arXiv subjects

Joseph T. Meyer

Publications and source records attributed to Joseph T. Meyer.

5 recordsLinked to original sources

arXiv2024

Consistency of Random Forest Type Algorithms under a Probabilistic Impurity Decrease Condition

This paper derives a unifying theorem establishing consistency results for a broad class of tree-based algorithms. It improves current results in two aspects. First of all, it can be applied to algorithms that vary from traditional Random Forests due to additional randomness for choosing splits, extending split options, allowing partitions into more than two cells in a single iteration step, and combinations of those. In particular, we prove consistency for Extremely Randomized Trees, Interaction Forests and Oblique Regression Trees using our general theorem. Secondly, it can be used to demonstrate consistency for a larger function class compared to previous results on Random Forests if one allows for additional random splits. Our results are based on the extension of the recently introduced notion of sufficient impurity decrease to a probabilistic sufficient impurity decrease condition.

math.ST↗

arXiv2023

Random Planted Forest: a directly interpretable tree ensemble

We introduce a novel interpretable tree based algorithm for prediction in a regression setting. Our motivation is to estimate the unknown regression function from a functional decomposition perspective in which the functional components correspond to lower order interaction terms. The idea is to modify the random forest algorithm by keeping certain leaves after they are split instead of deleting them. This leads to non-binary trees which we refer to as planted trees. An extension to a forest leads to our random planted forest algorithm. Additionally, the maximum number of covariates which can interact within a leaf can be bounded. If we set this interaction bound to one, the resulting estimator is a sum of one-dimensional functions. In the other extreme case, if we do not set a limit, the resulting estimator and corresponding model place no restrictions on the form of the regression function. In a simulation study we find encouraging prediction and visualisation properties of our random planted forest method. We also develop theory for an idealized version of random planted forests in cases where the interaction bound is low. We show that if it is smaller than three, the idealized version achieves asymptotically optimal convergence rates up to a logarithmic factor. Code is available on GitHub https://github.com/PlantedML/randomPlantedForest.

stat.ML↗

arXiv2023

Unifying local and global model explanations by functional decomposition of low dimensional structures

We consider a global representation of a regression or classification function by decomposing it into the sum of main and interaction components of arbitrary order. We propose a new identification constraint that allows for the extraction of interventional SHAP values and partial dependence plots, thereby unifying local and global explanations. With our proposed identification, a feature's partial dependence plot corresponds to the main effect term plus the intercept. The interventional SHAP value of feature $k$ is a weighted sum of the main component and all interaction components that include $k$, with the weights given by the reciprocal of the component's dimension. This brings a new perspective to local explanations such as SHAP values which were previously motivated by game theory only. We show that the decomposition can be used to reduce direct and indirect bias by removing all components that include a protected feature. Lastly, we motivate a new measure of feature importance. In principle, our proposed functional decomposition can be applied to any machine learning model, but exact calculation is only feasible for low-dimensional structures or ensembles of those. We provide an algorithm and efficient implementation for gradient-boosted trees (xgboost) and random planted forest. Conducted experiments suggest that our method provides meaningful explanations and reveals interactions of higher orders. The proposed methods are implemented in an R package, available at \url{https://github.com/PlantedML/glex}.

cs.LG↗

arXiv2022

Optimal Convergence Rates of Deep Neural Networks in a Classification Setting

We establish optimal convergence rates up to a log-factor for a class of deep neural networks in a classification setting under a restraint sometimes referred to as the Tsybakov noise condition. We construct classifiers in a general setting where the boundary of the bayes-rule can be approximated well by neural networks. Corresponding rates of convergence are proven with respect to the misclassification error. It is then shown that these rates are optimal in the minimax sense if the boundary satisfies a smoothness condition. Non-optimal convergence rates already exist for this setting. Our main contribution lies in improving existing rates and showing optimality, which was an open problem. Furthermore, we show almost optimal rates under some additional restraints which circumvent the curse of dimensionality. For our analysis we require a condition which gives new insight on the restraint used. In a sense it acts as a requirement for the "correct noise exponent" for a class of functions.

math.ST↗

arXiv2022

Local linear smoothing in additive models as data projection

We discuss local linear smooth backfitting for additive non-parametric models. This procedure is well known for achieving optimal convergence rates under appropriate smoothness conditions. In particular, it allows for the estimation of each component of an additive model with the same asymptotic accuracy as if the other components were known. The asymptotic discussion of local linear smooth backfitting is rather complex because typically an overwhelming notation is required for a detailed discussion. In this paper we interpret the local linear smooth backfitting estimator as a projection of the data onto a linear space with a suitably chosen semi-norm. This approach simplifies both the mathematical discussion as well as the intuitive understanding of properties of this version of smooth backfitting.

math.ST↗