Decision Trees (DTs) are commonly used for many machine learning tasks due to their high degree of interpretability. However, learning a DT from data is a difficult optimization problem, as it is non-convex and non-differentiable. Therefore, common approaches learn DTs using a greedy growth algorithm that minimizes the impurity locally at each internal node. Unfortunately, this greedy procedure can lead to inaccurate trees. In this paper, we present a novel approach for learning hard, axis-aligned DTs with gradient descent. The proposed method uses backpropagation with a straight-through operator on a dense DT representation, to jointly optimize all tree parameters. Our approach outperforms existing methods on binary classification benchmarks and achieves competitive results for multi-class tasks. The method is available under: https://github.com/s-marton/GradTree

Model explainability is crucial for human users to be able to interpret how a proposed classifier assigns labels to data based on its feature values. We study generalized linear models constructed using sets of feature value rules, which can capture nonlinear dependencies and interactions. An inherent trade-off exists between rule set sparsity and its prediction accuracy. It is computationally expensive to find the right choice of sparsity -- e.g., via cross-validation -- with existing methods. We propose a new formulation to learn an ensemble of rule sets that simultaneously addresses these competing factors. Good generalization is ensured while keeping computational costs low by utilizing distributionally robust optimization. The formulation utilizes column generation to efficiently search the space of rule sets and constructs a sparse ensemble of rule sets, in contrast with techniques like random forests or boosting and their variants. We present theoretical results that motivate and justify the use of our distributionally robust formulation. Extensive numerical experiments establish that our method improves over competing methods -- on a large set of publicly available binary classification problem instances -- with respect to one or more of the following metrics: generalization quality, computational cost, and explainability.

We propose a prognostic stratum matching framework that addresses the deficiencies of Randomized trial data subgroup analysis and transforms ObservAtional Data to be used as if they were randomized, thus paving the road for precision medicine. Our approach counters the effects of unobserved confounding in observational data by correcting the estimated probabilities of the outcome under a treatment through a novel two-step process. These probabilities are then used to train Optimal Policy Trees (OPTs), which are decision trees that optimally assign treatments to subgroups of patients based on their characteristics. This facilitates the creation of clinically intuitive treatment recommendations. We applied our framework to observational data of patients with gastrointestinal stromal tumors (GIST) and validated the OPTs in an external cohort using the sensitivity and specificity metrics. We show that these recommendations outperformed those of experts in GIST. We further applied the same framework to randomized clinical trial (RCT) data of patients with extremity sarcomas. Remarkably, despite the initial trial results suggesting that all patients should receive treatment, our framework, after addressing imbalances in patient distribution due to the trial's small sample size, identified through the OPTs a subset of patients with unique characteristics who may not require treatment. Again, we successfully validated our recommendations in an external cohort.

Case-based reasoning is known to play an important role in several legal settings. In this paper we focus on a recent approach to case-based reasoning, supported by an instantiation of abstract argumentation whereby arguments represent cases and attack between arguments results from outcome disagreement between cases and a notion of relevance. In this context, relevance is connected to a form of specificity among cases. We explore how relevance can be learnt automatically in practice with the help of decision trees, and explore the combination of case-based reasoning with abstract argumentation (AA-CBR) and learning of case relevance for prediction in legal settings. Specifically, we show that, for two legal datasets, AA-CBR and decision-tree-based learning of case relevance perform competitively in comparison with decision trees. We also show that AA-CBR with decision-tree-based learning of case relevance results in a more compact representation than their decision tree counterparts, which could be beneficial for obtaining cognitively tractable explanations.

Network intrusion detection (NID) systems which leverage machine learning have been shown to have strong performance in practice when used to detect malicious network traffic. Decision trees in particular offer a strong balance between performance and simplicity, but require users of NID systems to have background knowledge in machine learning to interpret. In addition, they are unable to provide additional outside information as to why certain features may be important for classification. In this work, we explore the use of large language models (LLMs) to provide explanations and additional background knowledge for decision tree NID systems. Further, we introduce a new human evaluation framework for decision tree explanations, which leverages automatically generated quiz questions that measure human evaluators' understanding of decision tree inference. Finally, we show LLM generated decision tree explanations correlate highly with human ratings of readability, quality, and use of background knowledge while simultaneously providing better understanding of decision boundaries.

We establish stability of random forests under the mild condition that the squared response ($Y^2$) does not have a heavy tail. In particular, our analysis holds for the practical version of random forests that is implemented in popular packages like \texttt{randomForest} in \texttt{R}. Empirical results show that stability may persist even beyond our assumption and hold for heavy-tailed $Y^2$. Using the stability property, we prove a non-asymptotic lower bound for the coverage probability of prediction intervals constructed from the out-of-bag error of random forests. With another mild condition that is typically satisfied when $Y$ is continuous, we also establish a complementary upper bound, which can be similarly established for the jackknife prediction interval constructed from an arbitrary stable algorithm. We also discuss the asymptotic coverage probability under assumptions weaker than those considered in previous literature. Our work implies that random forests, with its stability property, is an effective machine learning method that can provide not only satisfactory point prediction but also justified interval prediction at almost no extra computational cost.

Joint feature selection and tree ensemble learning is a challenging task. Popular tree ensemble toolkits e.g., Gradient Boosted Trees and Random Forests support feature selection post-training based on feature importances, which are known to be misleading, and can significantly hurt performance. We propose Skinny Trees: a toolkit for feature selection in tree ensembles, such that feature selection and tree ensemble learning occurs simultaneously. It is based on an end-to-end optimization approach that considers feature selection in differentiable trees with Group $\ell_0 - \ell_2$ regularization. We optimize with a first-order proximal method and present convergence guarantees for a non-convex and non-smooth objective. Interestingly, dense-to-sparse regularization scheduling can lead to more expressive and sparser tree ensembles than vanilla proximal method. On 15 synthetic and real-world datasets, Skinny Trees can achieve $1.5\times$ - $620\times$ feature compression rates, leading up to $10\times$ faster inference over dense trees, without any loss in performance. Skinny Trees lead to superior feature selection than many existing toolkits e.g., in terms of AUC performance for $25\%$ feature budget, Skinny Trees outperforms LightGBM by $10.2\%$ (up to $37.7\%$), and Random Forests by $3\%$ (up to $12.5\%$).

Counterfactual explanations play an important role in detecting bias and improving the explainability of data-driven classification models. A counterfactual explanation (CE) is a minimal perturbed data point for which the decision of the model changes. Most of the existing methods can only provide one CE, which may not be achievable for the user. In this work we derive an iterative method to calculate robust CEs, i.e. CEs that remain valid even after the features are slightly perturbed. To this end, our method provides a whole region of CEs allowing the user to choose a suitable recourse to obtain a desired outcome. We use algorithmic ideas from robust optimization and prove convergence results for the most common machine learning methods including logistic regression, decision trees, random forests, and neural networks. Our experiments show that our method can efficiently generate globally optimal robust CEs for a variety of common data sets and classification models.

We propose a simple generalization of standard and empirically successful decision tree learning algorithms such as ID3, C4.5, and CART. These algorithms, which have been central to machine learning for decades, are greedy in nature: they grow a decision tree by iteratively splitting on the best attribute. Our algorithm, Top-$k$, considers the $k$ best attributes as possible splits instead of just the single best attribute. We demonstrate, theoretically and empirically, the power of this simple generalization. We first prove a {\sl greediness hierarchy theorem} showing that for every $k \in \mathbb{N}$, Top-$(k+1)$ can be dramatically more powerful than Top-$k$: there are data distributions for which the former achieves accuracy $1-\varepsilon$, whereas the latter only achieves accuracy $\frac1{2}+\varepsilon$. We then show, through extensive experiments, that Top-$k$ outperforms the two main approaches to decision tree learning: classic greedy algorithms and more recent "optimal decision tree" algorithms. On one hand, Top-$k$ consistently enjoys significant accuracy gains over greedy algorithms across a wide range of benchmarks. On the other hand, Top-$k$ is markedly more scalable than optimal decision tree algorithms and is able to handle dataset and feature set sizes that remain far beyond the reach of these algorithms.

There is an increasing interest in learning reward functions that model human preferences. However, many frameworks use blackbox learning methods that, while expressive, are difficult to interpret. We propose and evaluate a novel approach for learning expressive and interpretable reward functions from preferences using Differentiable Decision Trees (DDTs). Our experiments across several domains, including CartPole, Visual Gridworld environments and Atari games, provide evidence that that the tree structure of our learned reward function is useful in determining the extent to which the reward function is aligned with human preferences. We provide experimental evidence that reward DDTs can achieve competitive performance when compared with larger capacity deep neural network reward functions. We also observe that the choice between soft and hard (argmax) output of reward DDT reveals a tension between wanting highly shaped rewards to ensure good RL performance, while also wanting simpler, more interpretable rewards.