112,061 research outputs found
The Support Vector Machine and Mixed Integer Linear Programming: Ramp Loss SVM with L1-Norm Regularization
The support vector machine (SVM) is a flexible classification method that accommodates a kernel trick to learn nonlinear decision rules. The traditional formulation as an optimization problem is a quadratic program. In efforts to reduce computational complexity, some have proposed using an L1-norm regularization to create a linear program (LP). In other efforts aimed at increasing the robustness to outliers, investigators have proposed using the ramp loss which results in what may be expressed as a quadratic integer programming problem (QIP). In this paper, we consider combining these ideas for ramp loss SVM with L1-norm regularization. The result is four formulations for SVM that each may be expressed as a mixed integer linear program (MILP). We observe that ramp loss SVM with L1-norm regularization provides robustness to outliers with the linear kernel. We investigate the time required to find good solutions to the various formulations using a branch and bound solver
Multi-group support vector machines with measurement costs a biobjective approach
Support Vector Machine has shown to have good performance in many practical classification settings. In this paper we propose, for multi-group classification, a biobjective optimization model in which we consider not only the generalization ability (modelled through the margin maximization), but also costs associated with the features. This cost is not limited to an economical payment, but can also refer to risk, computational effort, space requirements, etc. We introduce a biobjective mixed integer problem, for which Pareto optimal solutions are obtained. Those Pareto optimal solutions correspond to different classification rules, among which the user would choose the one yielding the most appropriate compromise between the cost and the expected misclassification rate.Ministerio de Ciencia y TecnologíaPlan Andaluz de Investigació
Ordered Counterfactual Explanation by Mixed-Integer Linear Optimization
Post-hoc explanation methods for machine learning models have been widely
used to support decision-making. One of the popular methods is Counterfactual
Explanation (CE), also known as Actionable Recourse, which provides a user with
a perturbation vector of features that alters the prediction result. Given a
perturbation vector, a user can interpret it as an "action" for obtaining one's
desired decision result. In practice, however, showing only a perturbation
vector is often insufficient for users to execute the action. The reason is
that if there is an asymmetric interaction among features, such as causality,
the total cost of the action is expected to depend on the order of changing
features. Therefore, practical CE methods are required to provide an
appropriate order of changing features in addition to a perturbation vector.
For this purpose, we propose a new framework called Ordered Counterfactual
Explanation (OrdCE). We introduce a new objective function that evaluates a
pair of an action and an order based on feature interaction. To extract an
optimal pair, we propose a mixed-integer linear optimization approach with our
objective function. Numerical experiments on real datasets demonstrated the
effectiveness of our OrdCE in comparison with unordered CE methods.Comment: 20 pages, 5 figures, to appear in the 35th AAAI Conference on
Artificial Intelligence (AAAI 2021
Polynomiality for Bin Packing with a Constant Number of Item Types
We consider the bin packing problem with d different item sizes s_i and item
multiplicities a_i, where all numbers are given in binary encoding. This
problem formulation is also known as the 1-dimensional cutting stock problem.
In this work, we provide an algorithm which, for constant d, solves bin
packing in polynomial time. This was an open problem for all d >= 3.
In fact, for constant d our algorithm solves the following problem in
polynomial time: given two d-dimensional polytopes P and Q, find the smallest
number of integer points in P whose sum lies in Q.
Our approach also applies to high multiplicity scheduling problems in which
the number of copies of each job type is given in binary encoding and each type
comes with certain parameters such as release dates, processing times and
deadlines. We show that a variety of high multiplicity scheduling problems can
be solved in polynomial time if the number of job types is constant
Probabilistic Graphical Models on Multi-Core CPUs using Java 8
In this paper, we discuss software design issues related to the development
of parallel computational intelligence algorithms on multi-core CPUs, using the
new Java 8 functional programming features. In particular, we focus on
probabilistic graphical models (PGMs) and present the parallelisation of a
collection of algorithms that deal with inference and learning of PGMs from
data. Namely, maximum likelihood estimation, importance sampling, and greedy
search for solving combinatorial optimisation problems. Through these concrete
examples, we tackle the problem of defining efficient data structures for PGMs
and parallel processing of same-size batches of data sets using Java 8
features. We also provide straightforward techniques to code parallel
algorithms that seamlessly exploit multi-core processors. The experimental
analysis, carried out using our open source AMIDST (Analysis of MassIve Data
STreams) Java toolbox, shows the merits of the proposed solutions.Comment: Pre-print version of the paper presented in the special issue on
Computational Intelligence Software at IEEE Computational Intelligence
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