Trend Detection based Regret Minimization for Bandit Problems

We study a variation of the classical multi-armed bandits problem. In this problem, the learner has to make a sequence of decisions, picking from a fixed set of choices. In each round, she receives as feedback only the loss incurred from the chosen action. Conventionally, this problem has been studied when losses of the actions are drawn from an unknown distribution or when they are adversarial. In this paper, we study this problem when the losses of the actions also satisfy certain structural properties, and especially, do show a trend structure. When this is true, we show that using \textit{trend detection}, we can achieve regret of order $\tilde{O} (N \sqrt{TK})$ with respect to a switching strategy for the version of the problem where a single action is chosen in each round and $\tilde{O} (Nm \sqrt{TK})$ when $m$ actions are chosen each round. This guarantee is a significant improvement over the conventional benchmark. Our approach can, as a framework, be applied in combination with various well-known bandit algorithms, like Exp3. For both versions of the problem, we give regret guarantees also for the \textit{anytime} setting, i.e. when the length of the choice-sequence is not known in advance. Finally, we pinpoint the advantages of our method by comparing it to some well-known other strategies

Fast Adaptive Non-Monotone Submodular Maximization Subject to a Knapsack Constraint

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. Moreover, 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 versatility of our approach allows us to further transfer it to a stochastic version of the problem. There, we obtain a (9 + ε)-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

Fast Adaptive Non-Monotone Submodular Maximization Subject to a Knapsack Constraint

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. Moreover, 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 versatility of our approach allows us to further transfer it to a stochastic version of the problem. There, we obtain a (9 + ε)-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

Allocating Indivisible Goods to Strategic Agents: Pure Nash Equilibria and Fairness

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 preceding question. In particular, we study two algorithms that are known to produce such allocations in the nonstrategic setting: round-robin (EF1 allocations for any number of agents) and a cut-and-choose algorithm of Plaut and Roughgarden (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, whereas 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 nonstrategic 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

Submodular Maximization subject to a Knapsack Constraint: Combinatorial Algorithms with Near-optimal Adaptive Complexity *

International audienceThe 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, capturing the number of sequential rounds of parallel computation needed. 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: one needs to account for the total number of function evaluations (or value queries) as well. Our algorithm asksÕ(n 2) value queries, but can be modified to run with onlyÕ(n) instead, while retaining a low adaptive complexity of O(log 2 n). 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. Finally, we showcase our algorithms' applicability on real-world datasets

Selfishness and uncertainty: successful strategies in algorithmic game theory

The thesis consists of three parts, which come from somewhat different branches of algorithmic game theory and focus on aspects of selfishness and uncertainty to different degrees. We provide the following results: First, we present a simple and natural correlated-rounding technique to obtain truthful-in-expectation mechanisms for two-sided allocation problems with tree structure. From the domain of packing problems, we outline the technique for the generalized assignment problem, where it results in a 2-approximation for social welfare. In contrast to some previous work, we maintain exact truthfulness. From the covering domain, we apply the technique to the example of restricted-related scheduling, where it matches the best-known algorithm without truthfulness and again achieves a 2-approximation. Second, we consider the weighted bipartite matching problem, simply referred to as the assignment problem in the field of operations research. For its online version in the random order model, there exist strategies with small constant competitive ratios, including an optimal e-competitive algorithm. However, incentive-compatible mechanisms fall short of such near-optimal solutions, and the best-known guarantee here achieves a competitive ratio logarithmic in the number of vertices. We present a simple, truthful mechanism achieving optimal competitive ratio of e, and close this gap between online algorithms and mechanism design. Finally, we study a variation of the classic multi-armed bandits problem. In contrast to the common assumption that losses of actions are either stochasticor adversarial, we examine the case that they also show a trend structure. When this is true, we give a general framework combinable with various well-known bandit algorithms, like Exp3. We achieve a new, state-of-the-art regret guarantee with respect to a switching strategy for the classic problem, and a similar one for the problem variant where m actions are chosen per round. This guarantee is a significant improvement over the conventional benchmark

Efficient two-sided markets with limited information

Many important practical markets inherently involve the interaction of strategic buyers with strategic sellers. A fundamental impossibility result for such two-sided markets due to Myerson and Satterthwaite [33] establishes that even in the simplest such market, that of bilateral trade, it is impossible to design a mechanism that is individually rational, truthful, (weakly) budget balanced, and efficient. Even worse, it is known that the “second best” mechanism—the mechanism that maximizes social welfare subject to the other constraints—has to be carefully tailored to the Bayesian priors and is extremely complex. In light of this impossibility result it is very natural to seek “simple” mechanisms that are approximately optimal, and indeed a very active line of recent work has established a broad spectrum of constant-factor approximation guarantees, which apply to settings well beyond those for which (implicit) characterizations of the optimal (second best) mechanism are known. In this work, we go one step further and show that for many fundamental two-sided markets— e.g., bilateral trade, double auctions, and combinatorial double auctions—it is possible to design near-optimal mechanisms with provable, constant-factor approximation guarantees with just a single sample from the priors! In fact, most of our results in addition to requiring less information also improve upon the best known approximation guarantees for the respective setting