A Fast Distributed Stateless Algorithm for α\alpha-Fair Packing Problems

Over the past two decades, fair resource allocation problems have received considerable attention in a variety of application areas. However, little progress has been made in the design of distributed algorithms with convergence guarantees for general and commonly used $\alpha$-fair allocations. In this paper, we study weighted $\alpha$-fair packing problems, that is, the problems of maximizing the objective functions (i) $\sum_j w_j x_j^{1-\alpha}/(1-\alpha)$ when $\alpha > 0$, $\alpha \neq 1$ and (ii) $\sum_j w_j \ln x_j$ when $\alpha = 1$, over linear constraints $Ax \leq b$, $x\geq 0$, where $w_j$ are positive weights and $A$ and $b$ are non-negative. We consider the distributed computation model that was used for packing linear programs and network utility maximization problems. Under this model, we provide a distributed algorithm for general $\alpha$ that converges to an $\varepsilon-$approximate solution in time (number of distributed iterations) that has an inverse polynomial dependence on the approximation parameter $\varepsilon$ and poly-logarithmic dependence on the problem size. This is the first distributed algorithm for weighted $\alpha-$fair packing with poly-logarithmic convergence in the input size. The algorithm uses simple local update rules and is stateless (namely, it allows asynchronous updates, is self-stabilizing, and allows incremental and local adjustments). We also obtain a number of structural results that characterize $\alpha-$fair allocations as the value of $\alpha$ is varied. These results deepen our understanding of fairness guarantees in $\alpha-$fair packing allocations, and also provide insight into the behavior of $\alpha-$fair allocations in the asymptotic cases $\alpha\rightarrow 0$, $\alpha \rightarrow 1$, and $\alpha \rightarrow \infty$.Comment: Added structural results for asymptotic cases of \alpha-fairness (\alpha approaching 0, 1, or infinity), improved presentation, and revised throughou

Multi-Path Alpha-Fair Resource Allocation at Scale in Distributed Software Defined Networks

The performance of computer networks relies on how bandwidth is shared among different flows. Fair resource allocation is a challenging problem particularly when the flows evolve over time. To address this issue, bandwidth sharing techniques that quickly react to the traffic fluctuations are of interest, especially in large scale settings with hundreds of nodes and thousands of flows. In this context, we propose a distributed algorithm based on the Alternating Direction Method of Multipliers (ADMM) that tackles the multi-path fair resource allocation problem in a distributed SDN control architecture. Our ADMM-based algorithm continuously generates a sequence of resource allocation solutions converging to the fair allocation while always remaining feasible, a property that standard primal-dual decomposition methods often lack. Thanks to the distribution of all computer intensive operations, we demonstrate that we can handle large instances at scale

Resource Allocation in Wireless Networks: Theory and Applications

Limited wireless resources, such as spectrum and maximum power, give rise to various resource allocation problems that are interesting both from theoretical and application viewpoints. While the problems in some of the wireless networking applications are amenable to general resource allocation methods, others require a more specialized approach suited to their unique structural characteristics. We study both types of the problems in this thesis. We start with a general problem of alpha-fair packing, namely, the problem of maximizing sum_j {w_j f_α(x_j)}, where w_j > 0, ∀j, and (i) f_α(x_j)=ln(x_j), if α = 1, (ii) f_α(x_j)= {x_j^(1-α)}/{1-α}, if α ≠ 1,α > 0, subject to positive linear constraints of the form Ax ≤ b, x ≥ 0, where A and b are non-negative. This problem has broad applications within and outside wireless networking. We present a distributed algorithm for general alpha that converges to an epsilon-approximate solution in time (number of distributed iterations) that has an inverse polynomial dependence on the approximation parameter epsilon and poly-logarithmic dependence on the problem size. This is the first distributed algorithm for weighted alpha-fair packing with poly-logarithmic convergence in the input size. We also obtain structural results that characterize alpha-fair allocations as the value of alpha is varied. These results deepen our understanding of fairness guarantees in alpha-fair packing allocations, and also provide insights into the behavior of alpha-fair allocations in the asymptotic cases when alpha tends to zero, one, and infinity. With these general tools on hand, we consider an application in wireless networks where fairness is of paramount importance: rate allocation and routing in energy-harvesting networks. We discuss the importance of fairness in such networks and cases where our results on alpha-fair packing apply. We then turn our focus to rate allocation in energy harvesting networks with highly variable energy sources and that are used for applications such as monitoring and tracking. In such networks, it is essential to guarantee fairness over both the network nodes and the time slots and to be as fair as possible -- in particular, to require max-min fairness. We first develop an algorithm that obtains a max-min fair rate assignment for any routing that is specified at the input. Then, we consider the problem of determining a "good'' routing. We consider various routing types and either provide polynomial-time algorithms for finding such routings or prove that the problems are NP-hard. Our results reveal an interesting trade-off between the complexities of computation and implementation. The results can also be applied to other related fairness problems. The second part of the thesis is devoted to the study of resource allocation problems that require a specialized approach. The problems we focus on arise in wireless networks employing full-duplex communication -- the simultaneous transmission and reception on the same frequency channel. Our primary goal is to understand the benefits and complexities tied to using this novel wireless technology through the study of resource (power, time, and channel) allocation problems. Towards that goal, we introduce a new realistic model of a compact (e.g., smartphone) full-duplex receiver and demonstrate its accuracy via measurements. First, we focus on the resource allocation problems with the objective of maximizing the sum of uplink and downlink rates, possibly over multiple orthogonal channels. For the single-channel case, we quantify the rate improvement as a function of the remaining self-interference and signal-to-noise ratios and provide structural results that characterize the sum of uplink and downlink rates on a full-duplex channel. Building on these results, we consider the multi-channel case and develop a polynomial time algorithm which is nearly optimal in practice under very mild restrictions. To reduce the running time, we develop an efficient nearly-optimal algorithm under the high SINR approximation. Then, we study the achievable capacity regions of full-duplex links in the single- and multi-channel cases. We present analytical results that characterize the uplink and downlink capacity region and efficient algorithms for computing rate pairs at the region's boundary. We also provide near-optimal and heuristic algorithms that "convexify'' the capacity region when it is not convex. The convexified region corresponds to a combination of a few full-duplex rates (i.e., to time sharing between different operation modes). The analytical results provide insights into the properties of the full-duplex capacity region and are essential for future development of fair resource allocation and scheduling algorithms in Wi-Fi and cellular networks incorporating full-duplex

Real-Time Fair Resource Allocation in Distributed Software Defined Networks

International audienceThe performance of computer networks relies on how bandwidth is shared among different flows. Fair resource allocation is a challenging problem particularly when the flows evolve over time. To address this issue, bandwidth sharing techniques that quickly react to the traffic fluctuations are of interest, especially in large scale settings with hundreds of nodes and thousands of flows. In this context, we propose a distributed algorithm based on the Alternating Direction Method of Multipliers (ADMM) that tackles the fair resource allocation problem in a distributed SDN control architecture. Our ADMM-based algorithm continuously generates a sequence of resource allocation solutions converging to the fair allocation while always remaining feasible, a property that standard primal-dual decomposition methods often lack. Thanks to the distribution of all computer intensive operations, we demonstrate that we can handle large instances in real-time

Scalable optimization algorithms for recommender systems

Recommender systems have now gained significant popularity and been widely used in many e-commerce applications. Predicting user preferences is a key step to providing high quality recommendations. In practice, however, suggestions made to users must not only consider user preferences in isolation; a good recommendation engine also needs to account for certain constraints. For instance, an online video rental that suggests multimedia items (e.g., DVDs) to its customers should consider the availability of DVDs in stock to reduce customer waiting times for accepted recommendations. Moreover, every user should receive a small but sufficient number of suggestions that the user is likely to be interested in. This thesis aims to develop and implement scalable optimization algorithms that can be used (but are not restricted) to generate recommendations satisfying certain objectives and constraints like the ones above. State-of-the-art approaches lack efficiency and/or scalability in coping with large real-world instances, which may involve millions of users and items. First, we study large-scale matrix completion in the context of collaborative filtering in recommender systems. For such problems, we propose a set of novel shared-nothing algorithms which are designed to run on a small cluster of commodity nodes and outperform alternative approaches in terms of efficiency, scalability, and memory footprint. Next, we view our recommendation task as a generalized matching problem, and propose the first distributed solution for solving such problems at scale. Our algorithm is designed to run on a small cluster of commodity nodes (or in a MapReduce environment) and has strong approximation guarantees. Our matching algorithm relies on linear programming. To this end, we present an efficient distributed approximation algorithm for mixed packing-covering linear programs, a simple but expressive subclass of linear programs. Our approximation algorithm requires a poly-logarithmic number of passes over the input, is simple, and well-suited for parallel processing on GPUs, in shared-memory architectures, as well as on a small cluster of commodity nodes.Empfehlungssysteme haben eine beachtliche Popularität erreicht und werden in zahlreichen E-Commerce Anwendungen eingesetzt. Entscheidend für die Generierung hochqualitativer Empfehlungen ist die Vorhersage von Nutzerpräferenzen. Jedoch sollten in der Praxis nicht nur Vorschläge auf Basis von Nutzerpräferenzen gegeben werden, sondern gute Empfehlungssysteme müssen auch bestimmte Nebenbedingungen berücksichtigen. Zum Beispiel sollten online Videoverleihfirmen, welche ihren Kunden multimediale Produkte (z.B. DVDs) vorschlagen, die Verfügbarkeit von vorrätigen DVDs beachten, um die Wartezeit der Kunden für angenommene Empfehlungen zu reduzieren. Darüber hinaus sollte jeder Kunde eine kleine, aber ausreichende Anzahl an Vorschlägen erhalten, an denen er interessiert sein könnte. Diese Arbeit strebt an skalierbare Optimierungsalgorithmen zu entwickeln und zu implementieren, die (unter anderem) eingesetzt werden können Empfehlungen zu generieren, welche weitere Zielvorgaben und Restriktionen einhalten. Derzeit existierenden Ansätzen mangelt es an Effizienz und/oder Skalierbarkeit im Umgang mit sehr großen, durchaus realen Datensätzen von, beispielsweise Millionen von Nutzern und Produkten. Zunächst analysieren wir die Vervollständigung großskalierter Matrizen im Kontext von kollaborativen Filtern in Empfehlungssystemen. Für diese Probleme schlagen wir verschiedene neue, verteilte Algorithmen vor, welche konzipiert sind auf einer kleinen Anzahl von gängigen Rechnern zu laufen. Zudem können sie alternative Ansätze hinsichtlich der Effizienz, Skalierbarkeit und benötigten Speicherkapazität überragen. Als Nächstes haben wir die Empfehlungsproblematik als ein generalisiertes Zuordnungsproblem betrachtet und schlagen daher die erste verteilte Lösung für großskalierte Zuordnungsprobleme vor. Unser Algorithmus funktioniert auf einer kleinen Gruppe von gängigen Rechnern (oder in einem MapReduce-Programmierungsmodel) und erzielt gute Approximationsgarantien. Unser Zuordnungsalgorithmus beruht auf linearer Programmierung. Daher präsentieren wir einen effizienten, verteilten Approximationsalgorithmus für vermischte lineare Packungs- und Überdeckungsprobleme, eine einfache aber expressive Unterklasse der linearen Programmierung. Unser Algorithmus benötigt eine polylogarithmische Anzahl an Scans der Eingabedaten. Zudem ist er einfach und sehr gut geeignet für eine parallele Verarbeitung mithilfe von Grafikprozessoren, unter einer gemeinsam genutzten Speicherarchitektur sowie auf einer kleinen Gruppe von gängigen Rechnern

Multi-Path Alpha-Fair Resource Allocation at Scale in Distributed Software Defined Networks

International audienceThe performance of computer networks relies on how bandwidth is shared among different flows. Fair resource allocation is a challenging problem particularly when the flows evolve over time. To address this issue, bandwidth sharing techniques that quickly react to the traffic fluctuations are of interest, especially in large scale settings with hundreds of nodes and thousands of flows. In this context, we propose a distributed algorithm based on the Alternating Direction Method of Multipliers (ADMM) that tackles the multi-path fair resource allocation problem in a distributed SDN control architecture. Our ADMM-based algorithm continuously generates a sequence of resource allocation solutions converging to the fair allocation while always remaining feasible, a property that standard primal-dual decomposition methods often lack. Thanks to the distribution of all computer intensive operations, we demonstrate that we can handle large instances at scale