Mixed-Integer Convex Nonlinear Optimization with Gradient-Boosted Trees Embedded

Decision trees usefully represent sparse, high dimensional and noisy data. Having learned a function from this data, we may want to thereafter integrate the function into a larger decision-making problem, e.g., for picking the best chemical process catalyst. We study a large-scale, industrially-relevant mixed-integer nonlinear nonconvex optimization problem involving both gradient-boosted trees and penalty functions mitigating risk. This mixed-integer optimization problem with convex penalty terms broadly applies to optimizing pre-trained regression tree models. Decision makers may wish to optimize discrete models to repurpose legacy predictive models, or they may wish to optimize a discrete model that particularly well-represents a data set. We develop several heuristic methods to find feasible solutions, and an exact, branch-and-bound algorithm leveraging structural properties of the gradient-boosted trees and penalty functions. We computationally test our methods on concrete mixture design instance and a chemical catalysis industrial instance

A Diversity-Accuracy Measure for Homogenous Ensemble Selection

Several selection methods in the literature are essentially based on an evaluation function that determines whether a model M contributes positively to boost the performances of the whole ensemble. In this paper, we propose a method called DIversity and ACcuracy for Ensemble Selection (DIACES) using an evaluation function based on both diversity and accuracy. The method is applied on homogenous ensembles composed of C4.5 decision trees and based on a hill climbing strategy. This allows selecting ensembles with the best compromise between maximum diversity and minimum error rate. Comparative studies show that in most cases the proposed method generates reduced size ensembles with better performances than usual ensemble simplification methods

An analysis of ensemble pruning techniques based on ordered aggregation

Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. G. Martínez-Muñoz, D. Hernández-Lobato and A. Suárez, "An analysis of ensemble pruning techniques based on ordered aggregation", IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 31, no. 2, pp. 245-249, February 2009Several pruning strategies that can be used to reduce the size and increase the accuracy of bagging ensembles are analyzed. These heuristics select subsets of complementary classifiers that, when combined, can perform better than the whole ensemble. The pruning methods investigated are based on modifying the order of aggregation of classifiers in the ensemble. In the original bagging algorithm, the order of aggregation is left unspecified. When this order is random, the generalization error typically decreases as the number of classifiers in the ensemble increases. If an appropriate ordering for the aggregation process is devised, the generalization error reaches a minimum at intermediate numbers of classifiers. This minimum lies below the asymptotic error of bagging. Pruned ensembles are obtained by retaining a fraction of the classifiers in the ordered ensemble. The performance of these pruned ensembles is evaluated in several benchmark classification tasks under different training conditions. The results of this empirical investigation show that ordered aggregation can be used for the efficient generation of pruned ensembles that are competitive, in terms of performance and robustness of classification, with computationally more costly methods that directly select optimal or near-optimal subensembles.The authors acknowledge support form the Spanish Ministerio de Educación y Ciencia under Project TIN2007-66862-C02-0

Explainable online ensemble of deep neural network pruning for time series forecasting

Both the complex and evolving nature of time series data make forecasting among one of the most challenging tasks in machine learning. Typical methods for forecasting are designed to model time-evolving dependencies between data observations. However, it is generally accepted that none of them are universally valid for every application. Therefore, methods for learning heterogeneous ensembles by combining a diverse set of forecasters together appears as a promising solution to tackle this task. While several approaches in the context of time series forecasting have focused on how to combine individual models in an ensemble, ranging from simple and enhanced averaging tactics to applying meta-learning methods, few works have tackled the task of ensemble pruning, i.e. individual model selection to take part in the ensemble. In addition, in classical ML literature, ensemble pruning techniques are mostly restricted to operate in a static manner. To deal with changes in the relative performance of models as well as changes in the data distribution, we employ gradient-based saliency maps for online ensemble pruning of deep neural networks. This method consists of generating individual models’ performance saliency maps that are subsequently used to prune the ensemble by taking into account both aspects of accuracy and diversity. In addition, the saliency maps can be exploited to provide suitable explanations for the reason behind selecting specific models to construct an ensemble that plays the role of a forecaster at a certain time interval or instant. An extensive empirical study on many real-world datasets demonstrates that our method achieves excellent or on par results in comparison to the state-of-the-art approaches as well as several baselines. Our code is available on Github (https://github.com/MatthiasJakobs/os-pgsm/tree/ecml_journal_2022)