An Exponential Lower Bound on the Complexity of Regularization Paths

For a variety of regularized optimization problems in machine learning, algorithms computing the entire solution path have been developed recently. Most of these methods are quadratic programs that are parameterized by a single parameter, as for example the Support Vector Machine (SVM). Solution path algorithms do not only compute the solution for one particular value of the regularization parameter but the entire path of solutions, making the selection of an optimal parameter much easier. It has been assumed that these piecewise linear solution paths have only linear complexity, i.e. linearly many bends. We prove that for the support vector machine this complexity can be exponential in the number of training points in the worst case. More strongly, we construct a single instance of n input points in d dimensions for an SVM such that at least \Theta(2^{n/2}) = \Theta(2^d) many distinct subsets of support vectors occur as the regularization parameter changes.Comment: Journal version, 28 Pages, 5 Figure

A Partial parametric path algorithm for multiclass classification

The objective functions of Support Vector Machine methods (SVMs) often includeparameters to weigh the relative importance of margins and training accuracies.The values of these parameters have a direct effect both on the optimal accuraciesand the misclassification costs. Usually, a grid search is used to find appropriatevalues for them. This method requires the repeated solution of quadraticprograms for different parameter values, and it may imply a large computationalcost, especially in a setting of multiclass SVMs and large training datasets. Formulti-class classification problems, in the presence of different misclassificationcosts, identifying a desirable set of values for these parameters becomes evenmore relevant. In this paper, we propose a partial parametric path algorithm, basedon the property that the path of optimal solutions of the SVMs with respect tothe preceding parameters is piecewise linear. This partial parametric path algorithmrequires the solution of just one quadratic programming problem, and anumber of linear systems of equations. Thus it can significantly reduce the computationalrequirements of the algorithm. To systematically explore the differentweights to assign to the misclassification costs, we combine the partial parametricpath algorithm with a variable neighborhood search method. Our numerical experimentsshow the efficiency and reliability of the proposed partial parametricpath algorithm