Kearns et al. [2018] recently proposed a notion of rich subgroup fairness intended to bridge the gap between statistical and individual notions of fairness. Rich subgroup fairness picks a statistical fairness constraint (say, equalizing false positive rates across protected groups), but then asks that this constraint hold over an exponentially or infinitely large collection of subgroups defined by a class of functions with bounded VC dimension. They give an algorithm guaranteed to learn subject to this constraint, under the condition that it has access to oracles for perfectly learning absent a fairness constraint. In this paper, we undertake an extensive empirical evaluation of the algorithm of Kearns et al. On four real datasets for which fairness is a concern, we investigate the basic convergence of the algorithm when instantiated with fast heuristics in place of learning oracles, measure the tradeoffs between fairness and accuracy, and compare this approach with the recent algorithm of Agarwal et al. [2018], which implements weaker and more traditional marginal fairness constraints defined by individual protected attributes. We find that in general, the Kearns et al. algorithm converges quickly, large gains in fairness can be obtained with mild costs to accuracy, and that optimizing accuracy subject only to marginal fairness leads to classifiers with substantial subgroup unfairness. We also provide a number of analyses and visualizations of the dynamics and behavior of the Kearns et al. algorithm. Overall we find this algorithm to be effective on real data, and rich subgroup fairness to be a viable notion in practice.
We study the power of interactivity in local differential privacy. First, we focus on the difference between fully interactive and sequentially interactive protocols. Sequentially interactive protocols may query users adaptively in sequence, but they cannot return to previously queried users. The vast majority of existing lower bounds for local differential privacy apply only to sequentially interactive protocols, and before this paper it was not known whether fully interactive protocols were more powerful.We resolve this question. First, we classify locally private protocols by their compositionality, the multiplicative factor k ≥ 1 by which the sum of a protocol's single-round privacy parameters exceeds its overall privacy guarantee. We then show how to efficiently transform any fully interactive k-compositional protocol into an equivalent sequentially interactive protocol with an O(k) blowup in sample complexity. Next, we show that our reduction is tight by exhibiting a family of problems such that for any k, there is a fully interactive k-compositional protocol which solves the problem, while no sequentially interactive protocol can solve the problem without at least anΩ(k) factor more examples.We then turn our attention to hypothesis testing problems. We show that for a large class of compound hypothesis testing problems -which include all simple hypothesis testing problems as a special case -a simple noninteractive test is optimal among the class of all (possibly fully interactive) tests.
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.
We study fairness in linear bandit problems. Starting from the notion of meritocratic fairness introduced in Joseph et al. [2016], we carry out a more refined analysis of a more general problem, achieving better performance guarantees with fewer modelling assumptions on the number and structure of available choices as well as the number selected. We also analyze the previously-unstudied question of fairness in infinite linear bandit problems, obtaining instance-dependent regret upper bounds as well as lower bounds demonstrating that this instance-dependence is necessary. The result is a framework for meritocratic fairness in an online linear setting that is substantially more powerful, general, and realistic than the current state of the art.
Abstract. Bipartite, periodic, planar graphs known as brane tilings can be associated to a large class of quivers. This paper will explore new algebraic properties of the well-studied del Pezzo 3 quiver and geometric properties of its corresponding brane tiling. In particular, a factorization formula for the cluster variables arising from a large class of mutation sequences (called τ −mutation sequences) is proven; this factorization also gives a recursion on the cluster variables produced by such sequences. We can realize these sequences as walks in a triangular lattice using a correspondence between the generators of the affine symmetric groupà 2 and the mutations which generate τ −mutation sequences. Using this bijection, we obtain explicit formulae for the cluster that corresponds to a specific alcove in the lattice. With this lattice visualization in mind, we then express each cluster variable produced in a τ -mutation sequence as the sum of weighted perfect matchings of a new family of subgraphs of the dP3 brane tiling, which we call Aztec castles. Our main result generalizes previous work on a certain mutation sequence on the dP3 quiver in [Zha12], and forms part of the emerging story in combinatorics and theoretical high energy physics relating cluster variables to subgraphs of the associated brane tiling.
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