Federico Fusco

Paul Dütting, Federico Fusco, Silvio Lattanzi et al.

Consistency is an important property in dynamic submodular maximization and entails maintaining a near-optimal solution at all times, making only a small number of adjustments to the solution in each step. Prior work has explored this question for the insertion-only case, where the algorithm faces a stream of $n$ insertions, and has established lower and upper bounds for the cardinality-constrained version of the problem. We consider this question in the fully dynamic setting, where the stream of operations may contain both insertions and deletions. We develop a general framework for designing algorithms for this setting, and instantiate it to obtain the first constant-factor approximations with sublinear consistency. For cardinality constraints, we propose a $\frac 12 - O(\varepsilon)$ approximation that is $O\left(\frac{1}{\varepsilon^2}\right)$ consistent. For rank-$k$ matroid constraints, we construct a $\frac 14 - O(\varepsilon)$ approximation to the dynamic optimum that is $O\left(\frac{\log k}{\varepsilon^2}\right)$ consistent.

Simone Di Gregorio, Federico Fusco, Stefano Leonardi et al.

Bilateral trade models one of the most fundamental economic interactions: the intermediation between two strategic agents, a seller and a buyer, willing to trade a good. We consider the learning version of the problem, where the goal is to learn a mechanism from a sampled dataset of agents' valuations to maximize either profit or economic efficiency. While known learning algorithms are characterized by high sensitivity to the input dataset, we specifically study this problem through the lens of differential privacy, ensuring that each data point does not significantly affect the probability of learning any specific mechanism. For our results, we adopt the PAC-learning framework: with high probability, the learning algorithm should output a mechanism that is at most an additive $α$ away from optimal, in a $\varepsilon$-differentially private way. As a first result, we show that differential privacy and (near)-optimality are not achievable for general distributions. Surprisingly, assuming that the distribution underlying the agents' valuations is $σ$-smooth, we recover nearly optimal sample-complexity bounds for both economic efficiency and profit. For profit, we show how to construct in polynomial time an $α$-optimal and $\varepsilon$-differentially private mechanism using $\tildeΘ(\frac{1}{σ\varepsilonα^2})$ samples. For efficiency, measured by the gain from trade, we achieve the same result using $\tildeΘ(\frac{1}{\varepsilonα}+\frac{1}{α^2})$ samples. Notably, these bounds are essentially tight in the precision parameter $α$, since achieving $α$-optimality (ignoring differential privacy) requires at least $\frac{1}{α^2}$ samples.

Yossi Azar, Amos Fiat, Federico Fusco

We study sequential bilateral trade where sellers and buyers valuations are completely arbitrary (i.e., determined by an adversary). Sellers and buyers are strategic agents with private valuations for the good and the goal is to design a mechanism that maximizes efficiency (or gain from trade) while being incentive compatible, individually rational and budget balanced. In this paper we consider gain from trade which is harder to approximate than social welfare. We consider a variety of feedback scenarios and distinguish the cases where the mechanism posts one price and when it can post different prices for buyer and seller. We show several surprising results about the separation between the different scenarios. In particular we show that (a) it is impossible to achieve sublinear $α$-regret for any $α<2$, (b) but with full feedback sublinear $2$-regret is achievable (c) with a single price and partial feedback one cannot get sublinear $α$ regret for any constant $α$ (d) nevertheless, posting two prices even with one-bit feedback achieves sublinear $2$-regret, and (e) there is a provable separation in the $2$-regret bounds between full and partial feedback.

Nicolò Cesa-Bianchi, Tommaso Cesari, Roberto Colomboni et al.

We study repeated bilateral trade where an adaptive $σ$-smooth adversary generates the valuations of sellers and buyers. We provide a complete characterization of the regret regimes for fixed-price mechanisms under different feedback models in the two cases where the learner can post either the same or different prices to buyers and sellers. We begin by showing that the minimax regret after $T$ rounds is of order $\sqrt{T}$ in the full-feedback scenario. Under partial feedback, any algorithm that has to post the same price to buyers and sellers suffers worst-case linear regret. However, when the learner can post two different prices at each round, we design an algorithm enjoying regret of order $T^{3/4}$ ignoring log factors. We prove that this rate is optimal by presenting a surprising $T^{3/4}$ lower bound, which is the main technical contribution of the paper.

Nicolò Cesa-Bianchi, Tommaso Cesari, Roberto Colomboni et al.

We study the problem of regret minimization for a single bidder in a sequence of first-price auctions where the bidder discovers the item's value only if the auction is won. Our main contribution is a complete characterization, up to logarithmic factors, of the minimax regret in terms of the auction's \emph{transparency}, which controls the amount of information on competing bids disclosed by the auctioneer at the end of each auction. Our results hold under different assumptions (stochastic, adversarial, and their smoothed variants) on the environment generating the bidder's valuations and competing bids. These minimax rates reveal how the interplay between transparency and the nature of the environment affects how fast one can learn to bid optimally in first-price auctions.

Martino Bernasconi, Matteo Castiglioni, Andrea Celli et al.

Bilateral trade models the problem of intermediating between two rational agents -- a seller and a buyer -- both characterized by a private valuation for an item they want to trade. We study the online learning version of the problem, in which at each time step a new seller and buyer arrive and the learner has to set prices for them without any knowledge about their (adversarially generated) valuations. In this setting, known impossibility results rule out the existence of no-regret algorithms when budget balanced has to be enforced at each time step. In this paper, we introduce the notion of \emph{global budget balance}, which only requires the learner to fulfill budget balance over the entire time horizon. Under this natural relaxation, we provide the first no-regret algorithms for adversarial bilateral trade under various feedback models. First, we show that in the full-feedback model, the learner can guarantee $\tilde O(\sqrt{T})$ regret against the best fixed prices in hindsight, and that this bound is optimal up to poly-logarithmic terms. Second, we provide a learning algorithm guaranteeing a $\tilde O(T^{3/4})$ regret upper bound with one-bit feedback, which we complement with a $Ω(T^{5/7})$ lower bound that holds even in the two-bit feedback model. Finally, we introduce and analyze an alternative benchmark that is provably stronger than the best fixed prices in hindsight and is inspired by the literature on bandits with knapsacks.

Paul Dütting, Federico Fusco, Silvio Lattanzi et al.

Maximizing a submodular function is a fundamental task in machine learning and in this paper we study the deletion robust version of the problem under the classic matroids constraint. Here the goal is to extract a small size summary of the dataset that contains a high value independent set even after an adversary deleted some elements. We present constant-factor approximation algorithms, whose space complexity depends on the rank $k$ of the matroid and the number $d$ of deleted elements. In the centralized setting we present a $(4.597+O(\varepsilon))$-approximation algorithm with summary size $O( \frac{k+d}{\varepsilon^2}\log \frac{k}{\varepsilon})$ that is improved to a $(3.582+O(\varepsilon))$-approximation with $O(k + \frac{d}{\varepsilon^2}\log \frac{k}{\varepsilon})$ summary size when the objective is monotone. In the streaming setting we provide a $(9.435 + O(\varepsilon))$-approximation algorithm with summary size and memory $O(k + \frac{d}{\varepsilon^2}\log \frac{k}{\varepsilon})$; the approximation factor is then improved to $(5.582+O(\varepsilon))$ in the monotone case.

Gagan Aggarwal, Ashwinkumar Badanidiyuru, Paul Dütting et al.

Motivated by online retail, we consider the problem of selling one item (e.g., an ad slot) to two non-excludable buyers (say, a merchant and a brand). This problem captures, for example, situations where a merchant and a brand cooperatively bid in an auction to advertise a product, and both benefit from the ad being shown. A mechanism collects bids from the two and decides whether to allocate and which payments the two parties should make. This gives rise to intricate incentive compatibility constraints, e.g., on how to split payments between the two parties. We approach the problem of finding a revenue-maximizing incentive-compatible mechanism from an online learning perspective; this poses significant technical challenges. First, the action space (the class of all possible mechanisms) is huge; second, the function that maps mechanisms to revenue is highly irregular, ruling out standard discretization-based approaches. In the stochastic setting, we design an efficient learning algorithm achieving a regret bound of $O(T^{3/4})$. Our approach is based on an adaptive discretization scheme of the space of mechanisms, as any non-adaptive discretization fails to achieve sublinear regret. In the adversarial setting, we exploit the non-Lipschitzness of the problem to prove a strong negative result, namely that no learning algorithm can achieve more than half of the revenue of the best fixed mechanism in hindsight. We then consider the $σ$-smooth adversary; we construct an efficient learning algorithm that achieves a regret bound of $O(T^{2/3})$ and builds on a succinct encoding of exponentially many experts. Finally, we prove that no learning algorithm can achieve less than $Ω(\sqrt T)$ regret in both the stochastic and the smooth setting, thus narrowing the range where the minimax regret rates for these two problems lie.

Emmanuel Esposito, Federico Fusco, Dirk van der Hoeven et al.

The framework of feedback graphs is a generalization of sequential decision-making with bandit or full information feedback. In this work, we study an extension where the directed feedback graph is stochastic, following a distribution similar to the classical Erdős-Rényi model. Specifically, in each round every edge in the graph is either realized or not with a distinct probability for each edge. We prove nearly optimal regret bounds of order $\min\bigl\{\min_{\varepsilon} \sqrt{(α_\varepsilon/\varepsilon) T},\, \min_{\varepsilon} (δ_\varepsilon/\varepsilon)^{1/3} T^{2/3}\bigr\}$ (ignoring logarithmic factors), where $α_{\varepsilon}$ and $δ_{\varepsilon}$ are graph-theoretic quantities measured on the support of the stochastic feedback graph $\mathcal{G}$ with edge probabilities thresholded at $\varepsilon$. Our result, which holds without any preliminary knowledge about $\mathcal{G}$, requires the learner to observe only the realized out-neighborhood of the chosen action. When the learner is allowed to observe the realization of the entire graph (but only the losses in the out-neighborhood of the chosen action), we derive a more efficient algorithm featuring a dependence on weighted versions of the independence and weak domination numbers that exhibits improved bounds for some special cases.

Martino Bernasconi, Matteo Castiglioni, Andrea Celli et al.

The bandits with knapsack (BwK) framework models online decision-making problems in which an agent makes a sequence of decisions subject to resource consumption constraints. The traditional model assumes that each action consumes a non-negative amount of resources and the process ends when the initial budgets are fully depleted. We study a natural generalization of the BwK framework which allows non-monotonic resource utilization, i.e., resources can be replenished by a positive amount. We propose a best-of-both-worlds primal-dual template that can handle any online learning problem with replenishment for which a suitable primal regret minimizer exists. In particular, we provide the first positive results for the case of adversarial inputs by showing that our framework guarantees a constant competitive ratio $α$ when $B=Ω(T)$ or when the possible per-round replenishment is a positive constant. Moreover, under a stochastic input model, our algorithm yields an instance-independent $\tilde{O}(T^{1/2})$ regret bound which complements existing instance-dependent bounds for the same setting. Finally, we provide applications of our framework to some economic problems of practical relevance.

Romain Cosson, Federico Fusco, Anupam Gupta et al.

We study repeated bilateral trade when the valuations of the sellers and the buyers are contextual. More precisely, the agents' valuations are given by the inner product of a context vector with two unknown $d$-dimensional vectors -- one for the buyers and one for the sellers. At each time step $t$, the learner receives a context and posts two prices, one for the seller and one for the buyer, and the trade happens if both agents accept their price. We study two objectives for this problem, gain from trade and profit, proving no-regret with respect to a surprisingly strong benchmark: the best omniscient dynamic strategy. In the natural scenario where the learner observes \emph{separately} whether the agents accept their price -- the so-called \emph{two-bit} feedback -- we design algorithms that achieve $O(d\log d)$ regret for gain from trade, and $O(d \log\log T + d\log d)$ regret for profit maximization. Both results are tight, up to the $\log(d)$ factor, and implement per-step budget balance, meaning that the learner never incurs negative profit. In the less informative \emph{one-bit} feedback model, the learner only observes whether a trade happens or not. For this scenario, we show that the tight two-bit regret regimes are still attainable, at the cost of allowing the learner to possibly incur a small negative profit of order $O(d\log d)$, which is notably independent of the time horizon. As a final set of results, we investigate the combination of one-bit feedback and per-step budget balance. There, we design an algorithm for gain from trade that suffers regret independent of the time horizon, but \emph{exponential} in the dimension $d$. For profit maximization, we maintain this exponential dependence on the dimension, which gets multiplied by a $\log T$ factor.

Matteo Russo, Andrea Celli, Riccardo Colini Baldeschi et al.

In online learning, a decision maker repeatedly selects one of a set of actions, with the goal of minimizing the overall loss incurred. Following the recent line of research on algorithms endowed with additional predictive features, we revisit this problem by allowing the decision maker to acquire additional information on the actions to be selected. In particular, we study the power of \emph{best-action queries}, which reveal beforehand the identity of the best action at a given time step. In practice, predictive features may be expensive, so we allow the decision maker to issue at most $k$ such queries. We establish tight bounds on the performance any algorithm can achieve when given access to $k$ best-action queries for different types of feedback models. In particular, we prove that in the full feedback model, $k$ queries are enough to achieve an optimal regret of $Θ\left(\min\left\{\sqrt T, \frac Tk\right\}\right)$. This finding highlights the significant multiplicative advantage in the regret rate achievable with even a modest (sublinear) number $k \in Ω(\sqrt{T})$ of queries. Additionally, we study the challenging setting in which the only available feedback is obtained during the time steps corresponding to the $k$ best-action queries. There, we provide a tight regret rate of $Θ\left(\min\left\{\frac{T}{\sqrt k},\frac{T^2}{k^2}\right\}\right)$, which improves over the standard $Θ\left(\frac{T}{\sqrt k}\right)$ regret rate for label efficient prediction for $k \in Ω(T^{2/3})$.

Riccardo Colini Baldeschi, Simone Di Gregorio, Simone Fioravanti et al.

Consider the problem of finding the best matching in a weighted graph where we only have access to predictions of the actual stochastic weights, based on an underlying context. If the predictor is the Bayes optimal one, then computing the best matching based on the predicted weights is optimal. However, in practice, this perfect information scenario is not realistic. Given an imperfect predictor, a suboptimal decision rule may compensate for the induced error and thus outperform the standard optimal rule. In this paper, we propose multicalibration as a way to address this problem. This fairness notion requires a predictor to be unbiased on each element of a family of protected sets of contexts. Given a class of matching algorithms $\mathcal C$ and any predictor $γ$ of the edge-weights, we show how to construct a specific multicalibrated predictor $\hat γ$, with the following property. Picking the best matching based on the output of $\hat γ$ is competitive with the best decision rule in $\mathcal C$ applied onto the original predictor $γ$. We complement this result by providing sample complexity bounds.

Simone Di Gregorio, Paul Dütting, Federico Fusco et al.

Bilateral trade models the task of intermediating between two strategic agents, a seller and a buyer, who wish to trade a good. We study this problem from the perspective of a profit-maximizing broker within an online learning framework, where the agents' valuations are generated by a smooth adversary. We devise a learning algorithm that guarantees a $\tilde{O}(\sqrt{T})$ regret bound, which is tight in the time horizon $T$ up to poly-logarithmic factors. This matches the minimax rate for the stochastic i.i.d. case, and is also well separated from the adversarial setting, where sublinear-regret is unattainable. By extending the strong regret guarantees from the i.i.d. case to the smooth adversary, we significantly broaden the scope of settings where such fast rate is achievable, while closing an important gap in the regret landscape of this fundamental economic problem. To overcome the challenges posed by this adversary, we leverage a continuity property of smooth instances and combines this with a hierarchical net-construction of the broker's action space, which is analyzed via algorithmic chaining. We showcase the applicability of these techniques by deriving a similarly tight $\tilde{O}(\sqrt{T})$ regret bound for a related mechanism design model: the joint ads problem.

Paul Dütting, Federico Fusco, Silvio Lattanzi et al.

In this work, we study online submodular maximization, and how the requirement of maintaining a stable solution impacts the approximation. In particular, we seek bounds on the best-possible approximation ratio that is attainable when the algorithm is allowed to make at most a constant number of updates per step. We show a tight information-theoretic bound of $\tfrac{2}{3}$ for general monotone submodular functions, and an improved (also tight) bound of $\tfrac{3}{4}$ for coverage functions. Since both these bounds are attained by non poly-time algorithms, we also give a poly-time randomized algorithm that achieves a $0.51$-approximation. Combined with an information-theoretic hardness of $\tfrac{1}{2}$ for deterministic algorithms from prior work, our work thus shows a separation between deterministic and randomized algorithms, both information theoretically and for poly-time algorithms.

Martino Bernasconi, Andrea Celli, Riccardo Colini-Baldeschi et al.

In the random-order model for online learning, the sequence of losses is chosen upfront by an adversary and presented to the learner after a random permutation. Any random-order input is \emph{asymptotically} equivalent to a stochastic i.i.d. one, but, for finite times, it may exhibit significant {\em non-stationarity}, which can hinder the performance of stochastic learning algorithms. While algorithms for adversarial inputs naturally maintain their regret guarantees in random order, simple no-regret algorithms exist for the stochastic model that fail against random-order instances. In this paper, we propose a general template to adapt stochastic learning algorithms to the random-order model without substantially affecting their regret guarantees. This allows us to recover improved regret bounds for prediction with delays, online learning with constraints, and bandits with switching costs. Finally, we investigate online classification and prove that, in random order, learnability is characterized by the VC dimension rather than the Littlestone dimension, thus providing a further separation from the general adversarial model.

Simone Di Gregorio, Paul Dütting, Federico Fusco et al.

Bilateral trade models the task of intermediating between two strategic agents, a seller and a buyer, willing to trade a good for which they hold private valuations. We study this problem from the perspective of a broker, in a regret minimization framework. At each time step, a new seller and buyer arrive, and the broker has to propose a mechanism that is incentive-compatible and individually rational, with the goal of maximizing profit. We propose a learning algorithm that guarantees a nearly tight $\tilde{O}(\sqrt{T})$ regret in the stochastic setting when seller and buyer valuations are drawn i.i.d. from a fixed and possibly correlated unknown distribution. We further show that it is impossible to achieve sublinear regret in the non-stationary scenario where valuations are generated upfront by an adversary. Our ambitious benchmark for these results is the best incentive-compatible and individually rational mechanism. This separates us from previous works on efficiency maximization in bilateral trade, where the benchmark is a single number: the best fixed price in hindsight. A particular challenge we face is that uniform convergence for all mechanisms' profits is impossible. We overcome this difficulty via a careful chaining analysis that proves convergence for a provably near-optimal mechanism at (essentially) optimal rate. We further showcase the broader applicability of our techniques by providing nearly optimal results for the joint ads problem.

Paul Dütting, Federico Fusco, Silvio Lattanzi et al.

Maximizing monotone submodular functions under a matroid constraint is a classic algorithmic problem with multiple applications in data mining and machine learning. We study this classic problem in the fully dynamic setting, where elements can be both inserted and deleted in real-time. Our main result is a randomized algorithm that maintains an efficient data structure with an $\tilde{O}(k^2)$ amortized update time (in the number of additions and deletions) and yields a $4$-approximate solution, where $k$ is the rank of the matroid.

Marwa El Halabi, Federico Fusco, Ashkan Norouzi-Fard et al.

Streaming submodular maximization is a natural model for the task of selecting a representative subset from a large-scale dataset. If datapoints have sensitive attributes such as gender or race, it becomes important to enforce fairness to avoid bias and discrimination. This has spurred significant interest in developing fair machine learning algorithms. Recently, such algorithms have been developed for monotone submodular maximization under a cardinality constraint. In this paper, we study the natural generalization of this problem to a matroid constraint. We give streaming algorithms as well as impossibility results that provide trade-offs between efficiency, quality and fairness. We validate our findings empirically on a range of well-known real-world applications: exemplar-based clustering, movie recommendation, and maximum coverage in social networks.

Paul Dütting, Federico Fusco, Silvio Lattanzi et al.

Maximizing a monotone submodular function is a fundamental task in machine learning. In this paper, we study the deletion robust version of the problem under the classic matroids constraint. Here the goal is to extract a small size summary of the dataset that contains a high value independent set even after an adversary deleted some elements. We present constant-factor approximation algorithms, whose space complexity depends on the rank $k$ of the matroid and the number $d$ of deleted elements. In the centralized setting we present a $(3.582+O(\varepsilon))$-approximation algorithm with summary size $O(k + \frac{d \log k}{\varepsilon^2})$. In the streaming setting we provide a $(5.582+O(\varepsilon))$-approximation algorithm with summary size and memory $O(k + \frac{d \log k}{\varepsilon^2})$. We complement our theoretical results with an in-depth experimental analysis showing the effectiveness of our algorithms on real-world datasets.

Georgios Amanatidis, Georgios Birmpas, Federico Fusco et al.

We consider the problem of fairly allocating a set of indivisible goods to a set of strategic agents with additive valuation functions. We assume no monetary transfers and, therefore, a mechanism in our setting is an algorithm that takes as input the reported -- rather than the true -- values of the agents. Our main goal is to explore whether there exist mechanisms that have pure Nash equilibria for every instance and, at the same time, provide fairness guarantees for the allocations that correspond to these equilibria. We focus on two relaxations of envy-freeness, namely envy-freeness up to one good (EF1), and envy-freeness up to any good (EFX), and we positively answer the above question. In particular, we study two algorithms that are known to produce such allocations in the non-strategic setting: Round-Robin (EF1 allocations for any number of agents) and a cut-and-choose algorithm of Plaut and Roughgarden [SIAM Journal of Discrete Mathematics, 2020] (EFX allocations for two agents). For Round-Robin we show that all of its pure Nash equilibria induce allocations that are EF1 with respect to the underlying true values, while for the algorithm of Plaut and Roughgarden we show that the corresponding allocations not only are EFX but also satisfy maximin share fairness, something that is not true for this algorithm in the non-strategic setting! Further, we show that a weaker version of the latter result holds for any mechanism for two agents that always has pure Nash equilibria which all induce EFX allocations.

Nicolò Cesa-Bianchi, Tommaso Cesari, Roberto Colomboni et al.

Bilateral trade, a fundamental topic in economics, models the problem of intermediating between two strategic agents, a seller and a buyer, willing to trade a good for which they hold private valuations. In this paper, we cast the bilateral trade problem in a regret minimization framework over $T$ rounds of seller/buyer interactions, with no prior knowledge on their private valuations. Our main contribution is a complete characterization of the regret regimes for fixed-price mechanisms with different feedback models and private valuations, using as a benchmark the best fixed-price in hindsight. More precisely, we prove the following tight bounds on the regret: - $Θ(\sqrt{T})$ for full-feedback (i.e., direct revelation mechanisms). - $Θ(T^{2/3})$ for realistic feedback (i.e., posted-price mechanisms) and independent seller/buyer valuations with bounded densities. - $Θ(T)$ for realistic feedback and seller/buyer valuations with bounded densities. - $Θ(T)$ for realistic feedback and independent seller/buyer valuations. - $Θ(T)$ for the adversarial setting.

Dirk van der Hoeven, Federico Fusco, Nicolò Cesa-Bianchi

We study the problem of online multiclass classification in a setting where the learner's feedback is determined by an arbitrary directed graph. While including bandit feedback as a special case, feedback graphs allow a much richer set of applications, including filtering and label efficient classification. We introduce Gappletron, the first online multiclass algorithm that works with arbitrary feedback graphs. For this new algorithm, we prove surrogate regret bounds that hold, both in expectation and with high probability, for a large class of surrogate losses. Our bounds are of order $B\sqrt{ρKT}$, where $B$ is the diameter of the prediction space, $K$ is the number of classes, $T$ is the time horizon, and $ρ$ is the domination number (a graph-theoretic parameter affecting the amount of exploration). In the full information case, we show that Gappletron achieves a constant surrogate regret of order $B^2K$. We also prove a general lower bound of order $\max\big\{B^2K,\sqrt{T}\big\}$ showing that our upper bounds are not significantly improvable. Experiments on synthetic data show that for various feedback graphs, our algorithm is competitive against known baselines.

Georgios Amanatidis, Federico Fusco, Philip Lazos et al.

Submodular maximization is a classic algorithmic problem with multiple applications in data mining and machine learning; there, the growing need to deal with massive instances motivates the design of algorithms balancing the quality of the solution with applicability. For the latter, an important measure is the adaptive complexity, which captures the number of sequential rounds of parallel computation needed by an algorithm to terminate. In this work we obtain the first constant factor approximation algorithm for non-monotone submodular maximization subject to a knapsack constraint with near-optimal $O(\log n)$ adaptive complexity. Low adaptivity by itself, however, is not enough: a crucial feature to account for is represented by the total number of function evaluations (or value queries). Our algorithm asks $\tilde{O}(n^2)$ value queries, but can be modified to run with only $\tilde{O}(n)$ instead, while retaining a low adaptive complexity of $O(\log^2n)$. Besides the above improvement in adaptivity, this is also the first combinatorial approach with sublinear adaptive complexity for the problem and yields algorithms comparable to the state-of-the-art even for the special cases of cardinality constraints or monotone objectives.

Nicolò Cesa-Bianchi, Tommaso Cesari, Roberto Colomboni et al.

Bilateral trade, a fundamental topic in economics, models the problem of intermediating between two strategic agents, a seller and a buyer, willing to trade a good for which they hold private valuations. Despite the simplicity of this problem, a classical result by Myerson and Satterthwaite (1983) affirms the impossibility of designing a mechanism which is simultaneously efficient, incentive compatible, individually rational, and budget balanced. This impossibility result fostered an intense investigation of meaningful trade-offs between these desired properties. Much work has focused on approximately efficient fixed-price mechanisms, i.e., Blumrosen and Dobzinski (2014; 2016), Colini-Baldeschi et al. (2016), which have been shown to fully characterize strong budget balanced and ex-post individually rational direct revelation mechanisms. All these results, however, either assume some knowledge on the priors of the seller/buyer valuations, or a black box access to some samples of the distributions, as in D{ü}tting et al. (2021). In this paper, we cast for the first time the bilateral trade problem in a regret minimization framework over rounds of seller/buyer interactions, with no prior knowledge on the private seller/buyer valuations. Our main contribution is a complete characterization of the regret regimes for fixed-price mechanisms with different models of feedback and private valuations, using as benchmark the best fixed price in hindsight. More precisely, we prove the following bounds on the regret: $\bullet$ $\widetildeΘ(\sqrt{T})$ for full-feedback (i.e., direct revelation mechanisms); $\bullet$ $\widetildeΘ(T^{2/3})$ for realistic feedback (i.e., posted-price mechanisms) and independent seller/buyer valuations with bounded densities; $\bullet$ $Θ(T)$ for realistic feedback and seller/buyer valuations with bounded densities; $\bullet$ $Θ(T)$ for realistic feedback and independent seller/buyer valuations; $\bullet$ $Θ(T)$ for the adversarial setting.

Georgios Amanatidis, Federico Fusco, Philip Lazos et al.

Constrained submodular maximization problems encompass a wide variety of applications, including personalized recommendation, team formation, and revenue maximization via viral marketing. The massive instances occurring in modern day applications can render existing algorithms prohibitively slow, while frequently, those instances are also inherently stochastic. Focusing on these challenges, we revisit the classic problem of maximizing a (possibly non-monotone) submodular function subject to a knapsack constraint. We present a simple randomized greedy algorithm that achieves a $5.83$ approximation and runs in $O(n \log n)$ time, i.e., at least a factor $n$ faster than other state-of-the-art algorithms. The robustness of our approach allows us to further transfer it to a stochastic version of the problem. There, we obtain a 9-approximation to the best adaptive policy, which is the first constant approximation for non-monotone objectives. Experimental evaluation of our algorithms showcases their improved performance on real and synthetic data.