Top-down induction of decision trees has been observed to suffer from the inadequate functioning of the pruning phase. In particular, it is known that the size of the resulting tree grows linearly with the sample size, even though the accuracy of the tree does not improve. Reduced Error Pruning is an algorithm that has been used as a representative technique in attempts to explain the problems of decision tree learning. In this paper we present analyses of Reduced Error Pruning in three different settings. First we study the basic algorithmic properties of the method, properties that hold independent of the input decision tree and pruning examples. Then we examine a situation that intuitively should lead to the subtree under consideration to be replaced by a leaf node, one in which the class label and attribute values of the pruning examples are independent of each other. This analysis is conducted under two different assumptions. The general analysis shows that the pruning probability of a node fitting pure noise is bounded by a function that decreases exponentially as the size of the tree grows. In a specific analysis we assume that the examples are distributed uniformly to the tree. This assumption lets us approximate the number of subtrees that are pruned because they do not receive any pruning examples. This paper clarifies the different variants of the Reduced Error Pruning algorithm, brings new insight to its algorithmic properties, analyses the algorithm with less imposed assumptions than before, and includes the previously overlooked empty subtrees to the analysis.

Stacked generalization is a general method of using a high-level model to combine lower-level models to achieve greater predictive accuracy. In this paper we address two crucial issues which have been considered to be a `black art' in classification tasks ever since the introduction of stacked generalization in 1992 by Wolpert: the type of generalizer that is suitable to derive the higher-level model, and the kind of attributes that should be used as its input. We find that best results are obtained when the higher-level model combines the confidence (and not just the predictions) of the lower-level ones. We demonstrate the effectiveness of stacked generalization for combining three different types of learning algorithms for classification tasks. We also compare the performance of stacked generalization with majority vote and published results of arcing and bagging.

We describe a four-week series of assignments in an undergraduate AI course at a liberal arts college developing a supervised learning solution to the problem of classifying Twitter accounts as either a person account or a non-person account (e.g. organization or spambot). This problem employs real data in an ongoing research project by the first author, yet is accessible to students with limited programming expertise.The students were able to experience a complete cycle of creating a machine learning solution: exploring raw data,creating a training set, engineering features, comparing different classifiers, evaluating the results, and performing erroranalysis. We received positive feedback from the students and intend to refine the assignment and make it available (together with the created training data) for use by the research community.

We present research in decision tree analysis that studies a data set and finds new patterns that were not obvious using statistical methods. Our method is applied to a database of accommodative esotropic patients. Accommodative esotropia is an eye disease that when left untreated leads to blindness. Patients whose muscles deteriorate often need corrective surgery, since less invasive methods of treatment tend to fail in these patients. Using a learn and prune methodology, decision tree analysis of 354 accommodative esotropic patients led to the discovery of two conjunctive variables that predicted deterioration in the initial year of treatment better than what was previously determined using standard statistical methods.

When choosing a suitable technique for regression and classification with multivariate predictor variables, one is often faced with a tradeoff between interpretability and high predictive accuracy. To give a classical example, classification and regression trees are easy to understand and interpret. Tree ensembles like Random Forests provide usually more accurate predictions. Yet tree ensembles are also more difficult to analyze than single trees and are often criticized, perhaps unfairly, as `black box' predictors. Node harvest is trying to reconcile the two aims of interpretability and predictive accuracy by combining positive aspects of trees and tree ensembles. Results are very sparse and interpretable and predictive accuracy is extremely competitive, especially for low signal-to-noise data. The procedure is simple: an initial set of a few thousand nodes is generated randomly. If a new observation falls into just a single node, its prediction is the mean response of all training observation within this node, identical to a tree-like prediction. A new observation falls typically into several nodes and its prediction is then the weighted average of the mean responses across all these nodes. The only role of node harvest is to `pick' the right nodes from the initial large ensemble of nodes by choosing node weights, which amounts in the proposed algorithm to a quadratic programming problem with linear inequality constraints. The solution is sparse in the sense that only very few nodes are selected with a nonzero weight. This sparsity is not explicitly enforced. Maybe surprisingly, it is not necessary to select a tuning parameter for optimal predictive accuracy. Node harvest can handle mixed data and missing values and is shown to be simple to interpret and competitive in predictive accuracy on a variety of data sets.

Boosting combines weak classifiers to form highly accurate predictors. Although the case of binary classification is well understood, in the multiclass setting, the "correct" requirements on the weak classifier, or the notion of the most efficient boosting algorithms are missing. In this paper, we create a broad and general framework, within which we make precise and identify the optimal requirements on the weak-classifier, as well as design the most effective, in a certain sense, boosting algorithms that assume such requirements.

Boosting combines weak classifiers to form highly accurate predictors. Although the case of binary classification is well understood, in the multiclass setting, the "correct" requirements on the weak classifier, or the notion of the most efficient boosting algorithms are missing. In this paper, we create a broad and general framework, within which we make precise and identify the optimal requirements on the weak-classifier, as well as design the most effective, in a certain sense, boosting algorithms that assume such requirements.

Undirected graphical models encode in a graph $G$ the dependency structure of a random vector $Y$. In many applications, it is of interest to model $Y$ given another random vector $X$ as input. We refer to the problem of estimating the graph $G(x)$ of $Y$ conditioned on $X=x$ as ``graph-valued regression''. In this paper, we propose a semiparametric method for estimating $G(x)$ that builds a tree on the $X$ space just as in CART (classification and regression trees), but at each leaf of the tree estimates a graph. We call the method ``Graph-optimized CART'', or Go-CART. We study the theoretical properties of Go-CART using dyadic partitioning trees, establishing oracle inequalities on risk minimization and tree partition consistency. We also demonstrate the application of Go-CART to a meteorological dataset, showing how graph-valued regression can provide a useful tool for analyzing complex data.

We combine random forest (RF) and conditional random field (CRF) into a new computational framework, called random forest random field (RF)^2. Inference of (RF)^2 uses the Swendsen-Wang cut algorithm, characterized by Metropolis-Hastings jumps. A jump from one state to another depends on the ratio of the proposal distributions, and on the ratio of the posterior distributions of the two states. Prior work typically resorts to a parametric estimation of these four distributions, and then computes their ratio. Our key idea is to instead directly estimate these ratios using RF. RF collects in leaf nodes of each decision tree the class histograms of training examples. We use these class histograms for a non-parametric estimation of the distribution ratios. We derive the theoretical error bounds of a two-class (RF)^2. (RF)^2 is applied to a challenging task of multiclass object recognition and segmentation over a random field of input image regions. In our empirical evaluation, we use only the visual information provided by image regions (e.g., color, texture, spatial layout), whereas the competing methods additionally use higher-level cues about the horizon location and 3D layout of surfaces in the scene. Nevertheless, (RF)^2 outperforms the state of the art on benchmark datasets, in terms of accuracy and computation time.

We propose a new nonparametric learning method based on multivariate dyadic regression trees (MDRTs). Unlike traditional dyadic decision trees (DDTs) or classification and regression trees (CARTs), MDRTs are constructed using penalized empirical risk minimization with a novel sparsity-inducing penalty. Theoretically, we show that MDRTs can simultaneously adapt to the unknown sparsity and smoothness of the true regression functions, and achieve the nearly optimal rates of convergence (in a minimax sense) for the class of $(\alpha, C)$-smooth functions. Empirically, MDRTs can simultaneously conduct function estimation and variable selection in high dimensions. To make MDRTs applicable for large-scale learning problems, we propose a greedy heuristics. The superior performance of MDRTs are demonstrated on both synthetic and real datasets.