Pandora's Box Problem with Order Constraints

The Pandora's Box Problem, originally formalized by Weitzman in 1979, models selection from set of random, alternative options, when evaluation is costly. This includes, for example, the problem of hiring a skilled worker, where only one hire can be made, but the evaluation of each candidate is an expensive procedure. Weitzman showed that the Pandora's Box Problem admits an elegant, simple solution, where the options are considered in decreasing order of reservation value,i.e., the value that reduces to zero the expected marginal gain for opening the box. We study for the first time this problem when order - or precedence - constraints are imposed between the boxes. We show that, despite the difficulty of defining reservation values for the boxes which take into account both in-depth and in-breath exploration of the various options, greedy optimal strategies exist and can be efficiently computed for tree-like order constraints. We also prove that finding approximately optimal adaptive search strategies is NP-hard when certain matroid constraints are used to further restrict the set of boxes which may be opened, or when the order constraints are given as reachability constraints on a DAG. We complement the above result by giving approximate adaptive search strategies based on a connection between optimal adaptive strategies and non-adaptive strategies with bounded adaptivity gap for a carefully relaxed version of the problem

Non trivial behavior of the linear response function in phase ordering kinetics

Drawing from exact, approximate and numerical results an overview of the properties of the out of equilibrium response function in phase ordering kinetics is presented. Focusing on the zero field cooled magnetization, emphasis is on those features of this quantity which display non trivial behavior when relaxation proceeds by coarsening. Prominent among these is the dimensionality dependence of the scaling exponent $a_{\chi}$ which leads to failure of the connection between static and dynamic properties at the lower dimensionality $d_L$, where $a_{\chi}=0$. We also analyse the mean spherical model as an explicit example of a stochastic unstable system, for which the connection between statics and dynamics fails at all dimensionalities.Comment: 10 pages, 2 figures. Contribution to the International Conference "Perspectives on Quantum Field Theory, Statistical Mechanics and Stochastics" in honour of the 60th birthday of Francesco Guerr

Platonic polyhedra, periodic orbits and chaotic motions in the N-body problem with non-Newtonian forces

We consider the $N$-body problem with interaction potential $U_alpha=rac{1}{ert x_i-x_jert^alpha}$ for alpha>1. We assume that the particles have all the same mass and that $N$ is the order $ertmathcal{R}ert$ of the rotation group $mathcal{R}$ of one of the five Platonic polyhedra. We study motions that, up to a relabeling of the $N$ particles, are invariant under $mathcal{R}$. By variational techniques we prove the existence of periodic and chaotic motions

Truthful Matching with Online Items and Offline Agents

We study truthful mechanisms for welfare maximization in online bipartite matching. In our (multi-parameter) setting, every buyer is associated with a (possibly private) desired set of items, and has a private value for being assigned an item in her desired set. Unlike most online matching settings, where agents arrive online, in our setting the items arrive online in an adversarial order while the buyers are present for the entire duration of the process. This poses a significant challenge to the design of truthful mechanisms, due to the ability of buyers to strategize over future rounds. We provide an almost full picture of the competitive ratios in different scenarios, including myopic vs. non-myopic agents, tardy vs. prompt payments, and private vs. public desired sets. Among other results, we identify the frontier up to which the celebrated e/(e-1) competitive ratio for the vertex-weighted online matching of Karp, Vazirani and Vazirani extends to truthful agents and online items

No-Regret Learning in Bilateral Trade via Global Budget Balance

Bilateral trade revolves around the challenge of facilitating transactions between two strategic agents -- a seller and a buyer -- both of whom have a private valuations for the item. We study the online version of the problem, in which at each time step a new seller and buyer arrive. The learner's task is to set a price for each agent, without any knowledge about their valuations. The sequence of sellers and buyers is chosen by an oblivious adversary. In this setting, known negative results rule out the possibility of designing algorithms with sublinear regret when the learner has to guarantee budget balance for each iteration. In this paper, we introduce the notion of global budget balance, which requires the agent to be budget balance only over the entire time horizon. By requiring global budget balance, we provide the first no-regret algorithms for bilateral trade with adversarial inputs 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, which is order-wise optimal. Then, in the case of partial feedback models, we provide an algorithm guaranteeing a $\tilde{O}(T^{3/4})$ regret upper bound with one-bit feedback, which we complement with a nearly-matching lower bound. Finally, we investigate how these results vary when measuring regret using an alternative benchmark

Multiple anti-predator mechanisms in the red-spotted Argentina Frog (Amphibia: Hylidae)

Anurans employ a wide variety of anti-predator mechanisms to defend themselves. In casque-headed hylids, defence is thought to be a complex combination of several anti-predator mechanisms. However, the defence traits of only a few species are known; some hypotheses have yet to be addressed, whereas others, already tested in some species, need to be tested in additional taxa. The anti-predator mechanism of the casque-headed frog, Argenteohyla siemersi, is described here. It is a complex mechanism consisting of (1) behavioural and ecological traits, including secretive and semi-phragmotic habits and posture; (2) morphological features, including cryptic and aposematic colourations, a skull covered with bony dermal spines and protuberances that are associated with two types of granular venom glands; and (3) physiological and chemical traits, such as a highly lethal skin secretion. Our results are compared with those of previous studies of defence mechanisms in casque-headed frogs in an effort to understand the mechanisms and evaluate their potential phylogenetic signal in this group of anurans.Fil: Cajade, Rodrigo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas, Naturales y Agrimensura. Departamento de Biología; ArgentinaFil: Hermida, Gladys Noemí. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Biodiversidad y Biología Experimental; ArgentinaFil: Piñeiro, Jose Miguel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas, Naturales y Agrimensura. Departamento de Biología; ArgentinaFil: Regueira, Eleonora. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Biodiversidad y Biología Experimental; ArgentinaFil: Alcalde, Leandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Limnología "Dr. Raúl A. Ringuelet". Universidad Nacional de La Plata. Facultad de Ciencias Naturales y Museo. Instituto de Limnología; ArgentinaFil: Fusco, Luciano Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas Naturales y Agrimensura. Departamento de Bioquímica; ArgentinaFil: Marangoni, Federico. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas, Naturales y Agrimensura. Departamento de Biología; Argentin

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 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

Online Revenue Maximization for Server Pricing

Efficient and truthful mechanisms to price resources on remote servers/machines has been the subject of much work in recent years due to the importance of the cloud market. This paper considers revenue maximization in the online stochastic setting with non-preemptive jobs and a unit capacity server. One agent/job arrives at every time step, with parameters drawn from an underlying unknown distribution. We design a posted-price mechanism which can be efficiently computed, and is revenue-optimal in expectation and in retrospect, up to additive error. The prices are posted prior to learning the agent's type, and the computed pricing scheme is deterministic, depending only on the length of the allotted time interval and on the earliest time the server is available. If the distribution of agent's type is only learned from observing the jobs that are executed, we prove that a polynomial number of samples is sufficient to obtain a near-optimal truthful pricing strategy

Learning on the Edge: Online Learning with Stochastic Feedback Graphs

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 Erd\H{o}s-R\'enyi 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 $\min\bigl\{\min_{\varepsilon} \sqrt{(\alpha_\varepsilon/\varepsilon) T},\, \min_{\varepsilon} (\delta_\varepsilon/\varepsilon)^{1/3} T^{2/3}\bigr\}$ (ignoring logarithmic factors), where $\alpha_{\varepsilon}$ and $\delta_{\varepsilon}$ 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