In recent years, a certain type of problems have become of interest where one wants to query a trained classifier. Specifically, one wants to find the closest instance to a given input instance such that the classifier's predicted label is changed in a desired way. Examples of these ``inverse classification'' problems are counterfactual explanations, adversarial examples and model inversion. All of them are fundamentally optimization problems over the input instance vector involving a fixed classifier, and it is of interest to achieve a fast solution for interactive or real-time applications. We focus on solving this problem efficiently for two of the most widely used classifiers: logistic regression and softmax classifiers. Owing to special properties of these models, we show that the optimization can be solved in closed form for logistic regression, and iteratively but extremely fast for the softmax classifier. This allows us to solve either case exactly (to nearly machine precision) in a runtime of milliseconds to around a second even for very high-dimensional instances and many classes.

通过闭合解法和快速迭代，本文对逻辑回归和 softmax 分类器两种常用分类器提出了高效的解决方案，以解决逆分类、反事实解释、对抗性示例和模型反演等最优化问题。

逻辑回归与Softmax分类器的逆向分类：高效优化