For the classical maximum coverage problem, the greedy algorithm achieves a worst-case 1-1/e approximation, which is optimal unless P = NP. The notion of coverage appears in a wide range of optimization tasks, where empirical evaluations indicate approximation ratios close to 1 for the greedy algorithm on real data. Random models have provided average-case justifications for the empirical performance of many well-known algorithms, but little is known about the average-case performance of greedy for maximum coverage. We analyze the expected approximation ratio of the greedy algorithm in a random model, which we call the left-regular random model. We first show that, for all parameter settings of this model, the expected approximation ratio of the greedy algorithm improves by a constant over its worst-case 1-1/e guarantee. We then identify two simple conditions, either of which ensures that the expected approximation ratio is close to 1 for sufficiently large graphs. Finally, we show that there is a regime where greedy does not achieve an expected approximation better than 0.94. To obtain these results, we develop analytical tools, including a novel application of the differential equation method and a connection to maximum matching in Erdős-Rényi graphs, which may be of independent interest for other random models.

In the correlation clustering problem, the input is a signed graph where the sign indicates whether pairs of nodes should be placed in the same cluster or not. The goal is to create a clustering that minimizes the number of disagreements with these signs. Correlation clustering is a key unsupervised learning problem with many practical applications. It has been widely studied in various settings, including versions with fairness constraints and cases where nodes arrive online. In this paper, we explore a problem that combines these two settings: nodes arrive online and reveal their membership in protected groups upon arrival. We are only allowed to output fair clusters, i.e., clusters where the representation of each protected group is upper bounded by a user-specified constant at the beginning of the arrival sequence. Our aim is to maintain approximately optimal fair clustering while minimizing a node’s worst-case recourse, i.e., the number of times it changes clusters. We present an algorithm that achieves worst-case logarithmic recourse per node while maintaining a constant-factor fair approximate clustering. Additionally, our approach simplifies the algorithm and analysis used in prior work by Cohen-Addad et al. (2022) in the online setting with recourse.

An important challenge in robust machine learning is when training data is provided by strategic sources who may intentionally report erroneous data for their own benefit. A line of work at the intersection of machine learning and mechanism design aims to deter strategic agents from reporting erroneous training data by designing learning algorithms that are strategyproof. Strategyproofness is a strong and desirable property, but it comes at a cost in the approximation ratio of even simple risk minimization problems. In this paper, we study strategyproof regression and classification problems in a model with advice. This model is part of a recent line on mechanism design with advice where the goal is to achieve both an improved approximation ratio when the advice is correct (consistency) and a bounded approximation ratio when the advice is incorrect (robustness). We provide the first non-trivial consistency-robustness tradeoffs for strategyproof regression and classification, which hold for simple yet interesting classes of functions. For classes of constant functions, we give a deterministic and strategyproof mechanism that is, for any $\gamma \in (0, 2]$, $1+\gamma$ consistent and $1 + 4/\gamma$ robust and provide a lower bound that shows that this tradeoff is optimal. We extend this mechanism and its guarantees to homogeneous linear regression over $\mathbb{R}$. In the binary classification problem of selecting from three or more labelings, we present strong impossibility results for both deterministic and randomized mechanism. Finally, we provide deterministic and randomized mechanisms for selecting from two labelings.

We study the problem of designing procurement auctions for the strategic uncapacitated facility location problem: a company needs to procure a set of facility locations in order to serve its customers and each facility location is owned by a strategic agent. Each owner has a private cost for providing access to their facility (e.g., renting it or selling it to the company) and needs to be compensated accordingly. The goal is to design truthful auctions that decide which facilities the company should procure and how much to pay the corresponding owners, aiming to minimize the total cost, i.e., the monetary cost paid to the owners and the connection cost suffered by the customers (their distance to the nearest facility). We evaluate the performance of these auctions using the \emph{frugality ratio}. We first analyze the performance of the classic VCG auction in this context and prove that its frugality ratio is exactly $3$. We then leverage the learning-augmented framework and design auctions that are augmented with predictions regarding the owners' private costs. Specifically, we propose a family of learning-augmented auctions that achieve significant payment reductions when the predictions are accurate, leading to much better frugality ratios. At the same time, we demonstrate that these auctions remain robust even if the predictions are arbitrarily inaccurate, and maintain reasonable frugality ratios even under adversarially chosen predictions. We finally provide a family of ``error-tolerant'' auctions that maintain improved frugality ratios even if the predictions are only approximately accurate, and we provide upper bounds on their frugality ratio as a function of the prediction error.

In dynamic submodular maximization, the goal is to maintain a high-value solution over a sequence of element insertions and deletions with a fast update time. Motivated by large-scale applications and the fact that dynamic data often exhibits patterns, we ask the following question: can predictions be used to accelerate the update time of dynamic submodular maximization algorithms? We consider the model for dynamic algorithms with predictions where predictions regarding the insertion and deletion times of elements can be used for preprocessing. Our main result is an algorithm with an $O(\text{poly}(\log \eta, \log w, \log k))$ amortized update time over the sequence of updates that achieves a $1/2 - \epsilon$ approximation for dynamic monotone submodular maximization under a cardinality constraint $k$, where the prediction error $\eta$ is the number of elements that are not inserted and deleted within $w$ time steps of their predicted insertion and deletion times. This amortized update time is independent of the length of the stream and instead depends on the prediction error.

In the strategic facility location problem, a set of agents report their locations in a metric space and the goal is to use these reports to open a new facility, minimizing an aggregate distance measure from the agents to the facility. However, agents are strategic and may misreport their locations to influence the facility’s placement in their favor. The aim is to design truthful mechanisms, ensuring agents cannot gain by misreporting. This problem was recently revisited through the learning-augmented framework, aiming to move beyond worst-case analysis and design truthful mechanisms that are augmented with (machine-learned) predictions. The focus of this prior work was on mechanisms that are deterministic and augmented with a prediction regarding the optimal facility location. In this paper, we provide a deeper understanding of this problem by exploring the power of randomization as well as the impact of different types of predictions on the performance of truthful learning-augmented mechanisms. We study both the single-dimensional and the Euclidean case and provide upper and lower bounds regarding the achievable approximation of the optimal egalitarian social cost.

Algorithms with predictions is a recent framework for decision-making under uncertainty that leverages the power of machine-learned predictions without making any assumption about their quality. The goal in this framework is for algorithms to achieve an improved performance when the predictions are accurate while maintaining acceptable guarantees when the predictions are erroneous. A serious concern with algorithms that use predictions is that these predictions can be biased and, as a result, cause the algorithm to make decisions that are deemed unfair. We show that this concern manifests itself in the classical secretary problem in the learning-augmented setting---the state-of-the-art algorithm can have zero probability of accepting the best candidate, which we deem unfair, despite promising to accept a candidate whose expected value is at least $\max\{\Omega (1) , 1 - O(\varepsilon)\}$ times the optimal value, where $\varepsilon$ is the prediction error. We show how to preserve this promise while also guaranteeing to accept the best candidate with probability $\Omega(1)$. Our algorithm and analysis are based on a new ``pegging'' idea that diverges from existing works and simplifies/unifies some of their results. Finally, we extend to the $k$-secretary problem and complement our theoretical analysis with experiments.

The assignment game, introduced by Shapley and Shubik [1971], is a classic model for two-sided matching markets between buyers and sellers. In the original assignment game, it is assumed that payments lead to transferable utility and that buyers have unit-demand valuations for the items being sold. There has since been substantial work studying various extensions of the assignment game. The first main area of extension is to imperfectly transferable utility, which is when frictions, taxes, or fees impede the transfer of money between agents. The second is with more complex valuation functions, in particular gross substitutes valuations, which describe substitutable goods. Multiple efficient algorithms have been proposed for computing a competitive equilibrium, the standard solution concept in assignment games, in each of these two settings. However, these lines of work have been mostly independent, with no algorithmic results combining the two.

Aiming to overcome some of the limitations of worst-case analysis, the recently proposed framework of "algorithms with predictions" allows algorithms to be augmented with a (possibly erroneous) machine-learned prediction that they can use as a guide. In this framework, the goal is to obtain improved guarantees when the prediction is correct, which is called consistency, while simultaneously guaranteeing some worst-case bounds even when the prediction is arbitrarily wrong, which is called robustness. The vast majority of the work on this framework has focused on a refined analysis of online algorithms augmented with predictions regarding the future input. A subsequent line of work has also successfully adapted this framework to mechanism design, where the prediction is regarding the private information of strategic agents. In this paper, we initiate the study of online mechanism design with predictions, which combines the challenges of online algorithms with predictions and mechanism design with predictions.

In their seminal paper that initiated the field of algorithmic mechanism design, \citet{NR99} studied the problem of designing strategyproof mechanisms for scheduling jobs on unrelated machines aiming to minimize the makespan. They provided a strategyproof mechanism that achieves an $n$-approximation and they made the bold conjecture that this is the best approximation achievable by any deterministic strategyproof scheduling mechanism. After more than two decades and several efforts, $n$ remains the best known approximation and very recent work by \citet{CKK21} has been able to prove an $Ω(\sqrt{n})$ approximation lower bound for all deterministic strategyproof mechanisms. This strong negative result, however, heavily depends on the fact that the performance of these mechanisms is evaluated using worst-case analysis. To overcome such overly pessimistic, and often uninformative, worst-case bounds, a surge of recent work has focused on the ``learning-augmented framework'', whose goal is to leverage machine-learned predictions to obtain improved approximations when these predictions are accurate (consistency), while also achieving near-optimal worst-case approximations even when the predictions are arbitrarily wrong (robustness). In this work, we study the classic strategic scheduling problem of~\citet{NR99} using the learning-augmented framework and give a deterministic polynomial-time strategyproof mechanism that is $6$-consistent and $2n$-robust. We thus achieve the ``best of both worlds'': an $O(1)$ consistency and an $O(n)$ robustness that asymptotically matches the best-known approximation. We then extend this result to provide more general worst-case approximation guarantees as a function of the prediction error. Finally, we complement our positive results by showing that any $1$-consistent deterministic strategyproof mechanism has unbounded robustness.

An important goal of modern scheduling systems is to efficiently manage power usage. In energy-efficient scheduling, the operating system controls the speed at which a machine is processing jobs with the dual objective of minimizing energy consumption and optimizing the quality of service cost of the resulting schedule. Since machine-learned predictions about future requests can often be learned from historical data, a recent line of work on learning-augmented algorithms aims to achieve improved performance guarantees by leveraging predictions. In particular, for energy-efficient scheduling, Bamas et. al. [NeurIPS '20] and Antoniadis et. al. [SWAT '22] designed algorithms with predictions for the energy minimization with deadlines problem and achieved an improved competitive ratio when the prediction error is small while also maintaining worst-case bounds even when the prediction error is arbitrarily large. In this paper, we consider a general setting for energy-efficient scheduling and provide a flexible learning-augmented algorithmic framework that takes as input an offline and an online algorithm for the desired energy-efficient scheduling problem. We show that, when the prediction error is small, this framework gives improved competitive ratios for many different energy-efficient scheduling problems, including energy minimization with deadlines, while also maintaining a bounded competitive ratio regardless of the prediction error. Finally, we empirically demonstrate that this framework achieves an improved performance on real and synthetic datasets.

In the online minimum cost matching problem, there are n servers and, at each of n time steps, a request arrives and must be irrevocably matched to a server that has not yet been matched, with the goal of minimizing the sum of the distances between the matched pairs. Online minimum cost matching is a central problem in applications such as ride-hailing platforms and food delivery services. Despite achieving a worst-case competitive ratio that is exponential in n even on the line, the simple greedy algorithm, which matches each request to its nearest available server, performs well in practice and has a number of attractive features such as strategyproofness. A major question is thus to explain greedy's strong empirical performance. In this paper, we aim to understand the performance of greedy on the line over instances that are at least partially random.

We study the problem of learning a hypergraph via edge detecting queries. In this problem, a learner queries subsets of vertices of a hidden hypergraph and observes whether these subsets contain an edge or not. In general, learning a hypergraph with m edges of maximum size d requires Omega((2m/d)^{d/2}) queries. In this paper, we aim to identify families of hypergraphs that can be learned without suffering from a query complexity that grows exponentially in the size of the edges. We show that hypermatchings and low-degree near-uniform hypergraphs with n vertices are learnable with poly(n) queries. For learning hypermatchings (hypergraphs of maximum degree Delta = 1), we give an O(log^3 n)-round algorithm with O(n log^5 n) queries. We complement this upper bound by showing that there are no algorithms with poly(n) queries that learn hypermatchings in o(log log n) adaptive rounds. For hypergraphs with maximum degree Delta and edge size ratio rho, we give a non-adaptive algorithm with O((2n)^{rho Delta+1} log^2 n) queries. To the best of our knowledge, these are the first algorithms with poly(n, m) query complexity for learning non-trivial families of hypergraphs that have a super-constant number of edges of arbitrarily large size.

In this work we introduce an alternative model for the design and analysis of strategyproof mechanisms that is motivated by the recent surge of work in "learning-augmented algorithms". Aiming to complement the traditional approach in computer science, which analyzes the performance of algorithms based on worst-case instances, this line of work has focused on the design and analysis of algorithms that are enhanced with machine-learned predictions regarding the optimal solution. The algorithms can use the predictions as a guide to inform their decisions, and the goal is to achieve much stronger performance guarantees when these predictions are accurate (consistency), while also maintaining near-optimal worst-case guarantees, even if these predictions are very inaccurate (robustness). So far, these results have been limited to algorithms, but in this work we argue that another fertile ground for this framework is in mechanism design.

We revisit the well-studied problem of budget-feasible procurement, where a buyer with a strict budget constraint seeks to acquire services from a group of strategic providers (the sellers). During the last decade, several strategyproof budget-feasible procurement auctions have been proposed, aiming to maximize the value of the buyer, while eliciting each seller's true cost for providing their service. These solutions predominantly take the form of randomized sealed-bid auctions: they ask the sellers to report their private costs and then use randomization to determine which subset of services will be procured and how much each of the chosen providers will be paid, ensuring that the total payment does not exceed the buyer's budget. Our main result in this paper is a novel method for designing budget-feasible auctions, leading to solutions that outperform the previously proposed auctions in multiple ways. First, our solutions take the form of descending clock auctions, and thus satisfy a list of very appealing properties, such as obvious strategyproofness, group strategyproofness, transparency, and unconditional winner privacy; this makes these auctions much more likely to be used in practice. Second, in contrast to previous results that heavily depend on randomization, our auctions are deterministic. As a result, we provide an affirmative answer to one of the main open questions in this literature, asking whether a deterministic strategyproof auction can achieve a constant approximation when the buyer's valuation function is submodular over the set of services. In addition to this, we also provide the first deterministic budget-feasible auction that matches the approximation bound of the best-known randomized auction for the class of subadditive valuations. Finally, using our method, we improve the best-known approximation factor for monotone submodular valuations, which has been the focus of most of the prior work.

In this article, we consider the following question: Can we optimize objective functions from the training data we use to learn them? We formalize this question through a novel framework we call optimization from samples ( OPS ). In OPS , we are given sampled values of a function drawn from some distribution and the objective is to optimize the function under some constraint. While there are interesting classes of functions that can be optimized from samples, our main result is an impossibility. We show that there are classes of functions that are statistically learnable and optimizable, but for which no reasonable approximation for optimization from samples is achievable. In particular, our main result shows that there is no constant factor approximation for maximizing coverage functions under a cardinality constraint using polynomially-many samples drawn from any distribution. We also show tight approximation guarantees for maximization under a cardinality constraint of several interesting classes of functions including unit-demand, additive, and general monotone submodular functions, as well as a constant factor approximation for monotone submodular functions with bounded curvature.

The predominant measure for the performance of an algorithm is its worst-case approximation guarantee. While worst-case approximations give desirable robustness guarantees, they can differ significantly from the performance of an algorithm in practice. For the problem of monotone submodular maximization under a cardinality constraint, the greedy algorithm is known to obtain a 1-1/e approximation guarantee, which is optimal for a polynomial-time algorithm. However, very little is known about the approximation achieved by greedy and other submodular maximization algorithms on real instances. We develop an algorithm that gives an instance-specific approximation for any solution of an instance of monotone submodular maximization under a cardinality constraint. This algorithm uses a novel dual approach to submodular maximization. In particular, it relies on the construction of a lower bound to the dual objective that can also be exactly minimized. We use this algorithm to show that on a wide variety of real-world datasets and objectives, greedy and other algorithms find solutions that approximate the optimal solution significantly better than the 1-1/e 0.63 worst-case approximation guarantee, often exceeding 0.9.

We study dynamic mechanisms for optimizing revenue in repeated auctions, that are robust to heterogeneous forward-looking and learning behavior of the buyers. Typically it is assumed that the buyers are either all myopic or are all infinite lookahead, and that buyers understand and trust the mechanism. These assumptions raise the following question: is it possible to design approximately revenue optimal mechanisms when the buyer pool is heterogeneous? We provide this fresh perspective on the problem by considering a heterogeneous population of buyers with an unknown mixture of k-lookahead buyers, myopic buyers, no-regret-learners and no-policy-regret learners. Facing this population, we design a simple state-based mechanism that achieves a constant fraction of the optimal achievable revenue.

In this paper we describe a new parallel algorithm called Fast Adaptive Sequencing Technique (FAST) for maximizing a monotone submodular function under a cardinality constraint k. This algorithm achieves the optimal 1-1/e approximation guarantee and is orders of magnitude faster than the state-of-the-art on a variety of experiments over real-world data sets. Following recent work by Balkanski and Singer (2018), there has been a great deal of research on algorithms whose theoretical parallel runtime is exponentially faster than algorithms used for submodular maximization over the past 40 years. However, while these new algorithms are fast in terms of asymptotic worst-case guarantees, it is computationally infeasible to use them in practice even on small data sets because the number of rounds and queries they require depend on large constants and high-degree polynomials in terms of precision and confidence. The design principles behind the FAST algorithm we present here are a significant departure from those of recent theoretically fast algorithms. Rather than optimize for asymptotic theoretical guarantees, the design of FAST introduces several new techniques that achieve remarkable practical and theoretical parallel runtimes. The approximation guarantee obtained by FAST is arbitrarily close to 1 - 1/e, and its asymptotic parallel runtime (adaptivity) is O(log(n) log^2(log k)) using O(n log log(k)) total queries. We show that FAST is orders of magnitude faster than any algorithm for submodular maximization we are aware of, including hyper-optimized parallel versions of state-of-the-art serial algorithms, by running experiments on large data sets.

In this paper, we study the adaptive complexity of maximizing a monotone gross substitutes function under a cardinality constraint. Our main result is an algorithm that achieves a 1-epsilon approximation in O(log n) adaptive rounds for any constant epsilon > 0, which is an exponential speedup in parallel running time compared to previously studied algorithms for gross substitutes functions. We show that the algorithmic results are tight in the sense that there is no algorithm that obtains a constant factor approximation in o(log n) rounds. Both the upper and lower bounds are under the assumption that queries are only on feasible sets (i.e., of size at most k). We also show that under a stronger model, where non-feasible queries are allowed, there is no non-adaptive algorithm that obtains an approximation better than 1/2 + epsilon. Both lower bounds extend to the class of OXS functions. Additionally, we conduct experiments on synthetic and real data sets to demonstrate the near-optimal performance and efficiency of the algorithm in practice.

In this paper, we study submodular function minimization in the adaptive complexity model. Seminal work by Gr'otschel, Lovász, and Schrijver shows that with oracle access to a function f, the problem of submodular minimization can be solved exactly with poly(n) queries to f. A long line of work has since then been dedicated to the acceleration of submodular minimization. In particular, recent work obtains a (strongly) polynomial time algorithm with Õ(n 3) query complexity. A natural way to accelerate computation is via parallelization, though very little is known about the extent to which submodular minimization can be parallelized.

We study a secretary problem which captures the task of ranking in online settings. We term this problem the secretary ranking problem: elements from an ordered set arrive in random order and instead of picking the maximum element, the algorithm is asked to assign a rank, or position, to each of the elements. The rank assigned is irrevocable and is given knowing only the pairwise comparisons with elements previously arrived. The goal is to minimize the distance of the rank produced to the true rank of the elements measured by the Kendall-Tau distance, which corresponds to the number of pairs that are inverted with respect to the true order. Our main result is a matching upper and lower bound for the secretary ranking problem. We present an algorithm that ranks n elements with only O(n^{3/2}) inversions in expectation, and show that any algorithm necessarily suffers \Omega(n^{3/2}) inversions when there are n available positions. In terms of techniques, the analysis of our algorithm draws connections to linear probing in the hashing literature, while our lower bound result relies on a general anti-concentration bound for a generic balls and bins sampling process. We also consider the case where the number of positions m can be larger than the number of secretaries n and provide an improved bound by showing a connection of this problem with random binary trees.

In this paper we study the adaptivity of submodular maximization. Adaptivity quantifies the number of sequential rounds that an algorithm makes when function evaluations can be executed in parallel. Adaptivity is a fundamental concept that is heavily studied across a variety of areas in computer science, largely due to the need for parallelizing computation. For the canonical problem of maximizing a monotone submodular function under a cardinality constraint, it is well known that a simple greedy algorithm achieves a 1 – 1/e approximation [NWF78] and that this approximation is optimal for polynomial-time algorithms [NW78]. Somewhat surprisingly, despite extensive efforts on submodular optimization for large-scale datasets, until very recently there was no known algorithm that achieves a constant factor approximation for this problem whose adaptivity is sublinear in the size of the ground set n. Recent work by [BS18] describes an algorithm that obtains an approximation arbitrarily close to 1/3 in O(log n) adaptive rounds and shows that no algorithm can obtain a constant factor approximation in õ(log n) adaptive rounds. This approach achieves an exponential speedup in adaptivity (and parallel running time) at the expense of approximation quality. In this paper we describe a novel approach that yields an algorithm whose approximation is arbitrarily close to the optimal 1 – 1/e guarantee in O(log n) adaptive rounds. This algorithm therefore achieves an exponential speedup in parallel running time for submodular maximization at the expense of an arbitrarily small loss in approximation quality. This guarantee is optimal in both approximation and adaptivity, up to lower order terms.

In this paper we study submodular maximization under a matroid constraint in the adaptive complexity model. This model was recently introduced in the context of submodular optimization to quantify the information theoretic complexity of black-box optimization in a parallel computation model. Informally, the adaptivity of an algorithm is the number of sequential rounds it makes when each round can execute polynomially-many function evaluations in parallel. Since submodular optimization is regularly applied on large datasets we seek algorithms with low adaptivity to enable speedups via parallelization. Consequently, a recent line of work has been devoted to designing constant factor approximation algorithms for maximizing submodular functions under various constraints in the adaptive complexity model.

In this paper we analyze an adaptive sampling approach for submodular maximization. Adaptive sampling is a technique that has recently been shown to achieve a constant factor approximation guarantee for submodular maximization under a cardinality constraint with exponentially fewer adaptive rounds than any previously studied constant factor approximation algorithm for this problem. Adaptivity quantifies the number of sequential rounds that an algorithm makes when function evaluations can be executed in parallel and is the parallel running time of an algorithm, up to low order terms. Adaptive sampling achieves its exponential speedup at the expense of approximation. In theory, it is guaranteed to produce a solution that is a 1/3 approximation to the optimum. Nevertheless, experiments show that adaptive sampling techniques achieve far better values in practice. In this paper we provide theoretical justification for this phenomenon. In particular, we show that under very mild conditions of curvature of a function, adaptive sampling techniques achieve an approximation arbitrarily close to 1/2 while maintaining their low adaptivity. Furthermore, we show that the approximation ratio approaches 1 in direct relationship to a homogeneity property of the submodular function. In addition, we conduct experiments on real data sets in which the curvature and homogeneity properties can be easily manipulated and demonstrate the relationship between approximation and curvature, as well as the effectiveness of adaptive sampling in practice.

Submodular functions have become a ubiquitous tool in machine learning. They are learnable from data, and can be optimized efficiently and with guarantees. Nonetheless, recent negative results show that optimizing learned surrogates of submodular functions can result in arbitrarily bad approximations of the true optimum. Our goal in this paper is to highlight the source of this hardness, and propose an alternative criterion for optimizing general combinatorial functions from sampled data. We prove a tight equivalence showing that a class of functions is optimizable if and only if it can be learned. We provide efficient and scalable optimization algorithms for several function classes of interest, and demonstrate their utility on the task of optimally choosing trending social media items.

In this paper we consider parallelization for applications whose objective can be expressed as maximizing a non-monotone submodular function under a cardinality constraint. Our main result is an algorithm whose approximation is arbitrarily close to 1/2e in O(log^2 n) adaptive rounds, where n is the size of the ground set. This is an exponential speedup in parallel running time over any previously studied algorithm for constrained non-monotone submodular maximization. Beyond its provable guarantees, the algorithm performs well in practice. Specifically, experiments on traffic monitoring and personalized data summarization applications show that the algorithm finds solutions whose values are competitive with state-of-the-art algorithms while running in exponentially fewer parallel iterations.

Gross substitutability is a central concept in Economics and is connected to important notions in Discrete Convex Analysis, Number Theory and the analysis of Greedy algorithms in Computer Science. Many different characterizations are known for this class, but providing a constructive description remains a major open problem. The construction problem asks how to construct all gross substitutes from a class of simpler functions using a set of operations. Since gross substitutes are a natural generalization of matroids to real-valued functions, matroid rank functions form a desirable such class of simpler functions. Shioura proved that a rich class of gross substitutes can be expressed as sums of matroid rank functions, but it is open whether all gross substitutes can be constructed this way. Our main result is a negative answer showing that some gross substitutes cannot be expressed as positive linear combinations of matroid rank functions. En route, we provide necessary and sufficient conditions for the sum to preserve substitutability, uncover a new operation preserving substitutability and fully describe all substitutes with at most 4 items.

In this paper we study the adaptive complexity of submodular optimization. Informally, the adaptive complexity of a problem is the minimal number of sequential rounds required to achieve a constant factor approximation when polynomially-many queries can be executed in parallel at each round. Adaptivity is a fundamental concept that is heavily studied in computer science, largely due to the need for parallelizing computation. Somewhat surprisingly, very little is known about adaptivity in submodular optimization. For the canonical problem of maximizing a monotone submodular function under a cardinality constraint, to the best of our knowledge, all that is known to date is that the adaptive complexity is between 1 and Ω(n).

In this paper we study optimization from samples of convex functions. There are many scenarios in which we do not know the function we wish to optimize but can learn it from data. In such cases, we are interested in bounding the number of samples required to optimize the function. Our main result shows that in general, the number of samples required to obtain a non-trivial approximation to the optimum of a convex function is exponential in its dimension, even when the function is PAC-learnable. We also obtain strong lower bounds for strongly convex and Lipschitz continuous functions. On the positive side, we show that there are interesting classes of functions and distributions for which the sample complexity is polynomial in the dimension of the function.

We consider the canonical problem of influence maximization in social networks. Since the seminal work of Kempe, Kleinberg, and Tardos there have been two, largely disjoint efforts on this problem. The first studies the problem associated with learning the generative model that produces cascades, and the second focuses on the algorithmic challenge of identifying a set of influencers, assuming the generative model is known. Recent results on learning and optimization imply that in general, if the generative model is not known but rather learned from training data, no algorithm for influence maximization can yield a constant factor approximation guarantee using polynomially-many samples, drawn from any distribution. In this paper we describe a simple algorithm for maximizing influence from training data. The main idea behind the algorithm is to leverage the strong community structure of social networks and identify a set of individuals who are influentials but whose communities have little overlap. Although in general, the approximation guarantee of such an algorithm is unbounded, we show that this algorithm performs well experimentally. To analyze its performance, we prove this algorithm obtains a constant factor approximation guarantee on graphs generated through the stochastic block model, traditionally used to model networks with community structure.

In this paper we consider the problem of minimizing a submodular function from training data. Submodular functions can be efficiently minimized and are conse- quently heavily applied in machine learning. There are many cases, however, in which we do not know the function we aim to optimize, but rather have access to training data that is used to learn the function. In this paper we consider the question of whether submodular functions can be minimized in such cases. We show that even learnable submodular functions cannot be minimized within any non-trivial approximation when given access to polynomially-many samples. Specifically, we show that there is a class of submodular functions with range in [0, 1] such that, despite being PAC-learnable and minimizable in polynomial-time, no algorithm can obtain an approximation strictly better than 1/2 − o(1) using polynomially-many samples drawn from any distribution. Furthermore, we show that this bound is tight using a trivial algorithm that obtains an approximation of 1/2.

We study the cost sharing problem for cooperative games in situations where the cost function C is not available via oracle queries, but must instead be learned from samples drawn from a distribution, represented as tuples (S, C(S)), for different subsets S of players. We formalize this approach, which we call statistical cost sharing, and consider the computation of the core and the Shapley value. Expanding on the work by Balcan et al, we give precise sample complexity bounds for computing cost shares that satisfy the core property with high probability for any function with a non-empty core. For the Shapley value, which has never been studied in this setting, we show that for submodular cost functions with curvature bounded curvature kappa it can be approximated from samples from the uniform distribution to a sqrt{1 - kappa} factor, and that the bound is tight. We then define statistical analogues of the Shapley axioms, and derive a notion of statistical Shapley value and that these can be approximated arbitrarily well from samples from any distribution and for any function.

In this paper we consider the following question: can we optimize objective functions from the training data we use to learn them? We formalize this question through a novel framework we call optimization from samples (OPS). In OPS, we are given sampled values of a function drawn from some distribution and the objective is to optimize the function under some constraint.

We consider the problem of learning sparse representations of data sets, where the goal is to reduce a data set in manner that optimizes multiple objectives. Motivated by applications of data summarization, we develop a new model which we refer to as the two-stage submodular maximization problem. This task can be viewed as a combinatorial analogue of representation learning problems such as dictionary learning and sparse regression. The two-stage problem strictly generalizes the problem of cardinality constrained submodular maximization, though the objective function is not submodular and the techniques for submodular maximization cannot be applied. We describe a continuous optimization method which achieves an approximation ratio which asymptotically approaches 1-1/e. For instances where the asymptotics do not kick in, we design a local-search algorithm whose approximation ratio is arbitrarily close to 1/2. We empirically demonstrate the effectiveness of our methods on two multi-objective data summarization tasks, where the goal is to construct summaries via sparse representative subsets w.r.t. to predefined objectives.

We consider the problem of optimization from samples of monotone submodular functions with bounded curvature. In numerous applications, the function optimized is not known a priori, but instead learned from data. What are the guarantees we have when optimizing functions from sampled data? In this paper we show that for any monotone submodular function with curvature c there is a (1 - c)/(1 + c - c^2) approximation algorithm for maximization under cardinality constraints when polynomially-many samples are drawn from the uniform distribution over feasible sets. Moreover, we show that this algorithm is optimal. That is, for any c < 1, there exists a submodular function with curvature c for which no algorithm can achieve a better approximation. The curvature assumption is crucial as for general monotone submodular functions no algorithm can obtain a constant-factor approximation for maximization under a cardinality constraint when observing polynomially-many samples drawn from any distribution over feasible sets, even when the function is statistically learnable.

We consider the problem of budget feasible mechanism design proposed by Singer, but in a Bayesian setting. A principal has a public value for hiring a subset of the agents and a budget, while the agents have private costs for being hired. We consider both additive and submodular value functions of the principal. We show that there are simple, practical, ex post budget balanced posted pricing mechanisms that approximate the value obtained by the Bayesian optimal mechanism that is budget balanced only in expectation. A main motivating application for this work is crowdsourcing, e.g., on Mechanical Turk, where workers are drawn from a large population and posted pricing is standard. Our analysis methods relate to contention resolution schemes in submodular optimization of Vondràk et al. and the correlation gap analysis of Yan.

We propose a new framework for optimization under fairness constraints. The problems we consider model procurement where the goal is to optimize a buyer's utility while paying sellers in a manner that reflects their contribution to the buyer's utility. The payment rules we consider are natural interpretations of fairness based on concepts such as Shapley values and the nucleolus from cooperative game theory. The question in this setting is whether the outcome (measured in terms of the buyer's utility) produced by mechanisms that enforce fair payments is competitive with the outcome of a mechanism that simply pays agents their costs and is not committed to fair payments. Our main result shows that there exists a mechanism which guarantees a solution whose value is at least one third of the optimal unfair solution for any submodular utility function, and that this ratio is optimal. We discuss several special cases for which this approximation ratio can be improved and natural extensions.

We introduce the simultaneous model for cake cutting (the fair allocation of a divisible good), in which agents simultaneously send messages containing a sketch of their preferences over the cake. We show that this model enables the computation of divisions that satisfy proportionality -- a popular fairness notion -- using a protocol that circumvents a standard lower bound via parallel information elicitation. Cake divisions satisfying another prominent fairness notion, envy-freeness, are impossible to compute in the simultaneous model, but admit arbitrarily good approximations.