Randomized decision trees and forests have a rich history in machine learning and have seen considerable success in application, perhaps particularly so for computer vision. However, they face a fundamental limitation: given enough data, the number of nodes in decision trees will grow exponentially with depth. For certain applications, for example on mobile or embedded processors, memory is a limited resource, and so the exponential growth of trees limits their depth, and thus their potential accuracy. This paper proposes decision jungles, revisiting the idea of ensembles of rooted decision directed acyclic graphs (DAGs), and shows these to be compact and powerful discriminative models for classification. Unlike conventional decision trees that only allow one path to every node, a DAG in a decision jungle allows multiple paths from the root to each leaf. We present and compare two new node merging algorithms that jointly optimize both the features and the structure of the DAGs efficiently. During training, node splitting and node merging are driven by the minimization of exactly the same objective function, here the weighted sum of entropies at the leaves. Results on varied datasets show that, compared to decision forests and several other baselines, decision jungles require dramatically less memory while considerably improving generalization.

We consider the problem of maintaining the data-structures of a partition-based regression procedure in a setting where the training data arrives sequentially over time.

Ensembles of randomized decision trees, usually referred to as random forests, are widely used for classification and regression tasks in machine learning and statistics. Random forests achieve competitive predictive performance and are computationally efficient to train and test, making them excellent candidates for real-world prediction tasks. The most popular random forest variants (such as Breiman's random forest and extremely randomized trees) operate on batches of training data. Online methods are now in greater demand. Existing online random forests, however, require more training data than their batch counterpart to achieve comparable predictive performance. In this work, we use Mondrian processes (Roy and Teh, 2009) to construct ensembles of random decision trees we call Mondrian forests. Mondrian forests can be grown in an incremental/online fashion and remarkably, the distribution of online Mondrian forests is the same as that of batch Mondrian forests. Mondrian forests achieve competitive predictive performance comparable with existing online random forests and periodically retrained batch random forests, while being more than an order of magnitude faster, thus representing a better computation vs accuracy tradeoff.

Even with the advent of more sophisticated, data-hungry methods, boosted decision trees remain extraordinarily successful for fast rigid object detection, achieving top accuracy on numerous datasets. While effective, most boosted detectors use decision trees with orthogonal (single feature) splits, and the topology of the resulting decision boundary may not be well matched to the natural topology of the data. Given highly correlated data, decision trees with oblique (multiple feature) splits can be effective. Use of oblique splits, however, comes at considerable computational expense. Inspired by recent work on discriminative decorrelation of HOG features, we instead propose an efficient feature transform that removes correlations in local neighborhoods. The result is an overcomplete but locally decorrelated representation ideally suited for use with orthogonal decision trees. In fact, orthogonal trees with our locally decorrelated features outperform oblique trees trained over the original features at a fraction of the computational cost. The overall improvement in accuracy is dramatic: on the Caltech Pedestrian Dataset, we reduce false positives nearly tenfold over the previous state-of-the-art.

We study the problem of multiclass classification with an extremely large number of classes (k), with the goal of obtaining train and test time complexity logarithmic in the number of classes. We develop top-down tree construction approaches for constructing logarithmic depth trees. On the theoretical front, we formulate a new objective function, which is optimized at each node of the tree and creates dynamic partitions of the data which are both pure (in terms of class labels) and balanced. We demonstrate that under favorable conditions, we can construct logarithmic depth trees that have leaves with low label entropy. However, the objective function at the nodes is challenging to optimize computationally. We address the empirical problem with a new online decision tree construction procedure. Experiments demonstrate that this online algorithm quickly achieves improvement in test error compared to more common logarithmic training time approaches, which makes it a plausible method in computationally constrained large-k applications.

Decision trees and randomized forests are widely used in computer vision and machine learning.

Deep distributed decision trees and tree ensembles have grown in importance due to the need to model increasingly large datasets.

We propose to prune a random forest (RF) for resource-constrained prediction. We first construct a RF and then prune it to optimize expected feature cost & accuracy. We pose pruning RFs as a novel 0-1 integer program with linear constraints that encourages feature re-use. We establish total unimodularity of the constraint set to prove that the corresponding LP relaxation solves the original integer program. We then exploit connections to combinatorial optimization and develop an efficient primal-dual algorithm, scalable to large datasets. In contrast to our bottom-up approach, which benefits from good RF initialization, conventional methods are top-down acquiring features based on their utility value and is generally intractable, requiring heuristics. Empirically, our pruning algorithm outperforms existing state-of-the-art resource-constrained algorithms.

Decision tree (and its extensions such as Gradient Boosting Decision Trees and Random Forest) is a widely used machine learning algorithm, due to its practical effectiveness and model interpretability. With the emergence of big data, there is an increasing need to parallelize the training process of decision tree. However, most existing attempts along this line suffer from high communication costs. In this paper, we propose a new algorithm, called Parallel Voting Decision Tree (PV-Tree), to tackle this challenge. After partitioning the training data onto a number of (e.g., M) machines, this algorithm performs both local voting and global voting in each iteration.

Machine learning has become a common approach to predicting the outcomes of soccer matches, and the body of literature in this domain has grown substantially in the past decade and a half. This chapter discusses available datasets, the types of models and features, and ways of evaluating model performance in this application domain. The aim of this chapter is to give a broad overview of the current state and potential future developments in machine learning for soccer match results prediction, as a resource for those interested in conducting future studies in the area. Our main findings are that while gradient-boosted tree models such as CatBoost, applied to soccer-specific ratings such as pi-ratings, are currently the best-performing models on datasets containing only goals as the match features, there needs to be a more thorough comparison of the performance of deep learning models and Random Forest on a range of datasets with different types of features. Furthermore, new rating systems using both player- and team-level information and incorporating additional information from, e.g., spatiotemporal tracking and event data, could be investigated further. Finally, the interpretability of match result prediction models needs to be enhanced for them to be more useful for team management.