arXiv subjects

Dusan Jakovetic

Publications and source records attributed to Dusan Jakovetic.

At least 37 records · Page 2Linked to original sources

arXiv2022

Nonlinear gradient mappings and stochastic optimization: A general framework with applications to heavy-tail noise

We introduce a general framework for nonlinear stochastic gradient descent (SGD) for the scenarios when gradient noise exhibits heavy tails. The proposed framework subsumes several popular nonlinearity choices, like clipped, normalized, signed or quantized gradient, but we also consider novel nonlinearity choices. We establish for the considered class of methods strong convergence guarantees assuming a strongly convex cost function with Lipschitz continuous gradients under very general assumptions on the gradient noise. Most notably, we show that, for a nonlinearity with bounded outputs and for the gradient noise that may not have finite moments of order greater than one, the nonlinear SGD's mean squared error (MSE), or equivalently, the expected cost function's optimality gap, converges to zero at rate~$O(1/t^ζ)$, $ζ\in (0,1)$. In contrast, for the same noise setting, the linear SGD generates a sequence with unbounded variances. Furthermore, for the nonlinearities that can be decoupled component wise, like, e.g., sign gradient or component-wise clipping, we show that the nonlinear SGD asymptotically (locally) achieves a $O(1/t)$ rate in the weak convergence sense and explicitly quantify the corresponding asymptotic variance. Experiments show that, while our framework is more general than existing studies of SGD under heavy-tail noise, several easy-to-implement nonlinearities from our framework are competitive with state of the art alternatives on real data sets with heavy tail noises.

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arXiv2022

A Hessian inversion-free exact second order method for distributed consensus optimization

We consider a standard distributed consensus optimization problem where a set of agents connected over an undirected network minimize the sum of their individual local strongly convex costs. Alternating Direction Method of Multipliers ADMM and Proximal Method of Multipliers PMM have been proved to be effective frameworks for design of exact distributed second order methods involving calculation of local cost Hessians. However, existing methods involve explicit calculation of local Hessian inverses at each iteration that may be very costly when the dimension of the optimization variable is large. In this paper we develop a novel method termed INDO Inexact Newton method for Distributed Optimization that alleviates the need for Hessian inverse calculation. INDO follows the PMM framework but unlike existing work approximates the Newton direction through a generic fixed point method, e.g., Jacobi Overrelaxation, that does not involve Hessian inverses. We prove exact global linear convergence of INDO and provide analytical studies on how the degree of inexactness in the Newton direction calculation affects the overall methods convergence factor. Numerical experiments on several real data sets demonstrate that INDOs speed is on par or better as state of the art methods iterationwise hence having a comparable communication cost. At the same time, for sufficiently large optimization problem dimensions n (even at n on the order of couple of hundreds), INDO achieves savings in computational cost by at least an order of magnitude.

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arXiv2022

Personalized Federated Learning via Convex Clustering

We propose a parametric family of algorithms for personalized federated learning with locally convex user costs. The proposed framework is based on a generalization of convex clustering in which the differences between different users' models are penalized via a sum-of-norms penalty, weighted by a penalty parameter $λ$. The proposed approach enables "automatic" model clustering, without prior knowledge of the hidden cluster structure, nor the number of clusters. Analytical bounds on the weight parameter, that lead to simultaneous personalization, generalization and automatic model clustering are provided. The solution to the formulated problem enables personalization, by providing different models across different clusters, and generalization, by providing models different than the per-user models computed in isolation. We then provide an efficient algorithm based on the Parallel Direction Method of Multipliers (PDMM) to solve the proposed formulation in a federated server-users setting. Numerical experiments corroborate our findings. As an interesting byproduct, our results provide several generalizations to convex clustering.

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arXiv2021

Accurate, Interpretable, and Fast Animation: An Iterative, Sparse, and Nonconvex Approach

Digital human animation relies on high-quality 3D models of the human face: rigs. A face rig must be accurate and, at the same time, fast to compute. One of the most common rigging models is the blendshape model. We propose a novel algorithm for solving the nonconvex inverse rig problem in facial animation. Our approach is model-based, but in contrast with previous model-based approaches, we use a quadratic instead of the linear approximation to the higher order rig model. This increases the accuracy of the solution by 8 percent on average and, confirmed by the empirical results, increases the sparsity of the resulting parameter vector -- an important feature for interpretability by animation artists. The proposed solution is based on a Levenberg-Marquardt (LM) algorithm, applied to a nonconvex constrained problem with sparsity regularization. In order to reduce the complexity of the iterates, a paradigm of Majorization Minimization (MM) is further invoked, which leads to an easy to solve problem that is separable in the parameters at each algorithm iteration. The algorithm is evaluated on a number of animation datasets, proprietary and open-source, and the results indicate the superiority of our method compared to the standard approach based on the linear rig approximation. Although our algorithm targets the specific problem, it might have additional signal processing applications.

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arXiv2021

Deep Learning Anomaly Detection for Cellular IoT with Applications in Smart Logistics

The number of connected Internet of Things (IoT) devices within cyber-physical infrastructure systems grows at an increasing rate. This poses significant device management and security challenges to current IoT networks. Among several approaches to cope with these challenges, data-based methods rooted in deep learning (DL) are receiving an increased interest. In this paper, motivated by the upcoming surge of 5G IoT connectivity in industrial environments, we propose to integrate a DL-based anomaly detection (AD) as a service into the 3GPP mobile cellular IoT architecture. The proposed architecture embeds autoencoder based anomaly detection modules both at the IoT devices (ADM-EDGE) and in the mobile core network (ADM-FOG), thereby balancing between the system responsiveness and accuracy. We design, integrate, demonstrate and evaluate a testbed that implements the above service in a real-world deployment integrated within the 3GPP Narrow-Band IoT (NB-IoT) mobile operator network.

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arXiv2020

EFIX: Exact Fixed Point Methods for Distributed Optimization

We consider strongly convex distributed consensus optimization over connected networks. EFIX, the proposed method, is derived using quadratic penalty approach. In more detail, we use the standard reformulation { transforming the original problem into a constrained problem in a higher dimensional space { to define a sequence of suitable quadratic penalty subproblems with increasing penalty parameters. For quadratic objectives, the corresponding sequence consists of quadratic penalty subproblems. For the generic strongly convex case, the objective function is approximated with a quadratic model and hence the sequence of the resulting penalty subproblems is again quadratic. EFIX is then derived by solving each of the quadratic penalty subproblems via a fixed point (R)-linear solver, e.g., Jacobi Over-Relaxation method. The exact convergence is proved as well as the worst case complexity of order O(epsilon^-1) for the quadratic case. In the case of strongly convex generic functions, a standard result for penalty methods is obtained. Numerical results indicate that the method is highly competitive with state-of-the-art exact first order methods, requires smaller computational and communication effort, and is robust to the choice of algorithm parameters.

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arXiv2020

Primal-dual methods for large-scale and distributed convex optimization and data analytics

The augmented Lagrangian method (ALM) is a classical optimization tool that solves a given "difficult" (constrained) problem via finding solutions of a sequence of "easier"(often unconstrained) sub-problems with respect to the original (primal) variable, wherein constraints satisfaction is controlled via the so-called dual variables. ALM is highly flexible with respect to how primal sub-problems can be solved, giving rise to a plethora of different primal-dual methods. The powerful ALM mechanism has recently proved to be very successful in various large scale and distributed applications. In addition, several significant advances have appeared, primarily on precise complexity results with respect to computational and communication costs in the presence of inexact updates and design and analysis of novel optimal methods for distributed consensus optimization. We provide a tutorial-style introduction to ALM and its variants for solving convex optimization problems in large scale and distributed settings. We describe control-theoretic tools for the algorithms' analysis and design, survey recent results, and provide novel insights in the context of two emerging applications: federated learning and distributed energy trading.

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arXiv2020

Distributed Fixed Point Method for Solving Systems of Linear Algebraic Equations

We present a class of iterative fully distributed fixed point methods to solve a system of linear equations, such that each agent in the network holds one of the equations of the system. Under a generic directed, strongly connected network, we prove a convergence result analogous to the one for fixed point methods in the classical, centralized, framework: the proposed method converges to the solution of the system of linear equations at a linear rate. We further explicitly quantify the rate in terms of the linear system and the network parameters. Next, we show that the algorithm provably works under time-varying directed networks provided that the underlying graph is connected over bounded iteration intervals, and we establish a linear convergence rate for this setting as well. A set of numerical results is presented, demonstrating practical benefits of the method over existing alternatives.

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arXiv2019

Distributed Nesterov gradient methods over arbitrary graphs

In this letter, we introduce a distributed Nesterov method, termed as $\mathcal{ABN}$, that does not require doubly-stochastic weight matrices. Instead, the implementation is based on a simultaneous application of both row- and column-stochastic weights that makes this method applicable to arbitrary (strongly-connected) graphs. Since constructing column-stochastic weights needs additional information (the number of outgoing neighbors at each agent), not available in certain communication protocols, we derive a variation, termed as FROZEN, that only requires row-stochastic weights but at the expense of additional iterations for eigenvector learning. We numerically study these algorithms for various objective functions and network parameters and show that the proposed distributed Nesterov methods achieve acceleration compared to the current state-of-the-art methods for distributed optimization.

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arXiv2019

Exact Spectral-Like Gradient Method for Distributed Optimization

Since the initial proposal in the late 80s, spectral gradient methods continue to receive significant attention, especially due to their excellent numerical performance on various large scale applications. However, to date, they have not been sufficiently explored in the context of distributed optimization. In this paper, we consider unconstrained distributed optimization problems where $n$ nodes constitute an arbitrary connected network and collaboratively minimize the sum of their local convex cost functions. In this setting, building from existing exact distributed gradient methods, we propose a novel exact distributed gradient method wherein nodes' step-sizes are designed according to the novel rules akin to those in spectral gradient methods. We refer to the proposed method as Distributed Spectral Gradient method (DSG). The method exhibits R-linear convergence under standard assumptions for the nodes' local costs and safeguarding on the algorithm step-sizes. We illustrate the method's performance through simulation examples.

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arXiv2018

Distributed second order methods with increasing number of working nodes

Recently, an idling mechanism has been introduced in the context of distributed \emph{first order} methods for minimization of a sum of nodes' local convex costs over a generic, connected network. With the idling mechanism, each node $i$, at each iteration $k$, is active -- updates its solution estimate and exchanges messages with its network neighborhood -- with probability $p_k$, and it stays idle with probability $1-p_k$, while the activations are independent both across nodes and across iterations. In this paper, we demonstrate that the idling mechanism can be successfully incorporated in \emph{distributed second order methods} also. Specifically, we apply the idling mechanism to the recently proposed Distributed Quasi Newton method (DQN). We first show theoretically that, when $p_k$ grows to one across iterations in a controlled manner, DQN with idling exhibits very similar theoretical convergence and convergence rates properties as the standard DQN method, thus achieving the same order of convergence rate (R-linear) as the standard DQN, but with significantly cheaper updates. Simulation examples confirm the benefits of incorporating the idling mechanism, demonstrate the method's flexibility with respect to the choice of the $p_k$'s, and compare the proposed idling method with related algorithms from the literature.

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arXiv2018

Communication-Efficient Distributed Strongly Convex Stochastic Optimization: Non-Asymptotic Rates

We examine fundamental tradeoffs in iterative distributed zeroth and first order stochastic optimization in multi-agent networks in terms of \emph{communication cost} (number of per-node transmissions) and \emph{computational cost}, measured by the number of per-node noisy function (respectively, gradient) evaluations with zeroth order (respectively, first order) methods. Specifically, we develop novel distributed stochastic optimization methods for zeroth and first order strongly convex optimization by utilizing a probabilistic inter-agent communication protocol that increasingly sparsifies communications among agents as time progresses. Under standard assumptions on the cost functions and the noise statistics, we establish with the proposed method the $O(1/(C_{\mathrm{comm}})^{4/3-ζ})$ and $O(1/(C_{\mathrm{comm}})^{8/9-ζ})$ mean square error convergence rates, for the first and zeroth order optimization, respectively, where $C_{\mathrm{comm}}$ is the expected number of network communications and $ζ>0$ is arbitrarily small. The methods are shown to achieve order-optimal convergence rates in terms of computational cost~$C_{\mathrm{comp}}$, $O(1/C_{\mathrm{comp}})$ (first order optimization) and $O(1/(C_{\mathrm{comp}})^{2/3})$ (zeroth order optimization), while achieving the order-optimal convergence rates in terms of iterations. Experiments on real-life datasets illustrate the efficacy of the proposed algorithms.

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arXiv2018

$\mathcal{CIRFE}$: A Distributed Random Fields Estimator

This paper presents a communication efficient distributed algorithm, $\mathcal{CIRFE}$ of the \emph{consensus}+\emph{innovations} type, to estimate a high-dimensional parameter in a multi-agent network, in which each agent is interested in reconstructing only a few components of the parameter. This problem arises for example when monitoring the high-dimensional distributed state of a large-scale infrastructure with a network of limited capability sensors and where each sensor is tasked with estimating some local components of the state. At each observation sampling epoch, each agent updates its local estimate of the parameter components in its interest set by simultaneously processing the latest locally sensed information~(\emph{innovations}) and the parameter estimates from agents~(\emph{consensus}) in its communication neighborhood given by a time-varying possibly sparse graph. Under minimal conditions on the inter-agent communication network and the sensing models, almost sure convergence of the estimate sequence at each agent to the components of the true parameter in its interest set is established. Furthermore, the paper establishes the performance of $\mathcal{CIRFE}$ in terms of asymptotic covariance of the estimate sequences and specifically characterizes the dependencies of the component wise asymptotic covariance in terms of the number of agents tasked with estimating it. Finally, simulation experiments demonstrate the efficacy of $\mathcal{CIRFE}$.

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arXiv2018

Convergence rates for distributed stochastic optimization over random networks

We establish the O($\frac{1}{k}$) convergence rate for distributed stochastic gradient methods that operate over strongly convex costs and random networks. The considered class of methods is standard each node performs a weighted average of its own and its neighbors solution estimates (consensus), and takes a negative step with respect to a noisy version of its local functions gradient (innovation). The underlying communication network is modeled through a sequence of temporally independent identically distributed (i.i.d.) Laplacian matrices connected on average, while the local gradient noises are also i.i.d. in time, have finite second moment, and possibly unbounded support. We show that, after a careful setting of the consensus and innovations potentials (weights), the distributed stochastic gradient method achieves a (order-optimal) O($\frac{1}{k}$) convergence rate in the mean square distance from the solution. This is the first order-optimal convergence rate result on distributed strongly convex stochastic optimization when the network is random and/or the gradient noises have unbounded support. Simulation examples confirm the theoretical findings.

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arXiv2018

Distributed Zeroth Order Optimization Over Random Networks: A Kiefer-Wolfowitz Stochastic Approximation Approach

We study a standard distributed optimization framework where $N$ networked nodes collaboratively minimize the sum of their local convex costs. The main body of existing work considers the described problem when the underling network is either static or deterministically varying, and the distributed optimization algorithm is of first or second order, i.e., it involves the local costs' gradients and possibly the local Hessians. In this paper, we consider the currently understudied but highly relevant scenarios when: 1) only noisy function values' estimates are available (no gradients nor Hessians can be evaluated); and 2) the underlying network is randomly varying (according to an independent, identically distributed process). For the described random networks-zeroth order optimization setting, we develop a distributed stochastic approximation method of the Kiefer-Wolfowitz type. Furthermore, under standard smoothness and strong convexity assumptions on the local costs, we establish the $O(1/k^{1/2})$ mean square convergence rate for the method -- the rate that matches that of the method's centralized counterpart under equivalent conditions.

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arXiv2018

Communication Optimality Trade-offs For Distributed Estimation

This paper proposes $\mathbf{C}$ommunication efficient $\mathbf{RE}$cursive $\mathbf{D}$istributed estimati$\mathbf{O}$n algorithm, $\mathcal{CREDO}$, for networked multi-worker setups without a central master node. $\mathcal{CREDO}$ is designed for scenarios in which the worker nodes aim to collaboratively estimate a vector parameter of interest using distributed online time-series data at the individual worker nodes. The individual worker nodes iteratively update their estimate of the parameter by assimilating latest locally sensed information and estimates from neighboring worker nodes exchanged over a (possibly sparse) time-varying communication graph. The underlying inter-worker communication protocol is adaptive, making communications increasingly (probabilistically) sparse as time progresses. Under minimal conditions on the inter-worker information exchange network and the sensing models, almost sure convergence of the estimate sequences at the worker nodes to the true parameter is established. Further, the paper characterizes the performance of $\mathcal{CREDO}$ in terms of asymptotic covariance of the estimate sequences and specifically establishes the achievability of optimal asymptotic covariance. The analysis reveals an interesting interplay between the algorithm's communication cost~$\mathcal{C}_{t}$ (over $t$ time-steps) and the asymptotic covariance. Most notably, it is shown that $\mathcal{CREDO}$ may be designed to achieve a $Θ\left(\mathcal{C}_{t}^{-2+ζ}\right)$ decay of the mean square error~($ζ>0$, arbitrarily small) at each worker node, which significantly improves over the existing $ Θ\left(\mathcal{C}_{t}^{-1}\right)$ rates. Simulation examples on both synthetic and real data sets demonstrate $\mathcal{CREDO}$'s communication efficiency.

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arXiv2017

A Unification and Generalization of Exact Distributed First Order Methods

Recently, there has been significant progress in the development of distributed first order methods. (At least) two different types of methods, designed from very different perspectives, have been proposed that achieve both exact and linear convergence when a constant step size is used -- a favorable feature that was not achievable by most prior methods. In this paper, we unify, generalize, and improve convergence speed of these exact distributed first order methods. We first carry out a novel unifying analysis that sheds light on how the different existing methods compare. The analysis reveals that a major difference between the methods is on how a past dual gradient of an associated augmented Lagrangian dual function is weighted. We then capitalize on the insights from the analysis to derive a novel method -- with a tuned past gradient weighting -- that improves upon the existing methods. We establish for the proposed generalized method global R-linear convergence rate under strongly convex costs with Lipschitz continuous gradients.

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arXiv2017

Cooperative Slotted ALOHA for Massive M2M Random Access Using Directional Antennas

Slotted ALOHA (SA) algorithms with Successive Interference Cancellation (SIC) decoding have received significant attention lately due to their ability to dramatically increase the throughput of traditional SA. Motivated by increased density of cellular radio access networks due to the introduction of small cells, and dramatic increase of user density in Machine-to-Machine (M2M) communications, SA algorithms with SIC operating cooperatively in multi base station (BS) scenario are recently considered. In this paper, we generalize our previous work on Slotted ALOHA with multiple-BS (SA-MBS) by considering users that use directional antennas. In particular, we focus on a simple randomized beamforming strategy where, for every packet transmission, a user orients its main beam in a randomly selected direction. We are interested in the total achievable system throughput for two decoding scenarios: i) non-cooperative scenario in which traditional SA operates at each BS independently, and ii) cooperative SA-MBS in which centralized SIC-based decoding is applied over all received user signals. For both scenarios, we provide upper system throughput limits and compare them against the simulation results. Finally, we discuss the system performance as a function of simple directional antenna model parameters applied in this paper.

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