We introduce a novel sensitivity analysis framework for large scale classification problems that can be used when a small number of instances are incrementally added or removed. For quickly updating the classifier in such a situation, incremental learning algorithms have been intensively studied in the literature. Although they are much more efficient than solving the optimization problem from scratch, their computational complexity yet depends on the entire training set size. It means that, if the original training set is large, completely solving an incremental learning problem might be still rather expensive. To circumvent this computational issue, we propose a novel framework that allows us to make an inference about the updated classifier without actually re-optimizing it. Specifically, the proposed framework can quickly provide a lower and an upper bounds of a quantity on the unknown updated classifier. The main advantage of the proposed framework is that the computational cost of computing these bounds depends only on the number of updated instances. This property is quite advantageous in a typical sensitivity analysis task where only a small number of instances are updated. In this paper we demonstrate that the proposed framework is applicable to various practical sensitivity analysis tasks, and the bounds provided by the framework are often sufficiently tight for making desired inferences.

Careful tuning of a regularization parameter is indispensable in many machine learning tasks because it has a significant impact on generalization performances.Nevertheless, current practice of regularization parameter tuning is more of an art than a science, e.g., it is hard to tell how many grid-points would be needed in cross-validation (CV) for obtaining a solution with sufficiently small CV error.In this paper we propose a novel framework for computing a lower bound of the CV errors as a function of the regularization parameter, which we call regularization path of CV error lower bounds.The proposed framework can be used for providing a theoretical approximation guarantee on a set of solutions in the sense that how far the CV error of the current best solution could be away from best possible CV error in the entire range of the regularization parameters.We demonstrate through numerical experiments that a theoretically guaranteed a choice of regularization parameter in the above sense is possible with reasonable computational costs.

In this paper, we claim that some of the non-support vectors (non-SVs) that have no influence on the classifier can be screened out prior to the training phase in pathwise SVM computation scenario, in which one is asked to train a sequence (or path) of SVM classifiers for different regularization parameters. Based on a recently proposed framework so-called safe screening rule, we derive a rule for screening out non-SVs in advance, and discuss how we can exploit the advantage of the rule in pathwise SVM computation scenario. Experiments indicate that our approach often substantially reduce the total pathwise computation cost.