Clément Bénard

Clément Bénard, Manon Arfib, Christophe Labreuche et al.

Explainability of deep learning algorithms is critical for computer-vision applications with high-stake decisions. Concept bottleneck models (CBM) have recently shown promising performance to provide explainable and accurate predictions for classification problems, based on a bottleneck of high-level concepts. Existing CBM methods rely on a linear aggregation of the concept scores to compute predictions. However, a large number of concepts is often used in this linear approach, which undermines explainability and favors information leakage. In general, the underlying relation between concepts and output logits is not linear. Therefore, we introduce Hoeffding Concept Bottleneck Models (HCBM), which build on the Hoeffding functional decomposition of gradient-boosted trees to provide non-linear and sparse aggregations of concept scores, and generate compact predictions using prime implicants. HCBM are proved to be robust to interconcept leakage, and outperform standard linear CBM in practice, as shown in extensive experiments. Beyond classification, HCBM can be adapted to object detection, and we focus on a challenging case with overhead images to show the high performance of HCBM in these settings.

Clément Bénard, Jeffrey Näf, Julie Josse

Distributional Random Forest (DRF) is a flexible forest-based method to estimate the full conditional distribution of a multivariate output of interest given input variables. In this article, we introduce a variable importance algorithm for DRFs, based on the well-established drop and relearn principle and MMD distance. While traditional importance measures only detect variables with an influence on the output mean, our algorithm detects variables impacting the output distribution more generally. We show that the introduced importance measure is consistent, exhibits high empirical performance on both real and simulated data, and outperforms competitors. In particular, our algorithm is highly efficient to select variables through recursive feature elimination, and can therefore provide small sets of variables to build accurate estimates of conditional output distributions.

Clément Bénard

Tree ensembles have demonstrated state-of-the-art predictive performance across a wide range of problems involving tabular data. Nevertheless, the black-box nature of tree ensembles is a strong limitation, especially for applications with critical decisions at stake. The Hoeffding or ANOVA functional decomposition is a powerful explainability method, as it breaks down black-box models into a unique sum of lower-dimensional functions, provided that input variables are independent. In standard learning settings, input variables are often dependent, and the Hoeffding decomposition is generalized through hierarchical orthogonality constraints. Such generalization leads to unique and sparse decompositions with well-defined main effects and interactions. However, the practical estimation of this decomposition from a data sample is still an open problem. Therefore, we introduce the TreeHFD algorithm to estimate the Hoeffding decomposition of a tree ensemble from a data sample. We show the convergence of TreeHFD, along with the main properties of orthogonality, sparsity, and causal variable selection. The high performance of TreeHFD is demonstrated through experiments on both simulated and real data, using our treehfd Python package (https://github.com/ThalesGroup/treehfd). Besides, we empirically show that the widely used TreeSHAP method, based on Shapley values, is strongly connected to the Hoeffding decomposition.

Clément Bénard, Sébastien da Veiga, Erwan Scornet

Variable importance measures are the main tools to analyze the black-box mechanisms of random forests. Although the mean decrease accuracy (MDA) is widely accepted as the most efficient variable importance measure for random forests, little is known about its statistical properties. In fact, the definition of MDA varies across the main random forest software. In this article, our objective is to rigorously analyze the behavior of the main MDA implementations. Consequently, we mathematically formalize the various implemented MDA algorithms, and then establish their limits when the sample size increases. This asymptotic analysis reveals that these MDA versions differ as importance measures, since they converge towards different quantities. More importantly, we break down these limits into three components: the first two terms are related to Sobol indices, which are well-defined measures of a covariate contribution to the response variance, widely used in the sensitivity analysis field, as opposed to the third term, whose value increases with dependence within covariates. Thus, we theoretically demonstrate that the MDA does not target the right quantity to detect influential covariates in a dependent setting, a fact that has already been noticed experimentally. To address this issue, we define a new importance measure for random forests, the Sobol-MDA, which fixes the flaws of the original MDA, and consistently estimates the accuracy decrease of the forest retrained without a given covariate, but with an efficient computational cost. The Sobol-MDA empirically outperforms its competitors on both simulated and real data for variable selection. An open source implementation in R and C++ is available online.

Clément Bénard, Gérard Biau, Sébastien da Veiga et al.

Interpretability of learning algorithms is crucial for applications involving critical decisions, and variable importance is one of the main interpretation tools. Shapley effects are now widely used to interpret both tree ensembles and neural networks, as they can efficiently handle dependence and interactions in the data, as opposed to most other variable importance measures. However, estimating Shapley effects is a challenging task, because of the computational complexity and the conditional expectation estimates. Accordingly, existing Shapley algorithms have flaws: a costly running time, or a bias when input variables are dependent. Therefore, we introduce SHAFF, SHApley eFfects via random Forests, a fast and accurate Shapley effect estimate, even when input variables are dependent. We show SHAFF efficiency through both a theoretical analysis of its consistency, and the practical performance improvements over competitors with extensive experiments. An implementation of SHAFF in C++ and R is available online.

Clément Bénard, Gérard Biau, Sébastien da Veiga et al.

We introduce SIRUS (Stable and Interpretable RUle Set) for regression, a stable rule learning algorithm which takes the form of a short and simple list of rules. State-of-the-art learning algorithms are often referred to as "black boxes" because of the high number of operations involved in their prediction process. Despite their powerful predictivity, this lack of interpretability may be highly restrictive for applications with critical decisions at stake. On the other hand, algorithms with a simple structure-typically decision trees, rule algorithms, or sparse linear models-are well known for their instability. This undesirable feature makes the conclusions of the data analysis unreliable and turns out to be a strong operational limitation. This motivates the design of SIRUS, which combines a simple structure with a remarkable stable behavior when data is perturbed. The algorithm is based on random forests, the predictive accuracy of which is preserved. We demonstrate the efficiency of the method both empirically (through experiments) and theoretically (with the proof of its asymptotic stability). Our R/C++ software implementation sirus is available from CRAN.

Clément Bénard, Gérard Biau, Sébastien da Veiga et al.

State-of-the-art learning algorithms, such as random forests or neural networks, are often qualified as "black-boxes" because of the high number and complexity of operations involved in their prediction mechanism. This lack of interpretability is a strong limitation for applications involving critical decisions, typically the analysis of production processes in the manufacturing industry. In such critical contexts, models have to be interpretable, i.e., simple, stable, and predictive. To address this issue, we design SIRUS (Stable and Interpretable RUle Set), a new classification algorithm based on random forests, which takes the form of a short list of rules. While simple models are usually unstable with respect to data perturbation, SIRUS achieves a remarkable stability improvement over cutting-edge methods. Furthermore, SIRUS inherits a predictive accuracy close to random forests, combined with the simplicity of decision trees. These properties are assessed both from a theoretical and empirical point of view, through extensive numerical experiments based on our R/C++ software implementation sirus available from CRAN.