Federated Learning (FL) is a distributed machine learning scheme that enables clients to train a shared global model without exchanging local data. The presence of label noise can severely degrade the FL performance, and some existing studies have focused on algorithm design for label denoising. However, they ignored the important issue that clients may not apply costly label denoising strategies due to them being self-interested and having heterogeneous valuations on the FL performance. To fill this gap, we model the clients' interactions as a novel label denoising game and characterize its equilibrium. We also analyze the price of stability, which quantifies the difference in the system performance (e.g., global model accuracy, social welfare) between the equilibrium outcome and the socially optimal solution. We prove that the equilibrium outcome always leads to a lower global model accuracy than the socially optimal solution does. We further design an efficient algorithm to compute the socially optimal solution. Numerical experiments on MNIST dataset show that the price of stability increases as the clients' data become noisier, calling for an effective incentive mechanism.
Outlier detection is critical in real applications to prevent financial fraud, defend network intrusions, or detecting imminent device failures. To reduce the human effort in evaluating outlier detection results and effectively turn the outliers into actionable insights, the users often expect a system to automatically produce interpretable summarizations of subgroups of outlier detection results. Unfortunately, to date no such systems exist. To fill this gap, we propose STAIR which learns a compact set of human understandable rules to summarize and explain the anomaly detection results. Rather than use the classical decision tree algorithms to produce these rules, STAIR proposes a new optimization objective to produce a small number of rules with least complexity, hence strong interpretability, to accurately summarize the detection results. The learning algorithm of STAIR produces a rule set by iteratively splitting the large rules and is optimal in maximizing this objective in each iteration. Moreover, to effectively handle high dimensional, highly complex data sets which are hard to summarize with simple rules, we propose a localized STAIR approach, called L-STAIR. Taking data locality into consideration, it simultaneously partitions data and learns a set of localized rules for each partition. Our experimental study on many outlier benchmark datasets shows that STAIR significantly reduces the complexity of the rules required to summarize the outlier detection results, thus more amenable for humans to understand and evaluate, compared to the decision tree methods.