We study an online linear classification problem, in which the data is generated by strategic agents who manipulate their features in an effort to change the classification outcome. In rounds, the learner deploys a classifier, and an adversarially chosen agent arrives, possibly manipulating her features to optimally respond to the learner. The learner has no knowledge of the agents' utility functions or "real" features, which may vary widely across agents. Instead, the learner is only able to observe their "revealed preferences" -i.e. the actual manipulated feature vectors they provide. For a broad family of agent cost functions, we give a computationally efficient learning algorithm that is able to obtain diminishing "Stackelberg regret" -a form of policy regret that guarantees that the learner is obtaining loss nearly as small as that of the best classifier in hindsight, even allowing for the fact that agents will best-respond differently to the optimal classifier. 1 The particulars of the models studied in Brückner and Scheffer (2011) and Hardt et al. (2016) differ. Brückner and Scheffer model a single data generation player who manipulates the data distribution, and experiences cost equal to the squared ℓ 2 distance of his manipulation. Hardt et al. study a model in which each agent can independently manipulate his own data point, but assume that all agents experience cost as a function of the same separable cost function, known to the learner.

Settings such as lending and policing can be modeled by a centralized agent allocating a scarce resource (e.g. loans or police officers) amongst several groups, in order to maximize some objective (e.g. loans given that are repaid, or criminals that are apprehended). Often in such problems fairness is also a concern. One natural notion of fairness, based on general principles of equality of opportunity, asks that conditional on an individual being a candidate for the resource in question, the probability of actually receiving it is approximately independent of the individual's group. For example, in lending this would mean that equally creditworthy individuals in different racial groups have roughly equal chances of receiving a loan. In policing it would mean that two individuals committing the same crime in different districts would have roughly equal chances of being arrested.In this paper, we formalize this general notion of fairness for allocation problems and investigate its algorithmic consequences. Our main technical results include an efficient learning algorithm that converges to an optimal fair allocation even when the allocator does not know the frequency of candidates (i.e. creditworthy individuals or criminals) in each group. This algorithm operates in a censored feedback model in which only the number of candidates who received the resource in a given allocation can be observed, rather than the true number of candidates in each group. This models the fact that we do not learn the creditworthiness of individuals we do not give loans to and do not learn about crimes committed if the police presence in a district is low.As an application of our framework and algorithm, we consider the predictive policing problem, in which the resource being allocated to each group is the number of police officers assigned to each district. The learning algorithm is trained on arrest data gathered from its own deployments on previous days, resulting in a potential feedback loop that our algorithm provably overcomes. In this case, the fairness constraint asks that the probability that an individual who has committed a crime is arrested should be independent of the district in which they live. We empirically investigate the performance of our learning algorithm on the Philadelphia Crime Incidents dataset.

Applications such as political redistricting demand quantitative measures of geometric compactness to distinguish between simple and contorted shapes. While the isoperimetric quotient, or ratio of area to perimeter squared, is commonly used in practice, it is sensitive to noisy data and irrelevant geographic features like coastline. These issues are addressed in theory by the isoperimetric profile, which plots the minimum perimeter needed to inscribe regions of different prescribed areas within the boundary of a shape. Efficient algorithms for computing this profile, however, are not known in practice. Hence, in this paper, we propose a convex Eulerian relaxation of the isoperimetric profile using total variation. We prove theoretical properties of our relaxation, showing that it still satisfies an isoperimetric inequality and yields a convex function of the prescribed area. Furthermore, we provide a discretization of the problem, an optimization technique, and experiments demonstrating the value of our relaxation.

We define a distance metric between partitions of a graph using machinery from optimal transport. Our metric is built from a linear assignment problem that matches partition components, with assignment cost proportional to transport distance over graph edges. We show that our distance can be computed using a single linear program without precomputing pairwise assignment costs and derive several theoretical properties of the metric. Finally, we provide experiments demonstrating these properties empirically, specifically focusing on the metric's value for new problems in ensemble-based analysis of political districting plans.

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