8,008 research outputs found

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

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    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

    New multicategory boosting algorithms based on multicategory Fisher-consistent losses

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    Fisher-consistent loss functions play a fundamental role in the construction of successful binary margin-based classifiers. In this paper we establish the Fisher-consistency condition for multicategory classification problems. Our approach uses the margin vector concept which can be regarded as a multicategory generalization of the binary margin. We characterize a wide class of smooth convex loss functions that are Fisher-consistent for multicategory classification. We then consider using the margin-vector-based loss functions to derive multicategory boosting algorithms. In particular, we derive two new multicategory boosting algorithms by using the exponential and logistic regression losses.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS198 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Formal Verification of Input-Output Mappings of Tree Ensembles

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    Recent advances in machine learning and artificial intelligence are now being considered in safety-critical autonomous systems where software defects may cause severe harm to humans and the environment. Design organizations in these domains are currently unable to provide convincing arguments that their systems are safe to operate when machine learning algorithms are used to implement their software. In this paper, we present an efficient method to extract equivalence classes from decision trees and tree ensembles, and to formally verify that their input-output mappings comply with requirements. The idea is that, given that safety requirements can be traced to desirable properties on system input-output patterns, we can use positive verification outcomes in safety arguments. This paper presents the implementation of the method in the tool VoTE (Verifier of Tree Ensembles), and evaluates its scalability on two case studies presented in current literature. We demonstrate that our method is practical for tree ensembles trained on low-dimensional data with up to 25 decision trees and tree depths of up to 20. Our work also studies the limitations of the method with high-dimensional data and preliminarily investigates the trade-off between large number of trees and time taken for verification

    Efficient Monte Carlo Integration Using Boosted Decision Trees and Generative Deep Neural Networks

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    New machine learning based algorithms have been developed and tested for Monte Carlo integration based on generative Boosted Decision Trees and Deep Neural Networks. Both of these algorithms exhibit substantial improvements compared to existing algorithms for non-factorizable integrands in terms of the achievable integration precision for a given number of target function evaluations. Large scale Monte Carlo generation of complex collider physics processes with improved efficiency can be achieved by implementing these algorithms into commonly used matrix element Monte Carlo generators once their robustness is demonstrated and performance validated for the relevant classes of matrix elements

    Optimization by gradient boosting

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    Gradient boosting is a state-of-the-art prediction technique that sequentially produces a model in the form of linear combinations of simple predictors---typically decision trees---by solving an infinite-dimensional convex optimization problem. We provide in the present paper a thorough analysis of two widespread versions of gradient boosting, and introduce a general framework for studying these algorithms from the point of view of functional optimization. We prove their convergence as the number of iterations tends to infinity and highlight the importance of having a strongly convex risk functional to minimize. We also present a reasonable statistical context ensuring consistency properties of the boosting predictors as the sample size grows. In our approach, the optimization procedures are run forever (that is, without resorting to an early stopping strategy), and statistical regularization is basically achieved via an appropriate L2L^2 penalization of the loss and strong convexity arguments
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