Distributionally Robust Machine Learning with Multi-source Data

Classical machine learning methods may lead to poor prediction performance when the target distribution differs from the source populations. This paper utilizes data from multiple sources and introduces a group distributionally robust prediction model defined to optimize an adversarial reward about explained variance with respect to a class of target distributions. Compared to classical empirical risk minimization, the proposed robust prediction model improves the prediction accuracy for target populations with distribution shifts. We show that our group distributionally robust prediction model is a weighted average of the source populations' conditional outcome models. We leverage this key identification result to robustify arbitrary machine learning algorithms, including, for example, random forests and neural networks. We devise a novel bias-corrected estimator to estimate the optimal aggregation weight for general machine-learning algorithms and demonstrate its improvement in the convergence rate. Our proposal can be seen as a distributionally robust federated learning approach that is computationally efficient and easy to implement using arbitrary machine learning base algorithms, satisfies some privacy constraints, and has a nice interpretation of different sources' importance for predicting a given target covariate distribution. We demonstrate the performance of our proposed group distributionally robust method on simulated and real data with random forests and neural networks as base-learning algorithms

Robust risk aggregation with neural networks

We consider settings in which the distribution of a multivariate random variable is partly ambiguous. We assume the ambiguity lies on the level of the dependence structure, and that the marginal distributions are known. Furthermore, a current best guess for the distribution, called reference measure, is available. We work with the set of distributions that are both close to the given reference measure in a transportation distance (e.g. the Wasserstein distance), and additionally have the correct marginal structure. The goal is to find upper and lower bounds for integrals of interest with respect to distributions in this set. The described problem appears naturally in the context of risk aggregation. When aggregating different risks, the marginal distributions of these risks are known and the task is to quantify their joint effect on a given system. This is typically done by applying a meaningful risk measure to the sum of the individual risks. For this purpose, the stochastic interdependencies between the risks need to be specified. In practice the models of this dependence structure are however subject to relatively high model ambiguity. The contribution of this paper is twofold: Firstly, we derive a dual representation of the considered problem and prove that strong duality holds. Secondly, we propose a generally applicable and computationally feasible method, which relies on neural networks, in order to numerically solve the derived dual problem. The latter method is tested on a number of toy examples, before it is finally applied to perform robust risk aggregation in a real world instance.Comment: Revised version. Accepted for publication in "Mathematical Finance