There has been substantial recent concern that automated pricing algorithms might learn to "collude." Supra-competitive prices can emerge as a Nash equilibrium of repeated pricing games, in which sellers play strategies which threaten to punish their competitors if they ever "defect" from a set of supra-competitive prices, and these strategies can be automatically learned. But threats are anti-competitive on their face. In fact, a standard economic intuition is that supra-competitive prices emerge from either the use of threats, or a failure of one party to correctly optimize their payoff. Is this intuition correct? Would explicitly preventing threats in algorithmic decision-making prevent supra-competitive prices when sellers are optimizing for their own revenue? No. We show that supra-competitive prices can robustly emerge even when both players are using algorithms which do not explicitly encode threats, and which optimize for their own revenue. Since deploying an algorithm is a form of commitment, we study sequential Bertrand pricing games (and a continuous variant) in which a first mover deploys an algorithm and then a second mover optimizes within the resulting environment. We show that if the first mover deploys any algorithm with a no-regret guarantee, and then the second mover even approximately optimizes within this now static environment, monopoly-like prices arise. The result holds for any no-regret learning algorithm deployed by the first mover and for any pricing policy of the second mover that obtains them profit at least as high as a random pricing would - and hence the result applies even when the second mover is optimizing only within a space of non-responsive pricing distributions which are incapable of encoding threats. In fact, there exists a set of strategies, neither of which explicitly encode threats that form a Nash equilibrium of the simultaneous pricing game in algorithm space, and lead to near monopoly prices. This suggests that the definition of "algorithmic collusion" may need to be expanded, to include strategies without explicitly encoded threats.

Given a non-negative n × n matrix viewed as a set of distances between n points, we consider the property testing problem of deciding if it is a metric. We also consider the same problem for two special classes of metrics, tree metrics and ultrametrics. For general metrics, our paper is the first to consider these questions. We prove an upper bound of O(n^2/3/ϵ^4/3) on the query complexity for this problem. Our algorithm is simple, but the analysis requires great care in bounding the variance on the number of violating triangles in a sample. When ϵ is a slowly decreasing function of n (rather than a constant, as is standard), we prove a lower bound of matching dependence on n of Ω (n^2/3), ruling out any property testers with o(n^2/3) query complexity unless their dependence on 1/ϵ is super-polynomial. Next, we turn to tree metrics and ultrametrics. While there were known upper and lower bounds, we considerably improve these bounds showing essentially tight bounds of Õ(1/ϵ ) on the sample complexity. We also show a lower bound of Ω ( 1/ϵ^4/3 ) on the query complexity. Our upper bounds are derived by doing a more careful analysis of a natural, simple algorithm. For the lower bounds, we construct distributions on NO instances, where it is hard to find a witness showing that these are not ultrametrics.

We study the problem of online prediction, in which at each time step $t$, an individual $x_t$ arrives, whose label we must predict. Each individual is associated with various groups, defined based on their features such as age, sex, race etc., which may intersect. Our goal is to make predictions that have regret guarantees not just overall but also simultaneously on each sub-sequence comprised of the members of any single group. Previous work such as [Blum & Lykouris] and [Lee et al] provide attractive regret guarantees for these problems; however, these are computationally intractable on large model classes. We show that a simple modification of the sleeping experts technique of [Blum & Lykouris] yields an efficient reduction to the well-understood problem of obtaining diminishing external regret absent group considerations. Our approach gives similar regret guarantees compared to [Blum & Lykouris]; however, we run in time linear in the number of groups, and are oracle-efficient in the hypothesis class. This in particular implies that our algorithm is efficient whenever the number of groups is polynomially bounded and the external-regret problem can be solved efficiently, an improvement on [Blum & Lykouris]'s stronger condition that the model class must be small. Our approach can handle online linear regression and online combinatorial optimization problems like online shortest paths. Beyond providing theoretical regret bounds, we evaluate this algorithm with an extensive set of experiments on synthetic data and on two real data sets -- Medical costs and the Adult income dataset, both instantiated with intersecting groups defined in terms of race, sex, and other demographic characteristics. We find that uniformly across groups, our algorithm gives substantial error improvements compared to running a standard online linear regression algorithm with no groupwise regret guarantees.

We present a stylized model with feedback loops for the evolution of a population's wealth over generations. Individuals have both talent and wealth: talent is a random variable distributed identically for everyone, but wealth is a random variable that is dependent on the population one is born into. Individuals then apply to a downstream agent, which we treat as a university throughout the paper (but could also represent an employer) who makes a decision about whether to admit them or not. The university does not directly observe talent or wealth, but rather a signal (representing e.g. a standardized test) that is a convex combination of both. The university knows the distributions from which an individual's type and wealth are drawn, and makes its decisions based on the posterior distribution of the applicant's characteristics conditional on their population and signal. Each population's wealth distribution at the next round then depends on the fraction of that population that was admitted by the university at the previous round. We study wealth dynamics in this model, and give conditions under which the dynamics have a single attracting fixed point (which implies population wealth inequality is transitory), and conditions under which it can have multiple attracting fixed points (which implies that population wealth inequality can be persistent). In the case in which there are multiple attracting fixed points, we study interventions aimed at eliminating or mitigating inequality, including increasing the capacity of the university to admit more people, aligning the signal generated by individuals with the preferences of the university, and making direct monetary transfers to the less wealthy population.

The problem of reconstructing evolutionary trees or phylogenies is of great interest in computational biology. A popular model for this problem assumes that we are given the set of leaves (current species) of an unknown binary tree and the results of `experiments' on triples of leaves (a,b,c), which return the pair with the deepest least common ancestor. If the tree is assumed to be an ultrametric (i.e., all root-leaf paths have the same length), the experiment can be equivalently seen to return the closest pair of leaves. In this model, efficient algorithms are known for tree reconstruction. In reality, since the data on which these `experiments' are run is itself generated by the stochastic process of evolution, these experiments are noisy. In all reasonable models of evolution, if the branches leading to the leaves in a triple separate from each other at common ancestors that are very close to each other in the tree, the result of the experiment should be close to uniformly random. Motivated by this, we consider a model where the noise on any triple is just dependent on the three pairwise distances (referred to as distance based noise). Our results are the following: 1. Suppose the length of every edge in the unknown tree is at least $\tilde{O}(\frac{1}{\sqrt n})$ fraction of the length of a root-leaf path. Then, we give an efficient algorithm to reconstruct the topology of the tree for a broad family of distance-based noise models. Further, we show that if the edges are asymptotically shorter, then topology reconstruction is information-theoretically impossible. 2. Further, for a specific distance-based noise model--which we refer to as the homogeneous noise model--we show that the edge weights can also be approximately reconstructed under the same quantitative lower bound on the edge lengths.

We introduce the pipeline intervention problem, defined by a layered directed acyclic graph and a set of stochastic matrices governing transitions between successive layers. The graph is a stylized model for how people from different populations are presented opportunities, eventually leading to some reward. In our model, individuals are born into an initial position (i.e., some node in the first layer of the graph) according to a fixed probability distribution and then stochastically progress through the graph according to the transition matrices until they reach a node in the final layer of the graph; each node in the final layer has a reward associated with it. The pipeline intervention problem asks how to best make costly changes to the transition matrices governing people’s stochastic transitions through the graph subject to a budget constraint. We consider two objectives: social welfare maximization and a fairness-motivated maximin objective that seeks to maximize the value to the population (starting node) with the least expected value. We consider two variants of the maximin objective that turn out to be distinct, depending on whether we demand a deterministic solution or allow randomization. For each objective, we give an efficient approximation algorithm (an additive fully polynomial-time approximation scheme) for constant-width networks. We also tightly characterize the “price of fairness” in our setting: the ratio between the highest achievable social welfare and the social welfare consistent with a maximin optimal solution. Finally, we show that, for polynomial-width networks, even approximating the maximin objective to any constant factor is NP hard even for networks with constant depth. This shows that the restriction on the width in our positive results is essential.

We study a game theoretic model of standardized testing for college admissions. Students are of two types; High and Low. There is a college that would like to admit the High type students. Students take a potentially costly standardized exam which provides a noisy signal of their type. The students come from two populations, which are identical in talent (i.e. the type distribution is the same), but differ in their access to resources: the higher resourced population can at their option take the exam multiple times, whereas the lower resourced population can only take the exam once. We study two models of score reporting, which capture existing policies used by colleges. The first policy (sometimes known as super-scoring) allows students to report the max of the scores they achieve. The other policy requires that all scores be reported. We find in our model that requiring that all scores be reported results in superior outcomes in equilibrium, both from the perspective of the college (the admissions rule is more accurate), and from the perspective of equity across populations: a student's probability of admission is independent of their population, conditional on their type. In particular, the false positive rates and false negative rates are identical in this setting, across the highly and poorly resourced student populations. This is the case despite the fact that the more highly resourced students can -- at their option -- either report a more accurate signal of their type, or pool with the lower resourced population under this policy.

Dependency parsing algorithms capable of producing the types of crossing dependencies seen in natural language sentences have traditionally been orders of magnitude slower than algorithms for projective trees. For 95.8–99.8% of dependency parses in various natural language treebanks, whenever an edge is crossed, the edges that cross it all have a common vertex. The optimal dependency tree that satisfies this 1-Endpoint-Crossing property can be found with an O( n 4 ) parsing algorithm that recursively combines forests over intervals with one exterior point. 1-Endpoint-Crossing trees also have natural connections to linguistics and another class of graphs that has been studied in NLP.

Existing network switches implement scheduling disciplines such as FIFO or deficit round robin that provide good utilization or fairness across flows, but do so at the expense of leaking a variety of information via timing side channels. To address this privacy breach, we propose a new scheduling mechanism for switches called indifferent-first scheduling (IFS). A salient aspect of IFS is that it provides privacy (a notion of strong isolation) to clients that opt-in, while preserving the (good) performance and utilization of FIFO or round robin for clients that are satisfied with the status quo. Such a hybrid scheduling mechanism addresses the main drawback of prior proposals such as time-division multiple access (TDMA) that provide strong isolation at the cost of low utilization and increased packet latency for all clients. We identify limitations of modern programmable switches which inhibit an implementation of IFS without compromising its privacy guarantees, and show that a version of IFS with full security can be implemented at line rate in the recently proposed push-in-first-out (PIFO) queuing architecture.

There is increasing regulatory interest in whether machine learning algorithms deployed in consequential domains (e.g. in criminal justice) treat different demographic groups "fairly." However, there are several proposed notions of fairness, typically mutually incompatible. Using criminal justice as an example, we study a model in which society chooses an incarceration rule. Agents of different demographic groups differ in their outside options (e.g. opportunity for legal employment) and decide whether to commit crimes. We show that equalizing type I and type II errors across groups is consistent with the goal of minimizing the overall crime rate; other popular notions of fairness are not.

This paper introduces a new cryptographic primitive called a private resource allocator (PRA) that can be used to allocate resources (e.g., network bandwidth, CPUs) to a set of clients without revealing to the clients whether any other clients received resources. We give several constructions of PRAs that provide guarantees ranging from information-theoretic to differential privacy. PRAs are useful in preventing a new class of attacks that we call allocation-based side-channel attacks. These attacks can be used, for example, to break the privacy guarantees of anonymous messaging systems that were designed specifically to defend against side-channel and traffic analysis attacks. Our implementation of PRAs in Alpenhorn, which is a recent anonymous messaging system, shows that PRAs increase the network resources required to start a conversation by up to 16× (can be made as low as 4× in some cases), but add no overhead once the conversation has been established.

We consider the multi-armed bandit setting with a twist. Rather than having just one decision maker deciding which arm to pull in each round, we have $n$ different decision makers (agents). In the simple stochastic setting, we show that a "free-riding" agent observing another "self-reliant" agent can achieve just $O(1)$ regret, as opposed to the regret lower bound of $\Omega (\log t)$ when one decision maker is playing in isolation. This result holds whenever the self-reliant agent's strategy satisfies either one of two assumptions: (1) each arm is pulled at least $\gamma \ln t$ times in expectation for a constant $\gamma$ that we compute, or (2) the self-reliant agent achieves $o(t)$ realized regret with high probability. Both of these assumptions are satisfied by standard zero-regret algorithms. Under the second assumption, we further show that the free rider only needs to observe the number of times each arm is pulled by the self-reliant agent, and not the rewards realized. In the linear contextual setting, each arm has a distribution over parameter vectors, each agent has a context vector, and the reward realized when an agent pulls an arm is the inner product of that agent's context vector with a parameter vector sampled from the pulled arm's distribution. We show that the free rider can achieve $O(1)$ regret in this setting whenever the free rider's context is a small (in $L_2$-norm) linear combination of other agents' contexts and all other agents pull each arm $\Omega (\log t)$ times with high probability. Again, this condition on the self-reliant players is satisfied by standard zero-regret algorithms like UCB. We also prove a number of lower bounds.

Suppose a graph G is stochastically created by uniformly sampling vertices along a line segment and connecting each pair of vertices with a probability that is a known decreasing function of their distance. We ask if it is possible to reconstruct the actual positions of the vertices in G by only observing the generated unlabeled graph. We study this question for two natural edge probability functions — one where the probability of an edge decays exponentially with the distance and another where this probability decays only linearly. We initiate our study with the weaker goal of recovering only the order in which vertices appear on the line segment. For a segment of length n and a precision parameter δ, we show that for both exponential and linear decay edge probability functions, there is an efficient algorithm that correctly recovers (up to reflection symmetry) the order of all vertices that are at least δ apart, using only samples (vertices). Building on this result, we then show that vertices (samples) are sufficient to additionally recover the location of each vertex on the line to within a precision of δ. We complement this result with an lower bound on samples needed for reconstructing positions (even by a computationally unbounded algorithm), showing that the task of recovering positions is information-theoretically harder than recovering the order. We give experimental results showing that our algorithm recovers the positions of almost all points with high accuracy.

When selecting locations for a set of facilities, standard clustering algorithms may place unfair burden on some individuals and neighborhoods. We formulate a fairness concept that takes local population densities into account. In particular, given $k$ facilities to locate and a population of size $n$, we define the "neighborhood radius" of an individual $i$ as the minimum radius of a ball centered at $i$ that contains at least $n/k$ individuals. Our objective is to ensure that each individual has a facility within at most a small constant factor of her neighborhood radius. We present several theoretical results: We show that optimizing this factor is NP-hard; we give an approximation algorithm that guarantees a factor of at most 2 in all metric spaces; and we prove matching lower bounds in some metric spaces. We apply a variant of this algorithm to real-world address data, showing that it is quite different from standard clustering algorithms and outperforms them on our objective function and balances the load between facilities more evenly.

Recent high-throughput structure-sensitive genome-wide sequencing-based assays have enabled large-scale studies of RNA structure, and robust transcriptome-wide computational prediction of individual RNA structures across RNA classes from these assays has potential to further improve the prediction accuracy. Here, we describe HiPR, a novel method for RNA structure prediction at single-nucleotide resolution that combines high-throughput structure probing data (DMS-seq, DMS-MaPseq) with a novel probabilistic folding algorithm. On validation data spanning a variety of RNA classes, HiPR often increases accuracy for predicting RNA structures, giving researchers new tools to study RNA structure.

The Monitoring and Checking (MaC) project gave rise to a framework for runtime monitoring with respect to formally specified properties, which later came to be known as runtime verification. The project also built a pioneering runtime verification tool, Java-MaC, that was an instantiation of the approach to check properties of Java programs. In this retrospective, we discuss decisions made in the design of the framework and summarize lessons learned in the course of the project.

Given a drawing of a read-once formula (called the blueprint), and a blackbox implementation with the same topology as the blueprint that purports to compute the formula, can we tell if it does? Under a fault model, where the only faults in the implementation are gates that complement their outputs, we show that there is an efficient algorithm that makes a linear number of probes to the blackbox implementation and determines if the blueprint and implementation are identical. We also show a matching lower bound. We further ask whether we can diagnose where the faults are, using blackbox testing. We prove that if the implementation has a property called polynomial balance , then it is possible to do this efficiently. To complement this result, we show that even if the blueprint is polynomially balanced and there are only logarithmically many errors in the implementation, the implementation could be unbalanced and the diagnosis problem provably requires super-polynomially many tests. We point out that this problem is one instance of a general class of problems of learning deviations from a blueprint, which we call conformance learning . Conformance learning seems worthy of further investigation in a broader context.

We study a two-stage model, in which students are 1) admitted to college on the basis of an entrance exam which is a noisy signal about their qualifications (type), and then 2) those students who were admitted to college can be hired by an employer as a function of their college grades, which are an independently drawn noisy signal of their type. Students are drawn from one of two populations, which might have different type distributions. We assume that the employer at the end of the pipeline is rational, in the sense that it computes a posterior distribution on student type conditional on all information that it has available (college admissions, grades, and group membership), and makes a decision based on posterior expectation. We then study what kinds of fairness goals can be achieved by the college by setting its admissions rule and grading policy. For example, the college might have the goal of guaranteeing equal opportunity across populations: that the probability of passing through the pipeline and being hired by the employer should be independent of group membership, conditioned on type. Alternately, the college might have the goal of incentivizing the employer to have a group blind hiring rule. We show that both goals can be achieved when the college does not report grades. On the other hand, we show that under reasonable conditions, these goals are impossible to achieve even in isolation when the college uses an (even minimally) informative grading policy.

Bandit learning is characterized by the tension between long-term exploration and short-term exploitation. However, as has recently been noted, in settings in which the choices of the learning algorithm correspond to important decisions about individual people (such as criminal recidivism prediction, lending, and sequential drug trials), exploration corresponds to explicitly sacrificing the well-being of one individual for the potential future benefit of others. This raises a fairness concern. In such settings, one might like to run a "greedy" algorithm, which always makes the (myopically) optimal decision for the individuals at hand - but doing this can result in a catastrophic failure to learn. In this paper, we consider the linear contextual bandit problem and revisit the performance of the greedy algorithm. We give a smoothed analysis, showing that even when contexts may be chosen by an adversary, small perturbations of the adversary's choices suffice for the algorithm to achieve "no regret", perhaps (depending on the specifics of the setting) with a constant amount of initial training data. This suggests that "generically" (i.e. in slightly perturbed environments), exploration and exploitation need not be in conflict in the linear setting.

We consider the problem of implementing an individually rational, asymptotically Pareto optimal allocation in a barter-exchange economy where agents are endowed with goods and preferences over the goods of others, but may not use money as a medium of exchange. Because one of the most important instantiations of such economies is kidney exchange -- where the "input" to the problem consists of sensitive patient medical records -- we ask to what extent such exchanges can be carried out while providing formal privacy guarantees to the participants. We show that individually rational allocations cannot achieve any non-trivial approximation to Pareto optimality if carried out under the constraint of differential privacy -- or even the relaxation of joint-differential privacy, under which it is known that asymptotically optimal allocations can be computed in two sided markets [Hsu et al. STOC 2014]. We therefore consider a further relaxation that we call marginal-differential privacy --which promises, informally, that the privacy of every agent i is protected from every other agent j ≠ i so long as j does not collude or share allocation information with other agents. We show that under marginal differential privacy, it is possible to compute an individually rational and asymptotically Pareto optimal allocation in such exchange economies.