Learning arguments is highly relevant to the field of explainable artificial intelligence. It is a family of symbolic machine learning techniques that is particularly human-interpretable. These techniques learn a set of arguments as an intermediate representation. Arguments are small rules with exceptions that can be chained to larger arguments for making predictions or decisions. We investigate the learning of arguments, specifically the learning of arguments from a 'case model' proposed by Verheij [34]. The case model in Verheij's approach are cases or scenarios in a legal setting. The number of cases in a case model are relatively low. Here, we investigate whether Verheij's approach can be used for learning arguments from other types of data sets with a much larger number of instances. We compare the learning of arguments from a case model with the HeRO algorithm [15] and learning a decision tree.

We review sufficient dimension reduction (SDR) estimators with multivariate response in this paper. A wide range of SDR methods are characterized as inverse regression SDR estimators or forward regression SDR estimators. The inverse regression family include pooled marginal estimators, projective resampling estimators, and distance-based estimators. Ordinary least squares, partial least squares, and semiparametric SDR estimators, on the other hand, are discussed as estimators from the forward regression family.

Tree-based models such as decision trees and random forests (RF) are a cornerstone of modern machine-learning practice. To mitigate overfitting, trees are typically regularized by a variety of techniques that modify their structure (e.g. pruning). We introduce Hierarchical Shrinkage (HS), a post-hoc algorithm that does not modify the tree structure, and instead regularizes the tree by shrinking the prediction over each node towards the sample means of its ancestors. The amount of shrinkage is controlled by a single regularization parameter and the number of data points in each ancestor. Since HS is a post-hoc method, it is extremely fast, compatible with any tree growing algorithm, and can be used synergistically with other regularization techniques. Extensive experiments over a wide variety of real-world datasets show that HS substantially increases the predictive performance of decision trees, even when used in conjunction with other regularization techniques. Moreover, we find that applying HS to each tree in an RF often improves accuracy, as well as its interpretability by simplifying and stabilizing its decision boundaries and SHAP values. We further explain the success of HS in improving prediction performance by showing its equivalence to ridge regression on a (supervised) basis constructed of decision stumps associated with the internal nodes of a tree. All code and models are released in a full-fledged package available on Github (github.com/csinva/imodels)

Machine learning is an integral part of many human-facing computer systems and is increasingly a key component of decisions that have profound effects on people's lives. There are many dangers that come with this. For instance, statistical models can easily be error-prone in regions of the input space that are not well-reflected in training data but that end up arising in practice. Or they can be excessively complicated in ways that impact their generalization ability. Or they might implicitly make their decisions based on criteria that would not considered acceptable by society. For all these reasons, and many others, it is crucial to have models that are understandable or can explain their predictions to humans [19]. Explanations of a classification system can take many forms, but should accurately reflect the classifier's inner workings.

We propose a general approach for distance based clustering, using the gradient of the cost function that measures clustering quality with respect to cluster assignments and cluster center positions. The approach is an iterative two step procedure (alternating between cluster assignment and cluster center updates) and is applicable to a wide range of functions, satisfying some mild assumptions. The main advantage of the proposed approach is a simple and computationally cheap update rule. Unlike previous methods that specialize to a specific formulation of the clustering problem, our approach is applicable to a wide range of costs, including non-Bregman clustering methods based on the Huber loss. We analyze the convergence of the proposed algorithm, and show that it converges to the set of appropriately defined fixed points, under arbitrary center initialization. In the special case of Bregman cost functions, the algorithm converges to the set of centroidal Voronoi partitions, which is consistent with prior works. Numerical experiments on real data demonstrate the effectiveness of the proposed method.

Simulation models, in particular agent-based models, are gaining popularity in economics. The considerable flexibility they offer, as well as their capacity to reproduce a variety of empirically observed behaviours of complex systems, give them broad appeal, and the increasing availability of cheap computing power has made their use feasible. Yet a widespread adoption in real-world modelling and decision-making scenarios has been hindered by the difficulty of performing parameter estimation for such models. In general, simulation models lack a tractable likelihood function, which precludes a straightforward application of standard statistical inference techniques. Several recent works have sought to address this problem through the application of likelihood-free inference techniques, in which parameter estimates are determined by performing some form of comparison between the observed data and simulation output. However, these approaches are (a) founded on restrictive assumptions, and/or (b) typically require many hundreds of thousands of simulations. These qualities make them unsuitable for large-scale simulations in economics and can cast doubt on the validity of these inference methods in such scenarios. In this paper, we investigate the efficacy of two classes of black-box approximate Bayesian inference methods that have recently drawn significant attention within the probabilistic machine learning community: neural posterior estimation and neural density ratio estimation. We present benchmarking experiments in which we demonstrate that neural network based black-box methods provide state of the art parameter inference for economic simulation models, and crucially are compatible with generic multivariate time-series data. In addition, we suggest appropriate assessment criteria for future benchmarking of approximate Bayesian inference procedures for economic simulation models.

The domain generalization (DG) problem setting challenges a model trained on multiple known data distributions to generalise well on unseen data distributions. Due to its practical importance, a large number of methods have been proposed to address this challenge. However much of the work in general purpose DG is heuristically motivated, as the DG problem is hard to model formally; and recent evaluations have cast doubt on existing methods' practical efficacy -- in particular compared to a well tuned empirical risk minimisation baseline. We present a novel learning-theoretic generalisation bound for DG that bounds unseen domain generalisation performance in terms of the model's Rademacher complexity. Based on this, we conjecture that existing methods' efficacy or lack thereof is largely determined by an empirical risk vs predictor complexity trade-off, and demonstrate that their performance variability can be explained in these terms. Algorithmically, this analysis suggests that domain generalisation should be achieved by simply performing regularised ERM with a leave-one-domain-out cross-validation objective. Empirical results on the DomainBed benchmark corroborate this.

Representing words by vectors, or embeddings, enables computational reasoning and is foundational to automating natural language tasks. For example, if word embeddings of similar words contain similar values, word similarity can be readily assessed, whereas judging that from their spelling is often impossible (e.g. cat /feline) and to predetermine and store similarities between all words is prohibitively time-consuming, memory intensive and subjective. We focus on word embeddings learned from text corpora and knowledge graphs. Several well-known algorithms learn word embeddings from text on an unsupervised basis by learning to predict those words that occur around each word, e.g. word2vec and GloVe. Parameters of such word embeddings are known to reflect word co-occurrence statistics, but how they capture semantic meaning has been unclear. Knowledge graph representation models learn representations both of entities (words, people, places, etc.) and relations between them, typically by training a model to predict known facts in a supervised manner. Despite steady improvements in fact prediction accuracy, little is understood of the latent structure that enables this. The limited understanding of how latent semantic structure is encoded in the geometry of word embeddings and knowledge graph representations makes a principled means of improving their performance, reliability or interpretability unclear. To address this: 1. we theoretically justify the empirical observation that particular geometric relationships between word embeddings learned by algorithms such as word2vec and GloVe correspond to semantic relations between words; and 2. we extend this correspondence between semantics and geometry to the entities and relations of knowledge graphs, providing a model for the latent structure of knowledge graph representation linked to that of word embeddings.

In logistic regression, it is often desirable to utilize regularization to promote sparse solutions, particularly for problems with a large number of features compared to available labels. In this paper, we present screening rules that safely remove features from logistic regression with $\ell_0-\ell_2$ regularization before solving the problem. The proposed safe screening rules are based on lower bounds from the Fenchel dual of strong conic relaxations of the logistic regression problem. Numerical experiments with real and synthetic data suggest that a high percentage of the features can be effectively and safely removed apriori, leading to substantial speed-up in the computations.

Iterative regularization exploits the implicit bias of an optimization algorithm to regularize ill-posed problems. Constructing algorithms with such built-in regularization mechanisms is a classic challenge in inverse problems but also in modern machine learning, where it provides both a new perspective on algorithms analysis, and significant speed-ups compared to explicit regularization. In this work, we propose and study the first iterative regularization procedure able to handle biases described by non smooth and non strongly convex functionals, prominent in low-complexity regularization. Our approach is based on a primal-dual algorithm of which we analyze convergence and stability properties, even in the case where the original problem is unfeasible. The general results are illustrated considering the special case of sparse recovery with the $\ell_1$ penalty. Our theoretical results are complemented by experiments showing the computational benefits of our approach.