Search

The rapid growth of UAV applications necessitates a robust communication and networking architecture capable of addressing the diverse requirements of various applications concurrently, rather than relying on application-specific solutions. This paper proposes a generic and reliable multi-UAV communication and networking architecture designed to support the varying demands of heterogeneous applications, including short-range and long-range communication, star and mesh topologies, different data rates, and multiple wireless standards. Our architecture accommodates both adhoc and infrastructure networks, ensuring seamless connectivity throughout the network. Additionally, we present the design of a multi-protocol UAV gateway that enables interoperability among various communication protocols. Furthermore, we introduce a data processing and service layer framework with a graphical user interface of a ground control station that facilitates remote control and monitoring from any location at any time. We practically implemented the proposed architecture and evaluated its performance using different metrics, demonstrating its effectiveness., Comment: 11 pages, 20 figures, Journal paper

This paper presents a communication and energy-aware Multi-UAV Coverage Path Planning (mCPP) method for scenarios requiring continuous inter-UAV communication, such as cooperative search and rescue and surveillance missions. Unlike existing mCPP solutions that focus on energy, time, or coverage efficiency, our approach generates coverage paths that require minimal the communication range to maintain inter-UAV connectivity while also optimizing energy consumption. The mCPP problem is formulated as a multi-objective optimization task, aiming to minimize both the communication range requirement and energy consumption. Our approach significantly reduces the communication range needed for maintaining connectivity while ensuring energy efficiency, outperforming state-of-the-art methods. Its effectiveness is validated through simulations on complex and arbitrary shaped regions of interests, including scenarios with no-fly zones. Additionally, real-world experiment demonstrate its high accuracy, achieving 99\% consistency between the estimated and actual communication range required during a multi-UAV coverage mission involving three UAVs.

This work considers the decentralized successive convex approximation (SCA) method for minimizing stochastic non-convex objectives subject to convex constraints, along with possibly non-smooth convex regularizers. Although SCA has been widely applied in decentralized settings, its stochastic first order (SFO) complexity is unknown, and it is thought to be slower than the centralized momentum-enhanced SCA variants. In this work, we advance the state-of-the-art for SCA methods by proposing an accelerated variant, namely the \textbf{D}ecentralized \textbf{M}omentum-based \textbf{S}tochastic \textbf{SCA} (\textbf{D-MSSCA}) and analyze its SFO complexity. The proposed algorithm entails creating a stochastic surrogate of the objective at every iteration, which is minimized at each node separately. Remarkably, the D-MSSCA achieves an SFO complexity of $\mathcal{O}(\epsilon^{-3/2})$ to reach an $\epsilon$-stationary point, which is at par with the SFO complexity lower bound for unconstrained stochastic non-convex optimization in centralized setting.

We consider stochastic optimization problems with functional constraints, such as those arising in trajectory generation, sparse approximation, and robust classification. To this end, we put forth a recursive momentum-based accelerated successive convex approximation (SCA) algorithm. At each iteration, the proposed algorithm entails constructing convex surrogates of the stochastic objective and the constraint functions, and solving the resulting convex optimization problem. A recursive update rule is employed to track the gradient of the stochastic objective function, which contributes to variance reduction and hence accelerates the algorithm convergence. A key ingredient of the proof is a new parameterized version of the standard Mangasarian-Fromowitz Constraints Qualification, that allows us to bound the dual variables and hence obtain problem-dependent bounds on the rate at which the iterates approach an $\epsilon$-stationary point. Remarkably, the proposed algorithm achieves near-optimal stochastic first order (SFO) complexity, almost at par with that achieved by state-of-the-art stochastic optimization algorithms for solving unconstrained problems. As an example, we detail a obstacle-avoiding trajectory optimization problem that can be solved using the proposed algorithm and show that its performance is superior to that of the existing algorithms used for trajectory optimization. The performance of the proposed algorithm is also shown to be comparable to that of a specialized sparse classification algorithm applied to a binary classification problem., Comment: 32 pages, 4 figures, journal submission

This work considers the problem of decentralized online learning, where the goal is to track the optimum of the sum of time-varying functions, distributed across several nodes in a network. The local availability of the functions and their gradients necessitates coordination and consensus among the nodes. We put forth the Generalized Gradient Tracking (GGT) framework that unifies a number of existing approaches, including the state-of-the-art ones. The performance of the proposed GGT algorithm is theoretically analyzed using a novel semidefinite programming-based analysis that yields the desired regret bounds under very general conditions and without requiring the gradient boundedness assumption. The results are applicable to the special cases of GGT, which include various state-of-the-art algorithms as well as new dynamic versions of various classical decentralized algorithms. To further minimize the regret, we consider a condensed version of GGT with only four free parameters. A procedure for offline tuning of these parameters using only the problem parameters is also detailed. The resulting optimized GGT (oGGT) algorithm not only achieves improved dynamic regret bounds, but also outperforms all state-of-the-art algorithms on both synthetic and real-world datasets., Comment: 30 pages, 6 Figures

The problem of minimizing the sum of $n$ functions in $d$ dimensions is ubiquitous in machine learning and statistics. In many applications where the number of observations $n$ is large, it is necessary to use incremental or stochastic methods, as their per-iteration cost is independent of $n$. Of these, Quasi-Newton (QN) methods strike a balance between the per-iteration cost and the convergence rate. Specifically, they exhibit a superlinear rate with $O(d^2)$ cost in contrast to the linear rate of first-order methods with $O(d)$ cost and the quadratic rate of second-order methods with $O(d^3)$ cost. However, existing incremental methods have notable shortcomings: Incremental Quasi-Newton (IQN) only exhibits asymptotic superlinear convergence. In contrast, Incremental Greedy BFGS (IGS) offers explicit superlinear convergence but suffers from poor empirical performance and has a per-iteration cost of $O(d^3)$. To address these issues, we introduce the Sharpened Lazy Incremental Quasi-Newton Method (SLIQN) that achieves the best of both worlds: an explicit superlinear convergence rate, and superior empirical performance at a per-iteration $O(d^2)$ cost. SLIQN features two key changes: first, it incorporates a hybrid strategy of using both classic and greedy BFGS updates, allowing it to empirically outperform both IQN and IGS. Second, it employs a clever constant multiplicative factor along with a lazy propagation strategy, which enables it to have a cost of $O(d^2)$. Additionally, our experiments demonstrate the superiority of SLIQN over other incremental and stochastic Quasi-Newton variants and establish its competitiveness with second-order incremental methods., Comment: 36 pages, 2 figures; Accepted to AISTATS 2024

This work puts forth low-complexity Riemannian subspace descent algorithms for the minimization of functions over the symmetric positive definite (SPD) manifold. Different from the existing Riemannian gradient descent variants, the proposed approach utilizes carefully chosen subspaces that allow the update to be written as a product of the Cholesky factor of the iterate and a sparse matrix. The resulting updates avoid the costly matrix operations like matrix exponentiation and dense matrix multiplication, which are generally required in almost all other Riemannian optimization algorithms on SPD manifold. We further identify a broad class of functions, arising in diverse applications, such as kernel matrix learning, covariance estimation of Gaussian distributions, maximum likelihood parameter estimation of elliptically contoured distributions, and parameter estimation in Gaussian mixture model problems, over which the Riemannian gradients can be calculated efficiently. The proposed uni-directional and multi-directional Riemannian subspace descent variants incur per-iteration complexities of $O(n)$ and $O(n^2)$ respectively, as compared to the $O(n^3)$ or higher complexity incurred by all existing Riemannian gradient descent variants. The superior runtime and low per-iteration complexity of the proposed algorithms is also demonstrated via numerical tests on large-scale covariance estimation and matrix square root problems. MATLAB code implementation is publicly available on GitHub : https://github.com/yogeshd-iitk/subspace_descent_over_SPD_manifold

We consider a smoothed online convex optimization (SOCO) problem with predictions, where the learner has access to a finite lookahead window of time-varying stage costs, but suffers a switching cost for changing its actions at each stage. Based on the Alternating Proximal Gradient Descent (APGD) framework, we develop Receding Horizon Alternating Proximal Descent (RHAPD) for proximable, non-smooth and strongly convex stage costs, and RHAPD-Smooth (RHAPD-S) for non-proximable, smooth and strongly convex stage costs. In addition to outperforming gradient descent-based algorithms, while maintaining a comparable runtime complexity, our proposed algorithms also allow us to solve a wider range of problems. We provide theoretical upper bounds on the dynamic regret achieved by the proposed algorithms, which decay exponentially with the length of the lookahead window. The performance of the presented algorithms is empirically demonstrated via numerical experiments on non-smooth regression, dynamic trajectory tracking, and {economic power dispatch} problems., Comment: 29 pages, 13 figures

Newton-type methods are popular in federated learning due to their fast convergence. Still, they suffer from two main issues, namely: low communication efficiency and low privacy due to the requirement of sending Hessian information from clients to parameter server (PS). In this work, we introduced a novel framework called FedNew in which there is no need to transmit Hessian information from clients to PS, hence resolving the bottleneck to improve communication efficiency. In addition, FedNew hides the gradient information and results in a privacy-preserving approach compared to the existing state-of-the-art. The core novel idea in FedNew is to introduce a two level framework, and alternate between updating the inverse Hessian-gradient product using only one alternating direction method of multipliers (ADMM) step and then performing the global model update using Newton's method. Though only one ADMM pass is used to approximate the inverse Hessian-gradient product at each iteration, we develop a novel theoretical approach to show the converging behavior of FedNew for convex problems. Additionally, a significant reduction in communication overhead is achieved by utilizing stochastic quantization. Numerical results using real datasets show the superiority of FedNew compared to existing methods in terms of communication costs.

Non-asymptotic analysis of quasi-Newton methods have gained traction recently. In particular, several works have established a non-asymptotic superlinear rate of $\mathcal{O}((1/\sqrt{t})^t)$ for the (classic) BFGS method by exploiting the fact that its error of Newton direction approximation approaches zero. Moreover, a greedy variant of BFGS was recently proposed which accelerates its convergence by directly approximating the Hessian, instead of the Newton direction, and achieves a fast local quadratic convergence rate. Alas, the local quadratic convergence of Greedy-BFGS requires way more updates compared to the number of iterations that BFGS requires for a local superlinear rate. This is due to the fact that in Greedy-BFGS the Hessian is directly approximated and the Newton direction approximation may not be as accurate as the one for BFGS. In this paper, we close this gap and present a novel BFGS method that has the best of both worlds in that it leverages the approximation ideas of both BFGS and Greedy-BFGS to properly approximate the Newton direction and the Hessian matrix simultaneously. Our theoretical results show that our method out-performs both BFGS and Greedy-BFGS in terms of convergence rate, while it reaches its quadratic convergence rate with fewer steps compared to Greedy-BFGS. Numerical experiments on various datasets also confirm our theoretical findings.

Missing data is a common problem in real-world sensor data collection. The performance of various approaches to impute data degrade rapidly in the extreme scenarios of low data sampling and noisy sampling, a case present in many real-world problems in the field of traffic sensing and environment monitoring, etc. However, jointly exploiting the spatiotemporal and periodic structure, which is generally not captured by classical matrix completion approaches, can improve the imputation performance of sensor data in such real-world conditions. We present a Bayesian approach towards spatiotemporal matrix completion wherein we estimate the underlying temporarily varying subspace using a Variational Bayesian technique. We jointly couple the low-rank matrix completion with the state space autoregressive framework along with a penalty function on the slowly varying subspace to model the temporal and periodic evolution in the data. A major advantage of our method is that a critical parameter like the rank of the model is automatically tuned using the automatic relevance determination (ARD) approach, unlike most matrix/tensor completion techniques. We also propose a robust version of the above formulation, which improves the performance of imputation in the presence of outliers. We evaluate the proposed Variational Bayesian Filtering with Subspace Information (VBFSI) method to impute matrices in real-world traffic and air pollution data. Simulation results demonstrate that the proposed method outperforms the recent state-of-the-art methods and provides a sufficiently accurate imputation for different sampling rates. In particular, we demonstrate that fusing the subspace evolution over days can improve the imputation performance with even 15% of the data sampling.

This work presents the first projection-free algorithm to solve stochastic bi-level optimization problems, where the objective function depends on the solution of another stochastic optimization problem. The proposed $\textbf{S}$tochastic $\textbf{Bi}$-level $\textbf{F}$rank-$\textbf{W}$olfe ($\textbf{SBFW}$) algorithm can be applied to streaming settings and does not make use of large batches or checkpoints. The sample complexity of SBFW is shown to be $\mathcal{O}(\epsilon^{-3})$ for convex objectives and $\mathcal{O}(\epsilon^{-4})$ for non-convex objectives. Improved rates are derived for the stochastic compositional problem, which is a special case of the bi-level problem, and entails minimizing the composition of two expected-value functions. The proposed $\textbf{S}$tochastic $\textbf{C}$ompositional $\textbf{F}$rank-$\textbf{W}$olfe ($\textbf{SCFW}$) is shown to achieve a sample complexity of $\mathcal{O}(\epsilon^{-2})$ for convex objectives and $\mathcal{O}(\epsilon^{-3})$ for non-convex objectives, at par with the state-of-the-art sample complexities for projection-free algorithms solving single-level problems. We demonstrate the advantage of the proposed methods by solving the problem of matrix completion with denoising and the problem of policy value evaluation in reinforcement learning., Comment: 34 Pages

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.

Federated Learning (FL) refers to the paradigm where multiple worker nodes (WNs) build a joint model by using local data. Despite extensive research, for a generic non-convex FL problem, it is not clear, how to choose the WNs' and the server's update directions, the minibatch sizes, and the local update frequency, so that the WNs use the minimum number of samples and communication rounds to achieve the desired solution. This work addresses the above question and considers a class of stochastic algorithms where the WNs perform a few local updates before communication. We show that when both the WN's and the server's directions are chosen based on a stochastic momentum estimator, the algorithm requires $\tilde{\mathcal{O}}(\epsilon^{-3/2})$ samples and $\tilde{\mathcal{O}}(\epsilon^{-1})$ communication rounds to compute an $\epsilon$-stationary solution. To the best of our knowledge, this is the first FL algorithm that achieves such {\it near-optimal} sample and communication complexities simultaneously. Further, we show that there is a trade-off curve between local update frequencies and local minibatch sizes, on which the above sample and communication complexities can be maintained. Finally, we show that for the classical FedAvg (a.k.a. Local SGD, which is a momentum-less special case of the STEM), a similar trade-off curve exists, albeit with worse sample and communication complexities. Our insights on this trade-off provides guidelines for choosing the four important design elements for FL algorithms, the update frequency, directions, and minibatch sizes to achieve the best performance.

Cloud computing facilitates the access of applications and data from any location by a distributed storage system. Erasure codes offer better data replication technique with reduced storage costs for more reliability. This paper considers the erasure-coded data center with multiple servers in a wireless network where each is equipped with a base-station. The cause of latency in the file retrieval process is mainly due to queuing delays at each server. This work puts forth a stochastic optimization framework for obtaining the optimal scheduling policy that maximizes users' quality of service (QoS) while adhering to the latency requirements. We further show that the problem has non-linear functions of expectations in objective and constraints and is impossible to solve with traditional SGD like algorithms. We propose a new algorithm that addresses compositional structure in the problem. Further, we show that the proposed algorithm achieves a faster convergence rate than the best-known results. Finally, we test the efficacy of the proposed method in a simulated environment., Comment: This work is accepted for publication in proceedings of IEEE ICC 2021

This work studies constrained stochastic optimization problems where the objective and constraint functions are convex and expressed as compositions of stochastic functions. The problem arises in the context of fair classification, fair regression, and the design of queuing systems. Of particular interest is the large-scale setting where an oracle provides the stochastic gradients of the constituent functions, and the goal is to solve the problem with a minimal number of calls to the oracle. Owing to the compositional form, the stochastic gradients provided by the oracle do not yield unbiased estimates of the objective or constraint gradients. Instead, we construct approximate gradients by tracking the inner function evaluations, resulting in a quasi-gradient saddle point algorithm. We prove that the proposed algorithm is guaranteed to find the optimal and feasible solution almost surely. We further establish that the proposed algorithm requires $\mathcal{O}(1/\epsilon^4)$ data samples in order to obtain an $\epsilon$-approximate optimal point while also ensuring zero constraint violation. The result matches the sample complexity of the stochastic compositional gradient descent method for unconstrained problems and improves upon the best-known sample complexity results for the constrained settings. The efficacy of the proposed algorithm is tested on both fair classification and fair regression problems. The numerical results show that the proposed algorithm outperforms the state-of-the-art algorithms in terms of the convergence rate., Comment: A part of this work is submitted in Asilomar Conference on Signals, Systems, and Computers

We consider the decentralized convex optimization problem, where multiple agents must cooperatively minimize a cumulative objective function, with each local function expressible as an empirical average of data-dependent losses. State-of-the-art approaches for decentralized optimization rely on gradient tracking, where consensus is enforced via a doubly stochastic mixing matrix. Construction of such mixing matrices is not straightforward and requires coordination even prior to the start of the optimization algorithm. This paper puts forth a primal-dual framework for decentralized stochastic optimization that obviates the need for such doubly stochastic matrices. Instead, dual variables are maintained to track the disagreement between neighbors. The proposed framework is flexible and is used to develop decentralized variants of SAGA, L-SVRG, SVRG++, and SEGA algorithms. Using a unified proof, we establish that the oracle complexity of these decentralized variants is $O(1/\epsilon)$, matching the complexity bounds obtained for the centralized variants. Additionally, we also present a decentralized primal-dual accelerated SVRG algorithm achieving $O(1/\sqrt{\epsilon})$ oracle complexity, again matching the bound for the centralized accelerated SVRG. Numerical tests on the algorithms establish their superior performance as compared to the variance-reduced gradient tracking algorithms., Comment: 31 pages, 6 Figures

This paper discusses distributed optimization over a directed graph. We begin with some well known algorithms which achieve consensus among agents including FROST [1], which possesses the quickest convergence to the optimum. It is a well known fact FROST has a linear convergence. However FROST works only over fixed topology of underlying network. Moreover the updates proposed therein require perfectly synchronized communication among nodes. Hence communication delays among nodes, which are inevitable in a realistic scenario, preclude the possibility of implementing FROST in real time. In this paper we introduce a co-operative control strategy which makes convergence to optimum robust to communication delays.

We consider expected risk minimization problems when the range of the estimator is required to be nonnegative, motivated by the settings of maximum likelihood estimation (MLE) and trajectory optimization. To facilitate nonlinear interpolation, we hypothesize that the search space is a Reproducing Kernel Hilbert Space (RKHS). We develop first and second-order variants of stochastic mirror descent employing (i) \emph{pseudo-gradients} and (ii) complexity-reducing projections. Compressive projection in the first-order scheme is executed via kernel orthogonal matching pursuit (KOMP), which overcomes the fact that the vanilla RKHS parameterization grows unbounded with the iteration index in the stochastic setting. Moreover, pseudo-gradients are needed when gradient estimates for cost are only computable up to some numerical error, which arise in, e.g., integral approximations. Under constant step-size and compression budget, we establish tradeoffs between the radius of convergence of the expected sub-optimality and the projection budget parameter, as well as non-asymptotic bounds on the model complexity. To refine the solution's precision, we develop a second-order extension which employs recursively averaged pseudo-gradient outer-products to approximate the Hessian inverse, whose convergence in mean is established under an additional eigenvalue decay condition on the Hessian of the optimal RKHS element, which is unique to this work. Experiments demonstrate favorable performance on inhomogeneous Poisson Process intensity estimation in practice.

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. Such extra-gradient information is not used by SGD, but has been shown to be useful, for instance in the context of stochastic expectation-maximization, stochastic majorization-minimization, and stochastic successive convex approximation (SCA) approaches. By constructing a stochastic strongly convex surrogates of the loss function at every iteration, the stochastic SCA algorithms can exploit the structural properties of the loss function and achieve superior empirical performance as compared to the SGD. 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 surrogates 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.

This paper considers stochastic convex optimization problems where the objective and constraint functions involve expectations with respect to the data indices or environmental variables, in addition to deterministic convex constraints on the domain of the variables. Although the setting is generic and arises in different machine learning applications, online and efficient approaches for solving such problems have not been widely studied. Since the underlying data distribution is unknown a priori, a closed-form solution is generally not available, and classical deterministic optimization paradigms are not applicable. State-of-the-art approaches, such as those using the saddle point framework, can ensure that the optimality gap as well as the constraint violation decay as $\O\left(T^{-\frac{1}{2}}\right)$ where $T$ is the number of stochastic gradients. The domain constraints are assumed simple and handled via projection at every iteration. In this work, we propose a novel conservative stochastic optimization algorithm (CSOA) that achieves zero constraint violation and $\O\left(T^{-\frac{1}{2}}\right)$ optimality gap. Further, the projection operation (for scenarios when calculating projection is expensive) in the proposed algorithm can be avoided by considering the conditional gradient or Frank-Wolfe (FW) variant of the algorithm. The state-of-the-art stochastic FW variants achieve an optimality gap of $\O\left(T^{-\frac{1}{3}}\right)$ after $T$ iterations, though these algorithms have not been applied to problems with functional expectation constraints. In this work, we propose the FW-CSOA algorithm that is not only projection-free but also achieves zero constraint violation with $\O\left(T^{-\frac{1}{4}}\right)$ decay of the optimality gap. The efficacy of the proposed algorithms is tested on two relevant problems: fair classification and structured matrix completion.

The importance of content delivery networks (CDN) continues to rise with the exponential increase in the generation and consumption of electronic media. In order to ensure a high quality of experience, CDNs often deploy cache servers that are capable of storing some of the popular files close to the user. Such edge caching solutions not only increase the content availability, but also result in higher download rates and lower latency at the user. We consider the problem of content placement from an optimization perspective. Different from the classical eviction-based algorithms, the present work formulates the content placement problem from an optimization perspective and puts forth an online algorithm for the same. In contrast to the existing optimization-based solutions, the proposed algorithm is incremental and incurs very low computation cost, while yielding storage allocations that are provably near-optimal. The proposed algorithm can handle time varying content popularity, thereby obviating the need for periodically estimating demand distribution. Using synthetic and real IPTV data, we show that the proposed policies outperform all the state of art caching techniques in terms of various metrics.

In this work, we propose a distributed algorithm for stochastic non-convex optimization. We consider a worker-server architecture where a set of $K$ worker nodes (WNs) in collaboration with a server node (SN) jointly aim to minimize a global, potentially non-convex objective function. The objective function is assumed to be the sum of local objective functions available at each WN, with each node having access to only the stochastic samples of its local objective function. In contrast to the existing approaches, we employ a momentum based "single loop" distributed algorithm which eliminates the need of computing large batch size gradients to achieve variance reduction. We propose two algorithms one with "adaptive" and the other with "non-adaptive" learning rates. We show that the proposed algorithms achieve the optimal computational complexity while attaining linear speedup with the number of WNs. Specifically, the algorithms reach an $\epsilon$-stationary point $x_a$ with $\mathbb{E}\| \nabla f(x_a) \| \leq \tilde{O}(K^{-1/3}T^{-1/2} + K^{-1/3}T^{-1/3})$ in $T$ iterations, thereby requiring $\tilde{O}(K^{-1} \epsilon^{-3})$ gradient computations at each WN. Moreover, our approach does not assume identical data distributions across WNs making the approach general enough for federated learning applications.

Gaussian processes provide a framework for nonlinear nonparametric Bayesian inference widely applicable across science and engineering. Unfortunately, their computational burden scales cubically with the training sample size, which in the case that samples arrive in perpetuity, approaches infinity. This issue necessitates approximations for use with streaming data, which to date mostly lack convergence guarantees. Thus, we develop the first online Gaussian process approximation that preserves convergence to the population posterior, i.e., asymptotic posterior consistency, while ameliorating its intractable complexity growth with the sample size. We propose an online compression scheme that, following each a posteriori update, fixes an error neighborhood with respect to the Hellinger metric centered at the current posterior, and greedily tosses out past kernel dictionary elements until its boundary is hit. We call the resulting method Parsimonious Online Gaussian Processes (POG). For diminishing error radius, exact asymptotic consistency is preserved (Theorem 1(i)) at the cost of unbounded memory in the limit. On the other hand, for constant error radius, POG converges to a neighborhood of the population posterior (Theorem 1(ii))but with finite memory at-worst determined by the metric entropy of the feature space (Theorem 2). Experimental results are presented on several nonlinear regression problems which illuminates the merits of this approach as compared with alternatives that fix the subspace dimension defining the history of past points.

Robotic calibration allows for the fusion of data from multiple sensors such as odometers, cameras, etc., by providing appropriate transformational relationships between the corresponding reference frames. For wheeled robots equipped with exteroceptive sensors, calibration entails learning the motion model of the sensor or the robot in terms of the odometric data, and must generally be performed prior to performing tasks such as simultaneous localization and mapping (SLAM). Within this context, the current trend is to carry out simultaneous calibration of odometry and sensor without the use of any additional hardware. Building upon the existing simultaneous calibration algorithms, we put forth a generalized calibration framework that can not only handle robots operating in 2D with arbitrary or unknown motion models but also handle outliers in an automated manner. We first propose an algorithm based on the alternating minimization framework applicable to two-wheel differential drive. Subsequently, for arbitrary but known drive configurations we put forth an iteratively re-weighted least squares methodology leveraging an intelligent weighing scheme. Different from the existing works, these proposed algorithms require no manual intervention and seamlessly handle outliers that arise due to both systematic and non-systematic errors. Finally, we put forward a novel Gaussian Process-based non-parametric approach for calibrating wheeled robots with arbitrary or unknown drive configurations. Detailed experiments are performed to demonstrate the accuracy, usefulness, and flexibility of the proposed algorithms., Comment: This manuscript has been submitted to 'Elsevier Journal of Robotics and Autonomous Systems' and is under review for possible publication. Based on IROS 2019 conference submission [arXiv:1910.11917]

This paper considers the problem of online trajectory design under time-varying environments. We formulate the general trajectory optimization problem within the framework of time-varying constrained convex optimization and proposed a novel version of the online gradient ascent algorithm for such problems. Moreover, the gradient feedback is noisy and allows us to use the proposed algorithm for a range of practical applications where it is difficult to acquire the true gradient. In contrast to the most available literature, we present the offline sublinear regret of the proposed algorithm up to the path length variations of the optimal offline solution, the cumulative gradient, and the error in the gradient variations. Furthermore, we establish a lower bound on the offline dynamic regret, which defines the optimality of any trajectory. To show the efficacy of the proposed algorithm, we consider two practical problems of interest. First, we consider a device to device (D2D) communications setting, and the goal is to design a user trajectory while maximizing its connectivity to the internet. The second problem is associated with the online planning of energy-efficient trajectories for unmanned surface vehicles (USV) under strong disturbances in ocean environments with both static and dynamic goal locations. The detailed simulation results demonstrate the significance of the proposed algorithm on synthetic and real data sets. Video on the real-world datasets can be found at {https://www.youtube.com/watch?v=FcRqqWtpf\_0}, Comment: arXiv admin note: text overlap with arXiv:1804.04860

In this paper, we propose a distributed algorithm for stochastic smooth, non-convex optimization. We assume a worker-server architecture where $N$ nodes, each having $n$ (potentially infinite) number of samples, collaborate with the help of a central server to perform the optimization task. The global objective is to minimize the average of local cost functions available at individual nodes. The proposed approach is a non-trivial extension of the popular parallel-restarted SGD algorithm, incorporating the optimal variance-reduction based SPIDER gradient estimator into it. We prove convergence of our algorithm to a first-order stationary solution. The proposed approach achieves the best known communication complexity $O(\epsilon^{-1})$ along with the optimal computation complexity. For finite-sum problems (finite $n$), we achieve the optimal computation (IFO) complexity $O(\sqrt{Nn}\epsilon^{-1})$. For online problems ($n$ unknown or infinite), we achieve the optimal IFO complexity $O(\epsilon^{-3/2})$. In both the cases, we maintain the linear speedup achieved by existing methods. This is a massive improvement over the $O(\epsilon^{-2})$ IFO complexity of the existing approaches. Additionally, our algorithm is general enough to allow non-identical distributions of data across workers, as in the recently proposed federated learning paradigm.

Robotic calibration allows for the fusion of data from multiple sensors such as odometers, cameras, etc., by providing appropriate relationships between the corresponding reference frames. For wheeled robots equipped with camera/lidar along with wheel encoders, calibration entails learning the motion model of the sensor or the robot in terms of the data from the encoders and generally carried out before performing tasks such as simultaneous localization and mapping (SLAM). This work puts forward a novel Gaussian Process-based non-parametric approach for calibrating wheeled robots with arbitrary or unknown drive configurations. The procedure is more general as it learns the entire sensor/robot motion model in terms of odometry measurements. Different from existing non-parametric approaches, our method relies on measurements from the onboard sensors and hence does not require the ground truth information from external motion capture systems. Alternatively, we propose a computationally efficient approach that relies on the linear approximation of the sensor motion model. Finally, we perform experiments to calibrate robots with un-modelled effects to demonstrate the accuracy, usefulness, and flexibility of the proposed approach., Comment: To be published in International Conference on Intelligent Robots and Systems (IROS), 2019

Batch training of machine learning models based on neural networks is now well established, whereas to date streaming methods are largely based on linear models. To go beyond linear in the online setting, nonparametric methods are of interest due to their universality and ability to stably incorporate new information via convexity or Bayes' Rule. Unfortunately, when used online, nonparametric methods suffer a "curse of dimensionality" which precludes their use: their complexity scales at least with the time index. We survey online compression tools which bring their memory under control and attain approximate convergence. The asymptotic bias depends on a compression parameter that trades off memory and accuracy. Further, the applications to robotics, communications, economics, and power are discussed, as well as extensions to multi-agent systems.

An open challenge in supervised learning is \emph{conceptual drift}: a data point begins as classified according to one label, but over time the notion of that label changes. Beyond linear autoregressive models, transfer and meta learning address drift, but require data that is representative of disparate domains at the outset of training. To relax this requirement, we propose a memory-efficient \emph{online} universal function approximator based on compressed kernel methods. Our approach hinges upon viewing non-stationary learning as online convex optimization with dynamic comparators, for which performance is quantified by dynamic regret. Prior works control dynamic regret growth only for linear models. In contrast, we hypothesize actions belong to reproducing kernel Hilbert spaces (RKHS). We propose a functional variant of online gradient descent (OGD) operating in tandem with greedy subspace projections. Projections are necessary to surmount the fact that RKHS functions have complexity proportional to time. For this scheme, we establish sublinear dynamic regret growth in terms of both loss variation and functional path length, and that the memory of the function sequence remains moderate. Experiments demonstrate the usefulness of the proposed technique for online nonlinear regression and classification problems with non-stationary data.

We consider learning in decentralized heterogeneous networks: agents seek to minimize a convex functional that aggregates data across the network, while only having access to their local data streams. We focus on the case where agents seek to estimate a regression \emph{function} that belongs to a reproducing kernel Hilbert space (RKHS). To incentivize coordination while respecting network heterogeneity, we impose nonlinear proximity constraints. To solve the constrained stochastic program, we propose applying a functional variant of stochastic primal-dual (Arrow-Hurwicz) method which yields a decentralized algorithm. To handle the fact that agents' functions have complexity proportional to time (owing to the RKHS parameterization), we project the primal iterates onto subspaces greedily constructed from kernel evaluations of agents' local observations. The resulting scheme, dubbed Heterogeneous Adaptive Learning with Kernels (HALK), when used with constant step-sizes, yields $\mathcal{O}(\sqrt{T})$ attenuation in sub-optimality and exactly satisfies the constraints in the long run, which improves upon the state of the art rates for vector-valued problems.

In this two-part work, we propose an algorithmic framework for solving non-convex problems whose objective function is the sum of a number of smooth component functions plus a convex (possibly non-smooth) or/and smooth (possibly non-convex) regularization function. The proposed algorithm incorporates ideas from several existing approaches such as alternate direction method of multipliers (ADMM), successive convex approximation (SCA), distributed and asynchronous algorithms, and inexact gradient methods. Different from a number of existing approaches, however, the proposed framework is flexible enough to incorporate a class of non-convex objective functions, allow distributed operation with and without a fusion center, and include variance reduced methods as special cases. Remarkably, the proposed algorithms are robust to uncertainties arising from random, deterministic, and adversarial sources. The part I of the paper develops two variants of the algorithm under very mild assumptions and establishes first-order convergence rate guarantees. The proof developed here allows for generic errors and delays, paving the way for different variance-reduced, asynchronous, and stochastic implementations, outlined and evaluated in part II.

Well-designed queuing systems form the backbone of modern communications, distributed computing, and content delivery architectures. Designs balancing infrastructure costs and user experience indices require tools from teletraffic theory and operations research. A standard approach to designing such systems involves formulating optimization problems that strive to maximize the pertinent utility functions while adhering to quality-of-service and other physical constraints. In many cases, formulating such problems necessitates making simplistic assumptions on arrival and departure processes to keep the problem simple. This work puts forth a stochastic optimization framework for designing queuing systems where the exogenous processes may have arbitrary and unknown distributions. We show that many such queuing design problems can generally be formulated as stochastic optimization problems where the objective and constraint are non-linear functions of expectations. The compositional structure obviates the use of classical stochastic approximation approaches where the stochastic gradients are often required to be unbiased. To this end, a constrained stochastic compositional gradient descent algorithm is proposed that utilizes a tracking step for the expected value functions. The non-asymptotic performance of the proposed algorithm is characterized via its iteration complexity. Numerical tests allow us to validate the theoretical results and demonstrate the efficacy of the proposed algorithm., Comment: This work has been submitted in "IEEE Transactions on Communication (TCOM)"

Matrix completion and robust principal component analysis have been widely used for the recovery of data suffering from missing entries or outliers. In many real-world applications however, the data is also time-varying, and the naive approach of per-snapshot recovery is both expensive and sub-optimal. This paper develops generative Bayesian models that fit sequential multivariate measurements arising from a low-dimensional time-varying subspace. A variational Bayesian subspace filtering approach is proposed that learns the underlying subspace and its state-transition matrix. Different from the plethora of deterministic counterparts, the proposed approach utilizes automatic relevance determination priors that obviate the need to tune key parameters such as rank and noise power. We also propose a forward-backward algorithm that allows the updates to be carried out at low complexity. Extensive tests over traffic and electricity data demonstrate the superior imputation, outlier rejection, and temporal prediction prowess of the proposed algorithm over the state-of-the-art matrix/tensor completion algorithms., Comment: 13 Pages

We consider the problem of tracking the minimum of a time-varying convex optimization problem over a dynamic graph. Motivated by target tracking and parameter estimation problems in intermittently connected robotic and sensor networks, the goal is to design a distributed algorithm capable of handling non-differentiable regularization penalties. The proposed proximal online gradient descent algorithm is built to run in a fully decentralized manner and utilizes consensus updates over possibly disconnected graphs. The performance of the proposed algorithm is analyzed by developing bounds on its dynamic regret in terms of the cumulative path length of the time-varying optimum. It is shown that as compared to the centralized case, the dynamic regret incurred by the proposed algorithm over $T$ time slots is worse by a factor of $\log(T)$ only, despite the disconnected and time-varying network topology. The empirical performance of the proposed algorithm is tested on the distributed dynamic sparse recovery problem, where it is shown to incur a dynamic regret that is close to that of the centralized algorithm.

Optimizing non-convex functions is of primary importance in the vast majority of machine learning algorithms. Even though many gradient descent based algorithms have been studied, successive convex approximation based algorithms have been recently empirically shown to converge faster. However, such successive convex approximation based algorithms can get stuck in a first-order stationary point. To avoid that, we propose an algorithm that perturbs the optimization variable slightly at the appropriate iteration. In addition to achieving the same convergence rate results as the non-perturbed version, we show that the proposed algorithm converges to a second order stationary point. Thus, the proposed algorithm escapes the saddle point efficiently and does not get stuck at the first order saddle points.

In this work, we address optimization problems where the objective function is a nonlinear function of an expected value, i.e., compositional stochastic {strongly convex programs}. We consider the case where the decision variable is not vector-valued but instead belongs to a reproducing Kernel Hilbert Space (RKHS), motivated by risk-aware formulations of supervised learning and Markov Decision Processes defined over continuous spaces. We develop the first memory-efficient stochastic algorithm for this setting, which we call Compositional Online Learning with Kernels (COLK). COLK, at its core a two-time-scale stochastic approximation method, addresses the fact that (i) compositions of expected value problems cannot be addressed by classical stochastic gradient due to the presence of the inner expectation; and (ii) the RKHS-induced parameterization has complexity which is proportional to the iteration index which is mitigated through greedily constructed subspace projections. We establish almost sure convergence of COLK with attenuating step-sizes, and linear convergence in mean to a neighborhood with constant step-sizes, as well as the fact that its complexity is at-worst finite. The experiments with robust formulations of supervised learning demonstrate that COLK reliably converges, attains consistent performance across training runs, and thus overcomes overfitting.

Traffic assignment is an integral part of urban city planning. Roads and freeways are constructed to cater to the expected demands of the commuters between different origin-destination pairs with the overall objective of minimising the travel cost. As compared to static traffic assignment problems where the traffic network is fixed over time, a dynamic traffic network is more realistic where the network's cost parameters change over time due to the presence of random congestion. In this paper, we consider a stochastic version of the traffic assignment problem where the central planner is interested in finding an optimal social flow in the presence of random users. These users are random and cannot be controlled by any central directives. We propose a Frank-Wolfe algorithm based stochastic algorithm to determine the socially optimal flow for the stochastic setting in an online manner. Further, simulation results corroborate the efficacy of the proposed algorithm.

This paper considers a single cell multi-antenna base station delivering content to multiple cache enabled single-antenna users. Coding strategies are developed that allow for decentralized placement in the wireless setting. Three different cases namely, max-min multicasting, linear combinations in the complex field, and linear combinations in the finite field, are considered and closed-form rate expressions are provided that hold with high probability. For the case of max-min fair multicasting delivery, we propose a new coding scheme that is capable of working with only two-user broadcasts. A cyclic-exchange protocol for efficient content delivery is proposed and shown to perform almost as well as the original multi-user broadcast scheme., Comment: Accepted in 56th Annual Allerton Conference 2018 on Communication, Control, and Computing at UIUC, IL,USA

We consider non-differentiable 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 non-differentiable 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 sub-sampled. 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.

This paper considers the problem of designing the user trajectory in a device-to-device communications setting. We consider a pair of pedestrians connected through a D2D link. The pedestrians seek to reach their respective destinations while using the D2D link for data exchange applications such as file transfer, video calling, and online gaming. In order to enable better D2D connectivity, the pedestrians are willing to deviate from their respective shortest paths, at the cost of reaching their destinations slightly late. A generic trajectory optimization problem is formulated and solved for the case when full information about the problem in known in advance. Motivated by the D2D user's need to keep their destinations private, we also formulate a regularized variant of the problem that can be used to develop a fully online algorithm. The proposed online algorithm is quite efficient, and is shown to achieve a sublinear \emph{offline} regret while satisfying the required mobility constraints exactly. The theoretical results are backed by detailed numerical tests that establish the efficacy of the proposed algorithms under various settings., Comment: To appear in IEEE INFOCOM 2018, April 2018, Honolulu HI

Multi-agent systems are being increasingly deployed in challenging environments for performing complex tasks such as multi-target tracking, search-and-rescue, and intrusion detection. Notwithstanding 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 non-strongly convex costs but also explicating 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 multi-agent multi-target tracking problem, while its flexibility is exemplified by generating online movie recommendations for Movielens $10$M dataset.

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 which consensus constraints are formulated is violated. Thus, we consider nonlinear network proximity constraints which incentivize nearby nodes to make decisions which are close to one another but 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 which operates by alternating primal stochastic descent steps and Lagrange multiplier updates which 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.

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 multi-cell coordinated beamforming is detailed, demonstrating the usefulness of the proposed algorithm.

This paper considers a general stochastic resource allocation problem that arises widely in wireless networks, cognitive radio, networks, smart-grid communications, and cross-layer design. The problem formulation involves expectations with respect to a collection of random variables with unknown distributions, representing exogenous quantities such as channel gain, user density, or spectrum occupancy. We consider the constant step-size stochastic dual subgradient descent (SDSD) method that has been widely used for online resource allocation in networks. The problem is solved in dual domain which results in a primal resource allocation subproblem at each time instant. The goal here is to characterize the non-asymptotic behavior of such stochastic resource allocations in an almost sure sense. It is well known that with a step size of $\epsilon$, {SDSD} converges to an $\mathcal{O}(\epsilon)$-sized neighborhood of the optimum. In practice however, there exists a trade-off between the rate of convergence and the choice of $\epsilon$. This paper establishes a convergence rate result for the SDSD algorithm that precisely characterizes this trade-off. {Towards this end, a novel stochastic bound on the gap between the objective function and the optimum is developed. The asymptotic behavior of the stochastic term is characterized in an almost sure sense, thereby generalizing the existing results for the {stochastic subgradient} methods.} For the stochastic resource allocation problem at hand, the result explicates the rate with which the allocated resources become near-optimal. As an application, the power and user-allocation problem in device-to-device networks is formulated and solved using the {SDSD} algorithm. Further intuition on the rate results is obtained from the verification of the regularity conditions and accompanying simulation results.

Multidimensional scaling (MDS) is a popular dimensionality reduction techniques that has been widely used for network visualization and cooperative localization. However, the traditional stress minimization formulation of MDS necessitates the use of batch optimization algorithms that are not scalable to large-sized problems. This paper considers an alternative stochastic stress minimization framework that is amenable to incremental and distributed solutions. A novel linear-complexity stochastic optimization algorithm is proposed that is provably convergent and simple to implement. The applicability of the proposed algorithm to localization and visualization tasks is also expounded. Extensive tests on synthetic and real datasets demonstrate the efficacy of the proposed algorithm.

We integrate a wireless powered communication network with a cooperative cognitive radio network, where multiple secondary users (SUs) powered wirelessly by a hybrid access point (HAP) help a primary user relay the data. As a reward for the cooperation, the secondary network gains the spectrum access where SUs transmit to HAP using time division multiple access. To maximize the sum-throughput of SUs, we present a secondary sum-throughput optimal resource allocation (STORA) scheme. Under the constraint of meeting target primary rate, the STORA scheme chooses the optimal set of relaying SUs and jointly performs the time and energy allocation for SUs. Specifically, by exploiting the structure of the optimal solution, we find the order in which SUs are prioritized to relay primary data. Since the STORA scheme focuses on the sum-throughput, it becomes inconsiderate towards individual SU throughput, resulting in low fairness. To enhance fairness, we investigate three resource allocation schemes, which are (i) equal time allocation, (ii) minimum throughput maximization, and (iii) proportional time allocation. Simulation results reveal the trade-off between sum-throughput and fairness. The minimum throughput maximization scheme is the fairest one as each SU gets the same throughput, but yields the least SU sum-throughput., Comment: Accepted in IEEE Transactions on Communications

This paper considers the distributed optimization of a sum of locally observable, non-convex functions. The optimization is performed over a multi-agent networked system, and each local function depends only on a subset of the variables. An asynchronous and distributed alternating directions method of multipliers (ADMM) method that allows the nodes to defer or skip the computation and transmission of updates is proposed in the paper. The proposed algorithm utilizes different approximations in the update step, resulting in proximal and majorized ADMM variants. Both variants are shown to converge to a local minimum, under certain regularity conditions. The proposed asynchronous algorithms are also applied to the problem of cooperative localization in wireless ad hoc networks, where it is shown to outperform the other state-of-the-art localization algorithms., Comment: Submitted to Transactions on signal and information processing over Networks