In this paper, we investigate the problem of optimization of multivariate performance measures, and propose a novel algorithm for it. Different from traditional machine learning methods which optimize simple loss functions to learn prediction function, the problem studied in this paper is how to learn effective hyper-predictor for a tuple of data points, so that a complex loss function corresponding to a multivariate performance measure can be minimized. We propose to present the tuple of data points to a tuple of sparse codes via a dictionary, and then apply a linear function to compare a sparse code against a given candidate class label. To learn the dictionary, sparse codes, and parameter of the linear function, we propose a joint optimization problem. In this problem, the both the reconstruction error and sparsity of sparse code, and the upper bound of the complex loss function are minimized. Moreover, the upper bound of the loss function is approximated by the sparse codes and the linear function parameter. To optimize this problem, we develop an iterative algorithm based on descent gradient methods to learn the sparse codes and hyper-predictor parameter alternately. Experiment results on some benchmark data sets show the advantage of the proposed methods over other state-of-the-art algorithms.
Joachims @cite_19 proposed a SVM method to optimize multivariate nonlinear performance measures, including F-score, AUC etc. This method takes a multivariate predictor, and gives an algorithm to train the a multivariate SVM in polynomial time for large classes so that the potentially non-linear performance measures can be optimized. Moreover, the translational SVM with hinge loss function can be treated as a special case of this method.
@cite_19
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This paper presents a Support Vector Method for optimizing multivariate nonlinear performance measures like the F 1 -score. Taking a multivariate prediction approach, we give an algorithm with which such multivariate SVMs can be trained in polynomial time for large classes of potentially non-linear performance measures, in particular ROCArea and all measures that can be computed from the contingency table. The conventional classification SVM arises as a special case of our method. Optimizing multivariate performance measure is an important task in Machine Learning. Joachims (2005) introduced a Support Vector Method whose underlying optimization problem is commonly solved by cutting plane methods (CPMs) such as SVM-Perf and BMRM. It can be shown that CPMs converge to an e accurate solution in O(1 λe) iterations, where λ is the trade-off parameter between the regularizer and the loss function. Motivated by the impressive convergence rate of CPM on a number of practical problems, it was conjectured that these rates can be further improved. We disprove this conjecture in this paper by constructing counter examples. However, surprisingly, we further discover that these problems are not inherently hard, and we develop a novel smoothing strategy, which in conjunction with Nesterov's accelerated gradient method, can find an e accurate solution in O* (min 1 e, 1 √λe ) iterations. Computationally, our smoothing technique is also particularly advantageous for optimizing multivariate performance scores such as precision recall break-even point and ROCArea; the cost per iteration remains the same as that of CPMs. Empirical evaluation on some of the largest publicly available data sets shows that our method converges significantly faster than CPMs without sacrificing generalization ability. In the interest of deriving classifiers that are robust to outlier observations, we present integer programming formulations of Vapnik's support vector machine (SVM) with the ramp loss and hard margin loss. The ramp loss allows a maximum error of 2 for each training observation, while the hard margin loss calculates error by counting the number of training observations that are in the margin or misclassified outside of the margin. SVM with these loss functions is shown to be a consistent estimator when used with certain kernel functions. In computational studies with simulated and real-world data, SVM with the robust loss functions ignores outlier observations effectively, providing an advantage over SVM with the traditional hinge loss when using the linear kernel. Despite the fact that training SVM with the robust loss functions requires the solution of a quadratic mixed-integer program (QMIP) and is NP-hard, while traditional SVM requires only the solution of a continuous quadratic program (QP), we are able to find good solutions and prove optimality for instances with up to 500 observations. Solution methods are presented for the new formulations that improve computational performance over industry-standard integer programming solvers alone. Recently, there has been a renewed interest in the machine learning community for variants of a sparse greedy approximation procedure for concave optimization known as the Frank-Wolfe (FW) method. In particular, this procedure has been successfully applied to train large-scale instances of non-linear Support Vector Machines (SVMs). Specializing FW to SVM training has allowed to obtain efficient algorithms, but also important theoretical results, including convergence analysis of training algorithms and new characterizations of model sparsity.In this paper, we present and analyze a novel variant of the FW method based on a new way to perform away steps, a classic strategy used to accelerate the convergence of the basic FW procedure. Our formulation and analysis is focused on a general concave maximization problem on the simplex. However, the specialization of our algorithm to quadratic forms is strongly related to some classic methods in computational geometry, namely the Gilbert and MDM algorithms.On the theoretical side, we demonstrate that the method matches the guarantees in terms of convergence rate and number of iterations obtained by using classic away steps. In particular, the method enjoys a linear rate of convergence, a result that has been recently proved for MDM on quadratic forms.On the practical side, we provide experiments on several classification datasets, and evaluate the results using statistical tests. Experiments show that our method is faster than the FW method with classic away steps, and works well even in the cases in which classic away steps slow down the algorithm. Furthermore, these improvements are obtained without sacrificing the predictive accuracy of the obtained SVM model.