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In this paper, we propose to extend the standard Convolutional Dictionary Learning problem to a tensor representation where the activations are constrained to be “low-rank” through a Canonical Polyadic decomposition. We show that this additional constraint increases the robustness of the CDL with respect to noise and improve the interpretability of the final results. In addition, we discuss in detail the advantages of this representation and introduce two algorithms, based on ADMM or FISTA, that efficiently solve this problem. We show that by exploiting the low rank property of activations, they achieve lower complexity than the main CDL algorithms. Finally, we evaluate our approach on a wide range of experiments, highlighting the modularity and the advantages of this tensorial low-rank formulation. [ABSTRACT FROM AUTHOR]

This paper studies the convergence properties of the well-known message-passing algorithm for convex optimisation. Under the assumption of pairwise separability and scaled diagonal dominance, asymptotic convergence is established and a simple bound for the convergence rate is provided for message-passing. In comparison with previous results, our results do not require the given convex program to have known convex pairwise components and that our bound for the convergence rate is tighter and simpler. When specialised to quadratic optimisation, we generalise known results by providing a very simple bound for the convergence rate. [ABSTRACT FROM AUTHOR]

We consider the problem of minimizing the $\ell _1$ norm of a linear map over the sphere, which arises in various machine learning applications such as orthogonal dictionary learning (ODL) and robust subspace recovery (RSR). The problem is numerically challenging due to its nonsmooth objective and nonconvex constraint, and its algorithmic aspects have not been well explored. In this paper, we show how the manifold structure of the sphere can be exploited to design fast algorithms with provable guarantees for tackling this problem. Specifically, our contribution is fourfold. First, we present a manifold proximal point algorithm (ManPPA) for the problem and show that it converges at a global sublinear rate. Furthermore, we show that ManPPA can achieve a local quadratic convergence rate when applied to sharp instances of the problem. Second, we develop a semismooth Newton-based inexact augmented Lagrangian method for computing the search direction in each iteration of ManPPA and show that it has an asymptotic superlinear convergence rate. Third, we propose a stochastic variant of ManPPA called StManPPA, which is well suited for large-scale computation, and establish its sublinear convergence rate. Both ManPPA and StManPPA have provably faster convergence rates than existing subgradient-type methods. Fourth, using ManPPA as a building block, we propose a new heuristic method for solving a matrix analog of the problem, in which the sphere is replaced by the Stiefel manifold. The results from our extensive numerical experiments on the ODL and RSR problems demonstrate the efficiency and efficacy of our proposed methods. [ABSTRACT FROM AUTHOR]

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]

The use of a large-scale antenna array (LSAA) has become an important characteristic of multi-antenna communication systems to achieve beamforming gains such as in designing millimeter-wave (mmWave) systems to combat severe propagation losses. In such applications, each antenna element has to be driven by a radio frequency (RF) chain for the implementation of fully-digital beamformers, significantly increasing the hardware cost, complexity, and power consumption. Therefore, constant-modulus analog beamforming (CMAB) becomes a viable solution. In this paper, we consider the scaled analog beamforming (SAB) or constant-modulus analog beamforming (CMAB) architecture and design the system parameters by solving two variants of beampattern matching problem. In the first case, both the magnitude and phase of the beampattern are matched to the given desired beampattern whereas in the second case, only the magnitude of the beampattern is matched. Both the beampattern matching problems are cast as a variant of the constant-modulus least-squares (CLS) problem. We provide efficient algorithms based on the alternating majorization-minimization (AMM) framework that combines the alternating minimization and the MM frameworks and the conventional-cyclic coordinate descent (C-CCD) algorithms to solve the problem in each case. We also propose algorithms based on a new modified-CCD (M-CCD) based approach. For all the developed algorithms we prove convergence to a Karush-Kuhn-Tucker (KKT) point (or a stationary point). Numerical results demonstrate that the proposed algorithms converge faster than the state-of-the-art solutions. Among all the algorithms, the M-CCD-based algorithms have faster convergence when evaluated in terms of the number of iterations and the AMM-based algorithms offer lower complexity. [ABSTRACT FROM AUTHOR]

While many distributed optimization algorithms have been proposed for solving smooth or convex problems over the networks, few of them can handle non-convex and non-smooth problems. Based on a proximal primal-dual approach, this paper presents a new (stochastic) distributed algorithm with Nesterov momentum for accelerated optimization of non-convex and non-smooth problems. Theoretically, we show that the proposed algorithm can achieve an $\epsilon$ -stationary solution under a constant step size with $\mathcal {O}(1/\epsilon ^2)$ computation complexity and $\mathcal {O}(1/\epsilon)$ communication complexity when the epigraph of the non-smooth term is a polyhedral set. When compared to the existing gradient tracking based methods, the proposed algorithm has the same order of computation complexity but lower order of communication complexity. To the best of our knowledge, the presented result is the first stochastic algorithm with the $\mathcal {O}(1/\epsilon)$ communication complexity for non-convex and non-smooth problems. Numerical experiments for a distributed non-convex regression problem and a deep neural network based classification problem are presented to illustrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we consider the policy evaluation problem in reinforcement learning with agents on a decentralized and directed network. In order to evaluate the quality of a fixed policy in this decentralized setting, one option is for agents to run decentralized temporal-difference (TD) collaboratively. To account for the practical scenarios where the state and action spaces are large and malicious attacks emerge, we focus on the decentralized TD learning with linear function approximation in the presence of malicious agents (often termed as Byzantine agents). We propose a trimmed mean-based Byzantine-resilient decentralized TD algorithm to perform policy evaluation in this setting. We establish the finite-time convergence rate, as well as the asymptotic learning error that depends on the number of Byzantine agents. Numerical experiments corroborate the robustness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We design an efficient robust multi-group multicast beamforming scheme for massive multiple-input multiple-output (MIMO) systems. Assuming only estimates of the channel covariance matrices are available at the base station with a bounded error, we formulate the robust quality-of-service (QoS) problem, which is to minimize the transmit power subject to the worst-case minimum signal-to-interference-plus-noise-ratio (SINR) guarantee. We directly solve the worst-case SINR problem and convert the robust QoS constraint into a number of non-convex constraints. Based on the recent convergence result of the alternating direction method of multipliers (ADMM) for non-convex problems, we develop an ADMM-based fast algorithm to directly tackle the reformulated non-convex problem with a convergence guarantee. The algorithm contains two layers of ADMM procedures. We design the outer-layer ADMM to decompose the problem into three convex subproblems and solve them alternatingly. We further develop an inner-layer consensus-ADMM-based algorithm to efficiently solve one subproblem. By exploring each subproblem structure and developing the special optimization techniques, we obtain closed-form or semi-closed-form solutions to each subproblem. These results lead to a fast iterative algorithm, which is guaranteed to converge to a stationary point of the original robust QoS problem. Simulation shows that our proposed algorithm provides a favorable performance compared with existing alternative methods with magnitudes of computational complexity reduction. [ABSTRACT FROM AUTHOR]

We study distributed composite optimization over networks: agents minimize a sum of smooth (strongly) convex functions–the agents’ sum-utility–plus a nonsmooth (extended-valued) convex one. We propose a general unified algorithmic framework for such a class of problems and provide a convergence analysis leveraging the theory of operator splitting. Distinguishing features of our scheme are: (i) When each of the agent’s functions is strongly convex, the algorithm converges at a linear rate, whose dependence on the agents’ functions and network topology is decoupled; (ii) When the objective function is convex (but not strongly convex), similar decoupling as in (i) is established for the coefficient of the proved sublinear rate. This also reveals the role of function heterogeneity on the convergence rate. (iii) The algorithm can adjust the ratio between the number of communications and computations to achieve a rate (in terms of computations) independent on the network connectivity; and (iv) A by-product of our analysis is a tuning recommendation for several existing (non-accelerated) distributed algorithms yielding provably faster (worst-case) convergence rate for the class of problems under consideration. [ABSTRACT FROM AUTHOR]

In this work, a gradient-based primal-dual method of multipliers is proposed for solving a class of linearly constrained non-convex problems. We show that with random initialization of the primal and dual variables, the algorithm is able to compute second-order stationary points (SOSPs) with probability one. Further, we present applications of the proposed method in popular signal processing and machine learning problems such as decentralized matrix factorization and decentralized training of overparameterized neural networks. One of the key steps in the analysis is to construct a new loss function for these problems such that the required convergence conditions (especially the gradient Lipschitz conditions) can be satisfied without changing the global optimal points. [ABSTRACT FROM AUTHOR]

We focus on over-the-air (OTA) Federated Learning (FL), which has been suggested recently to reduce the communication overhead of FL due to the repeated transmissions of the model updates by a large number of users over the wireless channel. In OTA FL, all users simultaneously transmit their updates as analog signals over a multiple access channel, and the server receives a superposition of the analog transmitted signals. However, this approach results in the channel noise directly affecting the optimization procedure, which may degrade the accuracy of the trained model. We develop a Convergent OTA FL (COTAF) algorithm which enhances the common local stochastic gradient descent (SGD) FL algorithm, introducing precoding at the users and scaling at the server, which gradually mitigates the effect of noise. We analyze the convergence of COTAF to the loss minimizing model and quantify the effect of a statistically heterogeneous setup, i.e. when the training data of each user obeys a different distribution. Our analysis reveals the ability of COTAF to achieve a convergence rate similar to that achievable over error-free channels. Our simulations demonstrate the improved convergence of COTAF over vanilla OTA local SGD for training using non-synthetic datasets. Furthermore, we numerically show that the precoding induced by COTAF notably improves the convergence rate and the accuracy of models trained via OTA FL. [ABSTRACT FROM AUTHOR]