In the secretary problem, a set of secretary candidates arrive in a uniformly random order and reveal their values one by one. A company, who can only hire one candidate and hopes to maximize the expected value of its hire, needs to make irrevocable online decisions about whether to hire the current candidate. The classical framework of evaluating a policy is to compute its worst-case competitive ratio against the optimal solution in hindsight, and there the best policy – the “1/e law” – has a competitive ratio of 1/e. We propose an alternative evaluation framework through the lens of regret – the worst-case additive difference between the optimal hindsight solution and the expected performance of the policy, assuming that each value is normalized between 0 and 1. The 1/e law for the classical framework has a regret of 1 - 1/e ≈ 0.632; by contrast, we show that the class of “pricing curves” algorithms can guarantee a regret of at most 1/4 = 0.25 (which is tight within the class), and the class of “best-only pricing curves” algorithms can guarantee a regret of at most 0.190 (with a lower bound of 0.171). In addition, we show that in general, no policy can give a regret guarantee better than 0.152. Finally, we discuss other objectives in our regret-minimization framework, such as selecting the top-k candidates for k > 1, or maximizing revenue during the selection process.

We study the price of anarchy of the generalized second-price auction where bidders are value maximizers (i.e., autobidders). We show that in general the price of anarchy can be as bad as $0$. For comparison, the price of anarchy of running VCG is $1/2$ in the autobidding world. We further show a fined-grained price of anarchy with respect to the discount factors (i.e., the ratios of click probabilities between lower slots and the highest slot in each auction) in the generalized second-price auction, which highlights the qualitative relation between the smoothness of the discount factors and the efficiency of the generalized second-price auction.

Hierarchical Clustering is a popular unsupervised machine learning method with decades of history and numerous applications. We initiate the study of *differentially-private* approximation algorithms for hierarchical clustering under the rigorous framework introduced by Dasgupta (2016). We show strong lower bounds for the problem: that any $\epsilon$-DP algorithm must exhibit $O(|V|^2/ \epsilon)$-additive error for an input dataset $V$. Then, we exhibit a polynomial-time approximation algorithm with $O(|V|^{2.5}/ \epsilon)$-additive error, and an exponential-time algorithm that meets the lower bound. To overcome the lower bound, we focus on the stochastic block model, a popular model of graphs, and, with a separation assumption on the blocks, propose a private $1+o(1)$ approximation algorithm which also recovers the blocks exactly. Finally, we perform an empirical study of our algorithms and validate their performance.

In the classic contextual bandits problem, in each round t, a learner observes some context c, chooses some action i to perform, and receives some reward [Formula: see text]. We consider the variant of this problem in which in addition to receiving the reward [Formula: see text], the learner also learns the values of [Formula: see text] for some other contexts [Formula: see text] in set [Formula: see text], that is, the rewards that would be achieved by performing that action under different contexts [Formula: see text]. This variant arises in several strategic settings, such as learning how to bid in nontruthful repeated auctions, which has gained a lot of attention lately as many platforms have switched to running first price auctions. We call this problem the contextual bandits problem with cross-learning. The best algorithms for the classic contextual bandits problem achieve [Formula: see text] regret against all stationary policies, in which C is the number of contexts, K the number of actions, and T the number of rounds. We design and analyze new algorithms for the contextual bandits problem with cross-learning and show that their regret has better dependence on the number of contexts. Under complete cross-learning in which the rewards for all contexts are learned when choosing an action, that is, set [Formula: see text] contains all contexts, we show that our algorithms achieve regret [Formula: see text], removing the dependence on C. For any other cases, that is, under partial cross-learning in which [Formula: see text] for some context–action pair of (i, c), the regret bounds depend on how the sets [Formula: see text] impact the degree to which cross-learning between contexts is possible. We simulate our algorithms on real auction data from an ad exchange running first price auctions and show that they outperform traditional contextual bandit algorithms. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2022.1313 .

In this paper, we study the setting in which data owners train machine learning models collaboratively under a privacy notion called joint differential privacy [Kearns et al., 2018]. In this setting, the model trained for each data owner j uses j's data without privacy consideration and other owners' data with differential privacy guarantees. This setting was initiated in [Jain et al., 2021] with a focus on linear regressions. In this paper, we study this setting for stochastic convex optimization (SCO). We present an algorithm that is a variant of DP-SGD [Song et al., 2013; Abadi et al., 2016] and provides theoretical bounds on its population loss. We compare our algorithm to several baselines and discuss for what parameter setups our algorithm is more preferred. We also empirically study joint differential privacy in the multi-class classification problem over two public datasets. Our empirical findings are well-connected to the insights from our theoretical results.

Streaming computation plays an important role in large-scale data analysis. The sliding window model is a model of streaming computation which also captures the recency of the data. In this model, data arrives one item at a time, but only the latest $W$ data items are considered for a particular problem. The goal is to output a good solution at the end of the stream by maintaining a small summary during the stream. In this work, we propose a new algorithmic framework for designing efficient sliding window algorithms via bucketing-based sketches. Based on this new framework, we develop space-efficient sliding window algorithms for $k$-cover, $k$-clustering and diversity maximization problems. For each of the above problems, our algorithm achieves $(1\pm \varepsilon)$-approximation. Compared with the previous work, it improves both the approximation ratio and the space.

In this paper, we study the $r$-gather problem, a natural formulation of minimum-size clustering in metric spaces. The goal of $r$-gather is to partition $n$ points into clusters such that each cluster has size at least $r$, and the maximum radius of the clusters is minimized. This additional constraint completely changes the algorithmic nature of the problem, and many clustering techniques fail. Also previous dynamic and parallel algorithms do not achieve desirable complexity. We propose algorithms both in the Massively Parallel Computation (MPC) model and in the dynamic setting. Our MPC algorithm handles input points from the Euclidean space $\mathbb{R}^d$. It computes an $O(1)$-approximate solution of $r$-gather in $O(\log^{\varepsilon} n)$ rounds using total space $O(n^{1+\gamma}\cdot d)$ for arbitrarily small constants $\varepsilon,\gamma>0$. In addition our algorithm is fully scalable, i.e., there is no lower bound on the memory per machine. Our dynamic algorithm maintains an $O(1)$-approximate $r$-gather solution under insertions/deletions of points in a metric space with doubling dimension $d$. The update time is $r \cdot 2^{O(d)}\cdot \log^{O(1)}\Delta$ and the query time is $2^{O(d)}\cdot \log^{O(1)}\Delta$, where $\Delta$ is the ratio between the largest and the smallest distance.

Recently, Hierarchical Clustering (HC) has been considered through the lens of optimization. In particular, two maximization objectives have been defined. Moseley and Wang defined the \emph{Revenue} objective to handle similarity information given by a weighted graph on the data points (w.l.o.g., $[0,1]$ weights), while Cohen-Addad et al. defined the \emph{Dissimilarity} objective to handle dissimilarity information. In this paper, we prove structural lemmas for both objectives allowing us to convert any HC tree to a tree with constant number of internal nodes while incurring an arbitrarily small loss in each objective. Although the best-known approximations are 0.585 and 0.667 respectively, using our lemmas we obtain approximations arbitrarily close to 1, if not all weights are small (i.e., there exist constants $\epsilon, \delta$ such that the fraction of weights smaller than $\delta$, is at most $1 - \epsilon$); such instances encompass many metric-based similarity instances, thereby improving upon prior work. Finally, we introduce Hierarchical Correlation Clustering (HCC) to handle instances that contain similarity and dissimilarity information simultaneously. For HCC, we provide an approximation of 0.4767 and for complementary similarity/dissimilarity weights (analogous to $+/-$ correlation clustering), we again present nearly-optimal approximations.

In online advertising, advertisers purchase ad placements by participating in a long sequence of repeated auctions. One of the most important features advertising platforms often provide, and advertisers often use, is budget management, which allows advertisers to control their cumulative expenditures. Advertisers typically declare the maximum daily amount they are willing to pay, and the platform adjusts allocations and payments to guarantee that cumulative expenditures do not exceed budgets. There are multiple ways to achieve this goal, and each one, when applied to all budget-constrained advertisers simultaneously, steers the system toward a different equilibrium. While previous research focused on online stochastic optimization techniques or game-theoretic equilibria of such settings, our goal in this paper is to compare the "system equilibria" of a range of budget management strategies in terms of the seller's profit and buyers' utility. In particular, we consider six different budget management strategies including probabilistic throttling, thresholding, bid shading, reserve pricing, and multiplicative boosting. We show these methods admit a system equilibrium in a rather general setting, and prove dominance relations between them in a simplified setting. Our study sheds light on the impact of budget management strategies on the tradeoff between the seller's profit and buyers' utility. Finally, we also empirically compare the system equilibria of these strategies using real ad auction data in sponsored search and randomly generated bids. The empirical study confirms our theoretical findings about the relative performances of budget management strategies.

In this paper, we study a number of well-known combinatorial optimization problems that fit in the following paradigm: the input is a collection of (potentially inconsistent) local relationships between the elements of a ground set (e.g., pairwise comparisons, similar/dissimilar pairs, or ancestry structure of triples of points), and the goal is to aggregate this information into a global structure (e.g., a ranking, a clustering, or a hierarchical clustering) in a way that maximizes agreement with the input. Well-studied problems such as rank aggregation, correlation clustering, and hierarchical clustering with triplet constraints fall in this class of problems. We study these problems on stochastic instances with a hidden embedded ground truth solution. Our main algorithmic contribution is a unified technique that uses the maximum cut problem in graphs to approximately solve these problems. Using this technique, we can often get approximation guarantees in the stochastic setting that are better than the known worst case inapproximability bounds for the corresponding problem. On the negative side, we improve the worst case inapproximability bound on several hierarchical clustering formulations through a reduction to related ranking problems.

We propose a general framework for interactively learning combinatorial structures, such as (binary or non-binary) classifiers, orderings/rankings of items, or clusterings, when the underlying structure changes over time. Inspired by Angluin's equivalence query model, the algorithm proposes a structure in each round, and it either learns that its proposal is the true structure in this round, or it observes a specific mistake in the proposal. The feedback is correct only with probability 1 - p, and adversarially incorrect with probability p. The algorithm's goal is to minimize its number of mistakes over the course of R rounds. Our general framework is based on a graph representation of the structures and feedback in a static environment, proposed by Emamjomeh-Zadeh and Kempe (2017). To be able to learn efficiently, it is sufficient that there be a graph G whose nodes are the candidate structures and whose (weighted) edges capture the possible feedback, satisfying a certain natural shortest paths property. To model the evolution of the underlying structure, we consider two natural models, which we term the Shifting Target model and Drifting Target model. In the former, the true structure always belongs to a small pool of candidate structures. In the latter, the structure can change only by transitioning along the edges of a known evolution graph. In order to achieve non-trivial results, we bound the total number of times the underlying structure can change, denoted by B. We provide upper and lower bounds on the number of mistakes, which depend on the total number of changes B, the total number of structures n, and natural measures of complexity of the dynamic models: the size of the pool of candidate structures in the Shifting Target model, and the maximum degree of the evolution graph in the Drifting Target model.

Motivated by online advertising applications, we study incentive properties of stationary mechanisms that satisfy budget constraints in expectation at a stationary equilibrium. We consider a general repeated auction setting where a seller sells identical items to buyers with budget constraints and the buyers' value distributions can be arbitrarily correlated. We introduce the novel notion of budget-constrained incentive compatibility (BCIC) under which each buyer chooses an optimal bidding strategy among stationary budget-feasible bidding strategies. Armed with the notion of BCIC, we characterize Bayesian optimal mechanisms that satisfy the budget constraints in expectation with respect to the profit, utility and welfare objectives in the restricted setting where the buyers' value distributions are independent. Furthermore, in the general setting where the buyers' value distributions are correlated, we provide the first systematic study on the incentive properties of different budget management mechanisms, including those studied in the literature and, to the best of our knowledge, used in the industry. We explore the following mechanisms and show some popular mechanisms are not incentive compatible even when restricting attention to stationary budget-feasible deviations: throttling, thresholding, bid shading, reserve pricing and two versions of multiplicative boosting.

Hierarchical Clustering is an unsupervised data analysis method which has been widely used for decades. Despite its popularity, it had an underdeveloped analytical foundation and to address this, Dasgupta recently introduced an optimization viewpoint of hierarchical clustering with pairwise similarity information that spurred a line of work shedding light on old algorithms (e.g., Average-Linkage), but also designing new algorithms. Here, for the maximization dual of Dasgupta's objective (introduced by Moseley-Wang), we present polynomial-time .4246 approximation algorithms that use Max-Uncut Bisection as a subroutine. The previous best worst-case approximation factor in polynomial time was .336, improving only slightly over Average-Linkage which achieves 1/3. Finally, we complement our positive results by providing APX-hardness (even for 0-1 similarities), under the Small Set Expansion hypothesis.

In this paper, we study correlation clustering under fairness constraints. Fair variants of k-median and k-center clustering have been studied recently, and approximation algorithms using a notion called fairlet decomposition have been proposed. We obtain approximation algorithms for fair correlation clustering under several important types of fairness constraints. Our results hinge on obtaining a fairlet decomposition for correlation clustering by introducing a novel combinatorial optimization problem. We define a fairlet decomposition with cost similar to the k-median cost and this allows us to obtain approximation algorithms for a wide range of fairness constraints. We complement our theoretical results with an in-depth analysis of our algorithms on real graphs where we show that fair solutions to correlation clustering can be obtained with limited increase in cost compared to the state-of-the-art (unfair) algorithms.

Hybrid quantum-classical algorithms are among the most promising systems to implement quantum computing under the Noisy-Intermediate Scale Quantum (NISQ) technology. In this paper, at first, we investigate a quantum dynamics algorithm for the density matrix obeying the von Neumann equation using an efficient Lagrangian-based approach. And then, we consider the dynamics of the ensemble-averaged of disordered quantum systems which is described by Hamiltonian ensemble with a hybrid quantum-classical algorithm. In a recent work (Chen et al. in Phys Rev Lett 120:030403, 2018), the authors concluded that the dynamics of an open system could be simulated by a Hamiltonian ensemble because of nature of the disorder average. We investigate our algorithm to simulating incoherent dynamics (decoherence) of open system using an efficient variational quantum circuit in the form of master equations. Despite the non-unitary evolution of open systems, our method is applicable to a wide range of problems for incoherent dynamics with the unitary quantum operation.

As machine learning has become more prevalent, researchers have begun to recognize the necessity of ensuring machine learning systems are fair. Recently, there has been an interest in defining a notion of fairness that mitigates over-representation in traditional clustering. In this paper we extend this notion to hierarchical clustering, where the goal is to recursively partition the data to optimize a specific objective. For various natural objectives, we obtain simple, efficient algorithms to find a provably good fair hierarchical clustering. Empirically, we show that our algorithms can find a fair hierarchical clustering, with only a negligible loss in the objective.

In this paper we consider clustering problems in which each point is endowed with a color. The goal is to cluster the points to minimize the classical clustering cost but with the additional constraint that no color is over-represented in any cluster. This problem is motivated by practical clustering settings, e.g., in clustering news articles where the color of an article is its source, it is preferable that no single news source dominates any cluster. For the most general version of this problem, we obtain an algorithm that has provable guarantees of performance; our algorithm is based on finding a fractional solution using a linear program and rounding the solution subsequently. For the special case of the problem where no color has an absolute majority in any cluster, we obtain a simpler combinatorial algorithm also with provable guarantees. Experiments on real-world data shows that our algorithms are effective in finding good clustering without over-representation.

We study two well-known planar visibility problems, namely visibility testing and visibility counting, in a model where there is uncertainty about the input data. The standard versions of these problems are defined as follows: we are given a set S of n segments in R 2 , and we would like to preprocess S so that we can quickly answer queries of the form: is the given query segment s ∈ S visible from the given query point q ∈ R 2 (for visibility testing) and how many segments in S are visible from the given query point q ∈ R 2 (for visibility counting). In our model of uncertainty, each segment may or may not exist, and if it does, it is located in one of finitely many possible locations, given by a discrete probability distribution. In this setting, the probabilistic visibility testing problem (PVTP, for short) is to compute the probability that a given segment s ∈ S is visible from a given query point q and the probabilistic visibility counting problem (PVCP, for short) is to compute the expected number of segments in S that are visible from a query point q. We first show that PVTP is #P-complete. In the special case where uncertainty is only about whether segments exist and not about their location, we show that PVTP is solvable in O ( n log ⁡ n ) time. Our algorithm for PVTP combined with linearity of expectation gives an O ( n 2 log ⁡ n ) time algorithm for PVCP. Using the algorithm for PVTP, together with a few old tricks, we can show that one can preprocess S in O ( n 5 log ⁡ n ) time into a data structure of size O ( n 4 ) , so that each PVTP query for a fixed segment s can be answered in O ( log ⁡ n ) time. We also give a faster 2-approximation algorithm for this problem. At the end, we improve the approximation factor of the algorithm.

Companies like Google and Microsoft run billions of auctions every day to sell advertising opportunities. Any change to the rules of these auctions can have a tremendous effect on the revenue of the company and the welfare of the advertisers and the users. Therefore, any change requires careful evaluation of its potential impacts. Currently, such impacts are often evaluated by running simulations or small controlled experiments. This, however, misses the important factor that the advertisers respond to changes. Our goal is to build a theoretical framework for predicting the actions of an agent (the advertiser) that is optimizing her actions in an uncertain environment. We model this problem using a variant of the multi-armed bandit setting where playing an arm is costly. The cost of each arm changes over time and is publicly observable. The value of playing an arm is drawn stochastically from a static distribution and is observed by the agent and not by us. We, however, observe the actions of the agent. Our main result is that assuming the agent is playing a strategy with a regret of at most f(T) within the first T rounds, we can learn to play the multi-armed bandits game (without observing the rewards) in such a way that the regret of our selected actions is at most O(k^4(f(T)+1)log (T)) , where k is the number of arms.