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In many applications one is faced with the problem of inferring some functional relation between input and output variables from given data. Consider, for instance, the task of email spam filtering where one seeks to find a model which automatically assigns new, previously unseen emails to class spam or non-spam. Building such a predictive model based on observed training inputs (e.g., emails) with corresponding outputs (e.g., spam labels) is a major goal of machine learning. Many learning methods assume that these training data are governed by the same distribution as the test data which the predictive model will be exposed to at application time. That assumption is violated when the test data are generated in response to the presence of a predictive model. This becomes apparent, for instance, in the above example of email spam filtering. Here, email service providers employ spam filters and spam senders engineer campaign templates such as to achieve a high rate of successful deliveries despite any filters. Most of the existing work casts such situations as learning robust models which are unsusceptible against small changes of the data generation process. The models are constructed under the worst-case assumption that these changes are performed such to produce the highest possible adverse effect on the performance of the predictive model. However, this approach is not capable to realistically model the true dependency between the model-building process and the process of generating future data. We therefore establish the concept of prediction games: We model the interaction between a learner, who builds the predictive model, and a data generator, who controls the process of data generation, as an one-shot game. The game-theoretic framework enables us to explicitly model the players' interests, their possible actions, their level of knowledge about each other, and the order at which they decide for an action. We model the players' interests as minimizing their own cost function which both depend on both players' actions. The learner's action is to choose the model parameters and the data generator's action is to perturbate the training data which reflects the modification of the data generation process with respect to the past data. We extensively study three instances of prediction games which differ regarding the order in which the players decide for their action. We first assume that both player choose their actions simultaneously, that is, without the knowledge of their opponent's decision. We identify conditions under which this Nash prediction game has a meaningful solution, that is, a unique Nash equilibrium, and derive algorithms that find the equilibrial prediction model. As a second case, we consider a data generator who is potentially fully informed about the move of the learner. This setting establishes a Stackelberg competition. We derive a relaxed optimization criterion to determine the solution of this game and show that this Stackelberg prediction game generalizes existing prediction models. Finally, we study the setting where the learner observes the data generator's action, that is, the (unlabeled) test data, before building the predictive model. As the test data and the training data may be governed by differing probability distributions, this scenario reduces to learning under covariate shift. We derive a new integrated as well as a two-stage method to account for this data set shift. In case studies on email spam filtering we empirically explore properties of all derived models as well as several existing baseline methods. We show that spam filters resulting from the Nash prediction game as well as the Stackelberg prediction game in the majority of cases outperform other existing baseline methods.

We address classification problems for which the training instances are governed by an input distribution that is allowed to differ arbitrarily from the test distribution-problems also referred to as classification under covariate shift. We derive a solution that is purely discriminative: neither training nor test distribution are modeled explicitly. The problem of learning under covariate shift can be written as an integrated optimization problem. Instantiating the general optimization problem leads to a kernel logistic regression and an exponential model classifier for covariate shift. The optimization problem is convex under certain conditions; our findings also clarify the relationship to the known kernel mean matching procedure. We report on experiments on problems of spam filtering, text classification, and landmine detection.

The standard assumption of identically distributed training and test data is violated when the test data are generated in response to the presence of a predictive model. This becomes apparent, for example, in the context of email spam filtering. Here, email service providers employ spam filters, and spam senders engineer campaign templates to achieve a high rate of successful deliveries despite the filters. We model the interaction between the learner and the data generator as a static game in which the cost functions of the learner and the data generator are not necessarily antagonistic. We identify conditions under which this prediction game has a unique Nash equilibrium and derive algorithms that find the equilibrial prediction model. We derive two instances, the Nash logistic regression and the Nash support vector machine, and empirically explore their properties in a case study on email spam filtering.