arXiv subjects

Mikael Johansson

Publications and source records attributed to Mikael Johansson.

At least 73 records · Page 4Linked to original sources

arXiv2018

POLO: a POLicy-based Optimization library

We present POLO --- a C++ library for large-scale parallel optimization research that emphasizes ease-of-use, flexibility and efficiency in algorithm design. It uses multiple inheritance and template programming to decompose algorithms into essential policies and facilitate code reuse. With its clear separation between algorithm and execution policies, it provides researchers with a simple and powerful platform for prototyping ideas, evaluating them on different parallel computing architectures and hardware platforms, and generating compact and efficient production code. A C-API is included for customization and data loading in high-level languages. POLO enables users to move seamlessly from serial to multi-threaded shared-memory and multi-node distributed-memory executors. We demonstrate how POLO allows users to implement state-of-the-art asynchronous parallel optimization algorithms in just a few lines of code and report experiment results from shared and distributed-memory computing architectures. We provide both POLO and POLO.jl, a wrapper around POLO written in the Julia language, at https://github.com/pologrp under the permissive MIT license.

math.OC↗

arXiv2018

The Convergence of Sparsified Gradient Methods

Distributed training of massive machine learning models, in particular deep neural networks, via Stochastic Gradient Descent (SGD) is becoming commonplace. Several families of communication-reduction methods, such as quantization, large-batch methods, and gradient sparsification, have been proposed. To date, gradient sparsification methods - where each node sorts gradients by magnitude, and only communicates a subset of the components, accumulating the rest locally - are known to yield some of the largest practical gains. Such methods can reduce the amount of communication per step by up to three orders of magnitude, while preserving model accuracy. Yet, this family of methods currently has no theoretical justification. This is the question we address in this paper. We prove that, under analytic assumptions, sparsifying gradients by magnitude with local error correction provides convergence guarantees, for both convex and non-convex smooth objectives, for data-parallel SGD. The main insight is that sparsification methods implicitly maintain bounds on the maximum impact of stale updates, thanks to selection by magnitude. Our analysis and empirical validation also reveal that these methods do require analytical conditions to converge well, justifying existing heuristics.

cs.LG↗

arXiv2018

Relay-Pair Selection in Buffer-Aided Successive Opportunistic Relaying using a Multi-Antenna Source

We study a cooperative network with a buffer-aided multi-antenna source, multiple half-duplex (HD) buffer-aided relays and a single destination. Such a setup could represent a cellular downlink scenario, in which the source can be a more powerful wireless device with a buffer and multiple antennas, while a set of intermediate less powerful devices are used as relays to reach the destination. The main target is to recover the multiplexing loss of the network by having the source and a relay to simultaneously transmit their information to another relay and the destination, respectively. Successive transmissions in such a cooperative network, however, cause inter-relay interference (IRI). First, by assuming global channel state information (CSI), we show that the detrimental effect of IRI can be alleviated by precoding at the source, mitigating or even fully cancelling the interference. A cooperative relaying policy is proposed that employs a joint precoding design and relay-pair selection. Note that both fixed rate and adaptive rate transmissions can be considered. For the case when channel state information is only available at the receiver side (CSIR), we propose a relay selection policy that employs a phase alignment technique to reduce the IRI. The performance of the two proposed relay pair selection policies are evaluated and compared with other state-of-the-art relaying schemes in terms of outage and throughput. The results show that the use of a powerful source can provide considerable performance improvements.

cs.IT↗

arXiv2017

Stochastic Online Shortest Path Routing: The Value of Feedback

This paper studies online shortest path routing over multi-hop networks. Link costs or delays are time-varying and modeled by independent and identically distributed random processes, whose parameters are initially unknown. The parameters, and hence the optimal path, can only be estimated by routing packets through the network and observing the realized delays. Our aim is to find a routing policy that minimizes the regret (the cumulative difference of expected delay) between the path chosen by the policy and the unknown optimal path. We formulate the problem as a combinatorial bandit optimization problem and consider several scenarios that differ in where routing decisions are made and in the information available when making the decisions. For each scenario, we derive a tight asymptotic lower bound on the regret that has to be satisfied by any online routing policy. These bounds help us to understand the performance improvements we can expect when (i) taking routing decisions at each hop rather than at the source only, and (ii) observing per-link delays rather than end-to-end path delays. In particular, we show that (i) is of no use while (ii) can have a spectacular impact. Three algorithms, with a trade-off between computational complexity and performance, are proposed. The regret upper bounds of these algorithms improve over those of the existing algorithms, and they significantly outperform state-of-the-art algorithms in numerical experiments.

cs.NI↗

arXiv2017

Optimal control of linear systems with limited control actions: threshold-based event-triggered control

We consider a finite-horizon linear-quadratic optimal control problem where only a limited number of control messages are allowed for sending from the controller to the actuator. To restrict the number of control actions computed and transmitted by the controller, we employ a threshold-based event-triggering mechanism that decides whether or not a control message needs to be calculated and delivered. Due to the nature of threshold-based event-triggering algorithms, finding the optimal control sequence requires minimizing a quadratic cost function over a non-convex domain. In this paper, we firstly provide an exact solution to the non-convex problem mentioned above by solving an exponential number of quadratic programs. To reduce computational complexity, we, then, propose two efficient heuristic algorithms based on greedy search and the Alternating Direction Method of Multipliers (ADMM) method. Later, we consider a receding horizon control strategy for linear systems controlled by event-triggered controllers, and we also provide a complete stability analysis of receding horizon control that uses finite horizon optimization in the proposed class. Numerical examples testify to the viability of the presented design technique.

eess.SY↗

arXiv2016

Analysis and Implementation of an Asynchronous Optimization Algorithm for the Parameter Server

This paper presents an asynchronous incremental aggregated gradient algorithm and its implementation in a parameter server framework for solving regularized optimization problems. The algorithm can handle both general convex (possibly non-smooth) regularizers and general convex constraints. When the empirical data loss is strongly convex, we establish linear convergence rate, give explicit expressions for step-size choices that guarantee convergence to the optimum, and bound the associated convergence factors. The expressions have an explicit dependence on the degree of asynchrony and recover classical results under synchronous operation. Simulations and implementations on commercial compute clouds validate our findings.

math.OC↗

arXiv2016

Stability Analysis of Monotone Systems via Max-separable Lyapunov Functions

We analyze stability properties of monotone nonlinear systems via max-separable Lyapunov functions, motivated by the following observations: first, recent results have shown that asymptotic stability of a monotone nonlinear system implies the existence of a max-separable Lyapunov function on a compact set; second, for monotone linear systems, asymptotic stability implies the stronger properties of D-stability and insensitivity to time-delays. This paper establishes that for monotone nonlinear systems, equivalence holds between asymptotic stability, the existence of a max-separable Lyapunov function, D-stability, and insensitivity to bounded and unbounded time-varying delays. In particular, a new and general notion of D-stability for monotone nonlinear systems is discussed and a set of necessary and sufficient conditions for delay-independent stability are derived. Examples show how the results extend the state-of-the-art.

eess.SY↗

arXiv2015

Finite-time Convergent Gossiping

Gossip algorithms are widely used in modern distributed systems, with applications ranging from sensor networks and peer-to-peer networks to mobile vehicle networks and social networks. A tremendous research effort has been devoted to analyzing and improving the asymptotic rate of convergence for gossip algorithms. In this work we study finite-time convergence of deterministic gossiping. We show that there exists a symmetric gossip algorithm that converges in finite time if and only if the number of network nodes is a power of two, while there always exists an asymmetric gossip algorithm with finite-time convergence, independent of the number of nodes. For $n=2^m$ nodes, we prove that a fastest convergence can be reached in $nm=n\log_2 n$ node updates via symmetric gossiping. On the other hand, under asymmetric gossip among $n=2^m+r$ nodes with $0\leq r<2^m$, it takes at least $mn+2r$ node updates for achieving finite-time convergence. It is also shown that the existence of finite-time convergent gossiping often imposes strong structural requirements on the underlying interaction graph. Finally, we apply our results to gossip algorithms in quantum networks, where the goal is to control the state of a quantum system via pairwise interactions. We show that finite-time convergence is never possible for such systems.

cs.DC↗

arXiv2015

On Reconstructability of Quadratic Utility Functions from the Iterations in Gradient Methods

In this paper, we consider a scenario where an eavesdropper can read the content of messages transmitted over a network. The nodes in the network are running a gradient algorithm to optimize a quadratic utility function where such a utility optimization is a part of a decision making process by an administrator. We are interested in understanding the conditions under which the eavesdropper can reconstruct the utility function or a scaled version of it and, as a result, gain insight into the decision-making process. We establish that if the parameter of the gradient algorithm, i.e.,~the step size, is chosen appropriately, the task of reconstruction becomes practically impossible for a class of Bayesian filters with uniform priors. We establish what step-size rules should be employed to ensure this.

math.OC↗

arXiv2015

The Evolution of Beliefs over Signed Social Networks

We study the evolution of opinions (or beliefs) over a social network modeled as a signed graph. The sign attached to an edge in this graph characterizes whether the corresponding individuals or end nodes are friends (positive links) or enemies (negative links). Pairs of nodes are randomly selected to interact over time, and when two nodes interact, each of them updates its opinion based on the opinion of the other node and the sign of the corresponding link. This model generalizes DeGroot model to account for negative links: when two enemies interact, their opinions go in opposite directions. We provide conditions for convergence and divergence in expectation, in mean-square, and in almost sure sense, and exhibit phase transition phenomena for these notions of convergence depending on the parameters of the opinion update model and on the structure of the underlying graph. We establish a {\it no-survivor} theorem, stating that the difference in opinions of any two nodes diverges whenever opinions in the network diverge as a whole. We also prove a {\it live-or-die} lemma, indicating that almost surely, the opinions either converge to an agreement or diverge. Finally, we extend our analysis to cases where opinions have hard lower and upper limits. In these cases, we study when and how opinions may become asymptotically clustered to the belief boundaries, and highlight the crucial influence of (strong or weak) structural balance of the underlying network on this clustering phenomenon.

cs.SI↗

arXiv2015

Optimal Radio Frequency Energy Harvesting with Limited Energy Arrival Knowledge

In this paper, we develop optimal policies for deciding when a wireless node with radio frequency (RF) energy harvesting (EH) capabilities should try and harvest ambient RF energy. While the idea of RF-EH is appealing, it is not always beneficial to attempt to harvest energy; in environments where the ambient energy is low, nodes could consume more energy being awake with their harvesting circuits turned on than what they can extract from the ambient radio signals; it is then better to enter a sleep mode until the ambient RF energy increases. Towards this end, we consider a scenario with intermittent energy arrivals and a wireless node that wakes up for a period of time (herein called the time-slot) and harvests energy. If enough energy is harvested during the time-slot, then the harvesting is successful and excess energy is stored; however, if there does not exist enough energy the harvesting is unsuccessful and energy is lost. We assume that the ambient energy level is constant during the time-slot, and changes at slot boundaries. The energy level dynamics are described by a two-state Gilbert-Elliott Markov chain model, where the state of the Markov chain can only be observed during the harvesting action, and not when in sleep mode. Two scenarios are studied under this model. In the first scenario, we assume that we have knowledge of the transition probabilities of the Markov chain and formulate the problem as a Partially Observable Markov Decision Process (POMDP), where we find a threshold-based optimal policy. In the second scenario, we assume that we don't have any knowledge about these parameters and formulate the problem as a Bayesian adaptive POMDP; to reduce the complexity of the computations we also propose a heuristic posterior sampling algorithm. The performance of our approaches is demonstrated via numerical examples.

cs.IT↗

arXiv2015

Energy efficient D2D communications in dynamic TDD systems

Network-assisted device-to-device communication is a promising technology for improving the performance of proximity-based services. This paper demonstrates how the integration of device-to-device communications and dynamic time-division duplex can improve the energy efficiency of future cellular networks, leading to a greener system operation and a prolonged battery lifetime of mobile devices. We jointly optimize the mode selection, transmission period and power allocation to minimize the energy consumption (from both a system and a device perspective) while satisfying a certain rate requirement. The radio resource management problems are formulated as mixed-integer nonlinear programming problems. Although they are known to be NP-hard in general, we exploit the problem structure to design efficient algorithms that optimally solve several problem cases. For the remaining cases, a heuristic algorithm that computes near-optimal solutions while respecting practical constraints on execution times and signaling overhead is also proposed. Simulation results confirm that the combination of device-to-device and flexible time-division-duplex technologies can significantly enhance spectrum and energy-efficiency of next generation cellular systems.

cs.IT↗

arXiv2015

An Asynchronous Mini-Batch Algorithm for Regularized Stochastic Optimization

Mini-batch optimization has proven to be a powerful paradigm for large-scale learning. However, the state of the art parallel mini-batch algorithms assume synchronous operation or cyclic update orders. When worker nodes are heterogeneous (due to different computational capabilities or different communication delays), synchronous and cyclic operations are inefficient since they will leave workers idle waiting for the slower nodes to complete their computations. In this paper, we propose an asynchronous mini-batch algorithm for regularized stochastic optimization problems with smooth loss functions that eliminates idle waiting and allows workers to run at their maximal update rates. We show that by suitably choosing the step-size values, the algorithm achieves a rate of the order $O(1/\sqrt{T})$ for general convex regularization functions, and the rate $O(1/T)$ for strongly convex regularization functions, where $T$ is the number of iterations. In both cases, the impact of asynchrony on the convergence rate of our algorithm is asymptotically negligible, and a near-linear speedup in the number of workers can be expected. Theoretical results are confirmed in real implementations on a distributed computing infrastructure.

math.OC↗

arXiv2015

Convergence analysis of approximate primal solutions in dual first-order methods

Dual first-order methods are powerful techniques for large-scale convex optimization. Although an extensive research effort has been devoted to studying their convergence properties, explicit convergence rates for the primal iterates have only been established under global Lipschitz continuity of the dual gradient. This is a rather restrictive assumption that does not hold for several important classes of problems. In this paper, we demonstrate that primal convergence rate guarantees can also be obtained when the dual gradient is only locally Lipschitz. The class of problems that we analyze admits general convex constraints including nonlinear inequality, linear equality, and set constraints. As an approximate primal solution, we take the minimizer of the Lagrangian, computed when evaluating the dual gradient. We derive error bounds for this approximate primal solution in terms of the errors of the dual variables, and establish convergence rates of the dual variables when the dual problem is solved using a projected gradient or fast gradient method. By combining these results, we show that the suboptimality and infeasibility of the approximate primal solution at iteration $k$ are no worse than $O(1/\sqrt{k})$ when the dual problem is solved using a projected gradient method, and $O(1/k)$ when a fast dual gradient method is used.

math.OC↗

arXiv2015

On the trade-off between control performance and communication cost in event-triggered control

We consider a stochastic system where the communication between the controller and the actuator is triggered by a threshold-based rule. The communication is performed across an unreliable link that stochastically erases transmitted packets. To decrease the communication burden, and as a partial protection against dropped packets, the controller sends a sequence of control commands to the actuator in each packet. These commands are stored in a buffer and applied sequentially until the next control packet arrives. In this context, we study dead-beat control laws and compute the expected linear-quadratic loss of the closed-loop system for any given event-threshold. Furthermore, we provide analytical expressions that quantify the trade-off between the communication cost and the control performance of event-triggered control systems. Numerical examples demonstrate the effectiveness of the proposed framework.

eess.SY↗

arXiv2014

Global convergence of the Heavy-ball method for convex optimization

This paper establishes global convergence and provides global bounds of the convergence rate of the Heavy-ball method for convex optimization problems. When the objective function has Lipschitz-continuous gradient, we show that the Cesaro average of the iterates converges to the optimum at a rate of $O(1/k)$ where k is the number of iterations. When the objective function is also strongly convex, we prove that the Heavy-ball iterates converge linearly to the unique optimum.

math.OC↗

arXiv2014

Optimal scaling of the ADMM algorithm for distributed quadratic programming

This paper presents optimal scaling of the alternating directions method of multipliers (ADMM) algorithm for a class of distributed quadratic programming problems. The scaling corresponds to the ADMM step-size and relaxation parameter, as well as the edge-weights of the underlying communication graph. We optimize these parameters to yield the smallest convergence factor of the algorithm. Explicit expressions are derived for the step-size and relaxation parameter, as well as for the corresponding convergence factor. Numerical simulations justify our results and highlight the benefits of optimally scaling the ADMM algorithm.

math.OC↗

arXiv2014

Emergent Behaviors over Signed Random Dynamical Networks: Relative-State-Flipping Model

We study asymptotic dynamical patterns that emerge among a set of nodes interacting in a dynamically evolving signed random network, where positive links carry out standard consensus and negative links induce relative-state flipping. A sequence of deterministic signed graphs define potential node interactions that take place independently. Each node receives a positive recommendation consistent with the standard consensus algorithm from its positive neighbors, and a negative recommendation defined by relative-state flipping from its negative neighbors. After receiving these recommendations, each node puts a deterministic weight to each recommendation, and then encodes these weighted recommendations in its state update through stochastic attentions defined by two Bernoulli random variables. We establish a number of conditions regarding almost sure convergence and divergence of the node states. We also propose a condition for almost sure state clustering for essentially weakly balanced graphs, with the help of several martingale convergence lemmas. Some fundamental differences on the impact of the deterministic weights and stochastic attentions to the node state evolution are highlighted between the current relative-state-flipping model and the state-flipping model considered in Altafini 2013 and Shi et al. 2014.

cs.SI↗