, and multi-class classification data sets indicate that our newly proposed model has better performance than traditional trees in terms of accuracy while inducing trees of comparable size.Yayın Regularizing soft decision trees(Springer, 2013) Yıldız, Olcay Taner; Alpaydın, Ahmet İbrahim EthemRecently, we have proposed a new decision tree family called soft decision trees where a node chooses both its left and right children with different probabilities as given by a gating function, different from a hard decision node which chooses one of the two. In this paper, we extend the original algorithm by introducing local dimension reduction via L-1 and L-2 regularization for feature selection and smoother fitting. We compare our novel approach with the standard decision tree algorithms over 27 classification data sets. We see that both regularized versions have similar generalization ability with less complexity in terms of number of nodes, where L-2 seems to work slightly better than L-1.Yayın Bagging soft decision trees(Springer Verlag, 2016) Yıldız, Olcay Taner; İrsoy, Ozan; Alpaydın, Ahmet İbrahim EthemThe decision tree is one of the earliest predictive models in machine learning. In the soft decision tree, based on the hierarchical mixture of experts model, internal binary nodes take soft decisions and choose both children with probabilities given by a sigmoid gating function. Hence for an input, all the paths to all the leaves are traversed and all those leaves contribute to the final decision but with different probabilities, as given by the gating values on the path. Tree induction is incremental and the tree grows when needed by replacing leaves with subtrees and the parameters of the newly-added nodes are learned using gradient-descent. We have previously shown that such soft trees generalize better than hard trees; here, we propose to bag such soft decision trees for higher accuracy. On 27 two-class classification data sets (ten of which are from the medical domain), and 26 regression data sets, we show that the bagged soft trees generalize better than single soft trees and bagged hard trees. This contribution falls in the scope of research track 2 listed in the editorial, namely, machine learning algorithms.Yayın Statistical tests using hinge/ε-sensitive loss(Springer-Verlag, 2013) Yıldız, Olcay Taner; Alpaydın, Ahmet İbrahim EthemStatistical tests used in the literature to compare algorithms use the misclassification error which is based on the 0/1 loss and square loss for regression. Kernel-based, support vector machine classifiers (regressors) however are trained to minimize the hinge (ε-sensitive) loss and hence they should not be assessed or compared in terms of the 0/1 (square loss) but with the loss measure they are trained to minimize. We discuss how the paired t test can use the hinge (ε-sensitive) loss and show in our experiments that doing that, we can detect differences that the test on error cannot detect, indicating higher power in distinguishing between the behavior of kernel-based classifiers (regressors). Such tests can be generalized to compare L > 2 algorithms.
