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This paper considers stochastic convex optimization problems with two sets of constraints: (a) deterministic constraints on the domain of the optimization variable, which are difficult to project onto; and (b) deterministic or stochastic constraints that admit efficient projection. Problems of this form arise frequently in the context of semidefinite programming as well as when various NP-hard problems are solved approximately via semidefinite relaxation. Since projection onto the first set of constraints is difficult, it becomes necessary to explore projection-free algorithms, such as the stochastic Frank-Wolfe (FW) algorithm. On the other hand, the second set of constraints cannot be handled in the same way, and must be incorporated as an indicator function within the objective function, thereby complicating the application of FW methods. Similar problems have been studied before; however, they suffer from slow convergence rates. This work, equipped with momentum based gradient tracking technique, guarantees fast convergence rates on par with the best-known rates for problems without the second set of constraints. Zeroth-order variants of the proposed algorithms are also developed and again improve upon the state-of-the-art rate results. We further propose the novel trimmed FW variants that enjoy the same convergence rates as their classical counterparts, but are empirically shown to require significantly fewer calls to the linear minimization oracle speeding up the overall algorithm. The efficacy of the proposed algorithms is tested on relevant applications of sparse matrix estimation, clustering via semidefinite relaxation, and uniform sparsest cut problem. [ABSTRACT FROM AUTHOR]

Optimizing non-convex functions is of primary importance in modern pattern recognition because it underlies the training of deep networks and nonlinear dimensionality reduction. First-order algorithms under suitable randomized perturbations or step-size rules have been shown to be effective for such settings as their limit points can be guaranteed to be local extrema rather than saddle points. However, it is well-known that the practical convergence of first-order methods is slower than those which exploit additional structure. In particular, empirically, successive convex approximation (SCA) converges faster than first-order methods. However, to date, SCA in general non-convex settings converges to first-order stationary points, which could either be local extrema or saddle points whose performance is typically inferior. To mitigate this issue, we propose calibrated randomized perturbations of SCA, which exhibit the improved convergence rate as compared to the gradient descent counter part. In particular, our main technical contributions are to establish the non-asymptotic performance of SCA algorithm and its perturbed variant converges to an approximate second-order stationary point. Experiments on multi-dimensional scaling, a machine learning problem whose training objective is non-convex, substantiate the performance gains associated with employing random perturbations. [ABSTRACT FROM AUTHOR]

We consider stochastic optimization of a smooth non-convex loss function with a convex non-smooth regularizer. In the online setting, where a single sample of the stochastic gradient of the loss is available at every iteration, the problem can be solved using the proximal stochastic gradient descent (SGD) algorithm and its variants. However in many problems, especially those arising in communications and signal processing, information beyond the stochastic gradient may be available thanks to the structure of the loss function. By constructing a strongly convex surrogate of the loss function at every iteration, the stochastic SCA algorithms can exploit the structural properties of the loss function. In this work, we take a closer look at the stochastic SCA algorithm and develop its asynchronous variant which can be used for resource allocation in wireless networks. While the stochastic SCA algorithm is known to converge asymptotically, its iteration complexity has not been well-studied, and is the focus of the current work. The insights obtained from the non-asymptotic analysis allow us to develop a more practical asynchronous variant of the stochastic SCA algorithm which allows the use of surrogate calculated in earlier iterations. We characterize precise bound on the maximum delay the algorithm can tolerate, while still achieving the same convergence rate. We apply the algorithm to the problem of linear precoding in wireless sensor networks, where it can be implemented at low complexity but is shown to perform well in practice. [ABSTRACT FROM AUTHOR]

Recent decades have observed an exponential growth in network traffic, thanks to the increased popularity of real-time applications, such as live video chat and gaming. The resulting growth in the network infrastructure has made it difficult for the service providers to abide by the service level agreements, especially with regards to the quality-of-service guarantees. Predicting network latencies from noisy and missing measurements has therefore emerged as an important problem, and a plethora of solutions have been proposed for the same. Existing network latency predictions rely either on Euclidean embedding or matrix completion methods. This work considers the estimation and prediction of network latencies from a sequence of noisy and incomplete latency matrices collected over time. An adaptive matrix completion algorithm is proposed that can handle streaming data at low computational complexity. The performance of the proposed algorithm is characterized both in theory and using a real dataset, demonstrating its viability as a network monitoring tool. [ABSTRACT FROM AUTHOR]

We consider expected risk minimization in multi-agent systems comprised of distinct subsets of agents operating without a common time scale. Each individual in the network is charged with minimizing the global objective function, which is an average of sum of the statistical average loss function of each agent in the network. Since agents are not assumed to observe data from identical distributions, the hypothesis that all agents seek a common action is violated, and thus the hypothesis upon that consensus constraints are formulated is violated. Thus, we consider nonlinear network proximity constraints, which incentivize nearby nodes to make decisions that are close to one another but do not necessarily coincide. Moreover, agents are not assumed to receive their sequentially arriving observations on a common time index, and thus seek to learn in an asynchronous manner. An asynchronous stochastic variant of the Arrow–Hurwicz saddle point method is proposed to solve this problem that operates by alternating primal stochastic descent steps and Lagrange multiplier updates that penalize the discrepancies between agents. This tool leads to an implementation that allows for each agent to operate asynchronously with local information only and message passing with neighbors. Our main result establishes that the proposed method yields convergence in expectation both in terms of the primal sub-optimality and constraint violation to radii of sizes ${\mathcal O}(\sqrt{T})$ and ${\mathcal O}(T^{3/4})$ , respectively. Empirical evaluation on an asynchronously operating wireless network that manages user channel interference through an adaptive communications pricing mechanism demonstrates that our theoretical results translates well to practice. [ABSTRACT FROM AUTHOR]

We consider nondifferentiable dynamic optimization problems such as those arising in robotics and subspace tracking. Given the computational constraints and the time-varying nature of the problem, a low-complexity algorithm is desirable, while the accuracy of the solution may only increase slowly over time. We put forth the proximal online gradient descent (OGD) algorithm for tracking the optimum of a composite objective function comprising of a differentiable loss function and a nondifferentiable regularizer. An online learning framework is considered and the gradient of the loss function is allowed to be erroneous. Both, the gradient error as well as the dynamics of the function optimum or target are adversarial and the performance of the inexact proximal OGD is characterized in terms of its dynamic regret, expressed in terms of the cumulative error and path length of the target. The proposed inexact proximal OGD is generalized for application to large-scale problems where the loss function has a finite sum structure. In such cases, evaluation of the full gradient may not be viable and a variance reduced version is proposed that allows the component functions to be subsampled. The efficacy of the proposed algorithms is tested on the problem of formation control in robotics and on the dynamic foreground–background separation problem in video. [ABSTRACT FROM AUTHOR]

Stochastic network optimization problems entail finding resource allocation policies that are optimum on an average but must be designed in an online fashion. Such problems are ubiquitous in communication networks, where resources such as energy and bandwidth are divided among nodes to satisfy certain long-term objectives. This paper proposes an asynchronous incremental dual decent resource allocation algorithm that utilizes delayed stochastic gradients for carrying out its updates. The proposed algorithm is well-suited to heterogeneous networks as it allows the computationally-challenged or energy-starved nodes to, at times, postpone the updates. The asymptotic analysis of the proposed algorithm is carried out, establishing dual convergence under both, constant and diminishing step sizes. It is also shown that with constant step size, the proposed resource allocation policy is asymptotically near-optimal. An application involving multicell coordinated beamforming is detailed, demonstrating the usefulness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Multiagent systems are being increasingly deployed in challenging environments for performing complex tasks such as multitarget tracking, search-and-rescue, and intrusion detection. Not with standing the computational limitations of individual robots, such systems rely on collaboration to sense and react to the environment. This paper formulates the generic target tracking problem as a time-varying optimization problem and puts forth an inexact online gradient descent method for solving it sequentially. The performance of the proposed algorithm is studied by characterizing its dynamic regret, a notion common to the online learning literature. Building upon the existing results, we provide improved regret rates that not only allow nonstrongly convex costs but also explain the role of the cumulative gradient error. Two distinct classes of problems are considered: one in which the objective function adheres to a quadratic growth condition, and another where the objective function is convex but the variable belongs to a compact domain. For both cases, results are developed while allowing the error to be either adversarial or arising from a white noise process. Further, the generality of the proposed framework is demonstrated by developing online variants of existing stochastic gradient algorithms and interpreting them as special cases of the proposed inexact gradient method. The efficacy of the proposed inexact gradient framework is established on a multiagent multitarget tracking problem, while its flexibility is exemplified by generating online movie recommendations for Movielens 10M dataset. [ABSTRACT FROM PUBLISHER]

The low-rank matrix completion problem is fundamental to a number of tasks in data mining, machine learning, and signal processing. This paper considers the problem of adaptive matrix completion in time-varying scenarios. Given a sequence of incomplete and noise-corrupted matrices, the goal is to recover and track the underlying low rank matrices. Motivated from the classical least-mean square (LMS) algorithms for adaptive filtering, three LMS-like algorithms are proposed for estimating and tracking low-rank matrices. Performance of the proposed algorithms is provided in form of nonasymptotic bounds on the tracking mean-square error. Tracking performance of the algorithms is also studied via detailed simulations over real-world datasets. [ABSTRACT FROM PUBLISHER]

This paper considers the problem of cooperative localization in mobile networks. In static networks, node locations can be obtained from pairwise distance measurements using the classical multidimensional scaling (MDS) approach. This paper introduces a modified MDS framework that also incorporates relative velocity measurements if available in mobile networks. The proposed cost function is minimized via a low complexity majorization algorithm. A distributed variant of the proposed algorithm is also outlined for use in large resource-constrained multi-hop networks. The proposed algorithm incurs low computational and communication cost, and handles practical constraints such as missing measurements and variable node velocities. Simulation results corroborate the performance gains obtained by the proposed algorithm over state-of-the-art localization algorithms. [ABSTRACT FROM AUTHOR]

This paper deals with network-coded multicast for real-time and streaming-media applications where packets have explicit expiration deadlines. Most of the popular network coding approaches require asymptotically large block-lengths, thereby incurring long decoding delays. The present paper introduces a joint scheduling and network coding design that aims to maximize the average throughput while respecting the packet deadlines. The novel approach relies on a time-unwrapped graph expansion in order to construct the network codes. The resultant algorithm draws from the well-known augmenting-path algorithm, and is both distributed as well as scalable. For networks with primary interference, a lower-bound on the worst-case performance of the algorithm is provided. The associated optimization problem is also analyzed from an integer programming perspective, and a set of valid inequalities is derived to obtain an upper bound. [ABSTRACT FROM AUTHOR]