Subgradient Descent Method Research Articles

Energy-Efficient Resource Allocation for Dual-NOMA-UAV Assisted Internet of Things

Employing unmanned aerial vehicles (UAVs) characterized by low cost, high maneuverability, and on-demand deployment as aerial base stations (BSs) of Internet of Things (IoT) can guarantee communication performance in the absence of terrestrial BSs. However, the limited energy budget of UAV constrains its development. In this paper, a dual-UAV-assisted IoT using non-orthogonal multiple access (NOMA) is proposed to improve IoT capacity. To reduce energy consumption of the UAVs while ensuring a certain throughput, a joint resource optimization problem of communication scheduling, transmit power and motion parameters of UAVs is formulated to maximize energy efficiency of UAVs. To solve the proposed non-convex optimization problem, we present an alternating iterative optimization algorithm to alternately optimize three sub-problems: communication scheduling optimization, UAV transmit power optimization and UAV motion parameters optimization, each of which can be converted into convex optimization and solved using Lagrange multiplier method, subgradient descent method and successive convex approximation (SCA). The numerical results show that optimizing UAV motion parameters can effectively improve energy efficiency of UAVs, and the proposed dual-NOMA-UAV assisted IoT can achieve higher energy efficiency than the orthogonal multiple access (OMA)-UAV assisted IoT.

Energy-Efficient Utility Function-Based Channel Resource Allocation and Power Control for D2D Clusters With NOMA Enablement in Cellular Networks

Device-to-device(D2D) communication combined with non-orthogonal multiple access(NOMA) can further improve the quality of service for future communications systems and increase spectrum utilization and connection density in cellular networks. In this paper, an effective power allocation scheme is designed to optimize the performance of a D2D communication system based on NOMA for imperfect Successive Interference Cancellation(SIC) decoding. To solve the non-convexity of the problem due to integer constraints and coupling variables, an alternate optimization algorithm was designed to obtain the optimal solution of each subproblem by Lagrange duality analysis and the sub-gradient descent method. Finally, the numerical simulation results demonstrate the performance advantages of the proposed joint optimization algorithm for channel resource allocation and power control in energy efficiency.

Maximizing Average Throughput of Cooperative Cognitive Radio Networks Based on Energy Harvesting.

Energy harvesting (EH) and cooperative communication techniques have been widely used in cognitive radio networks. However, most studies on throughput in energy-harvesting cooperative cognitive radio networks (EH-CCRNs) are end-to-end, which ignores the overall working state of the network. For the above problems, under the premise of prioritizing the communication quality of short-range users, this paper focuses on the optimization of the EH-CCRN average throughput, with energy and transmission power as constraints. The formulated problem was an unsolved non-deterministic polynomial-time hardness (NP-hard) problem. To make it tractable to solve, a multi-user time-power resource allocation algorithm (M-TPRA) is proposed, which is based on sub-gradient descent and unary linear optimization methods. Simulation results show that the M-TPRA algorithm can improve the average throughput of the network. In addition, the energy consumed by executing the M-TPRA algorithm is analyzed.

Block-sparse sign algorithm and its performance analysis

System identification is often encountered in applications such as echo cancellation, active noise control, and channel equalization. If the unknown system is sparse, using sparsity-induced methods can improve the convergence performance. Recently, the l2,0-norm constraint was used to derive a block-sparse LMS (BS-LMS) algorithm to accelerate convergence for identifying multi-clustering sparse systems. In some cases, output of the unknown system is contaminated by impulsive noise, and BS-LMS performs poorly or even diverges when identifying such systems. To address this problem, this paper constructs a loss function by combining the absolute error and the l2,0-norm of adaptive filter weight vector, and then uses the subgradient descent method to develop a block-sparse sign algorithm (BS-SA). Its mean and mean-square performance is also analyzed based on the Gaussian-Bernoulli impulsive noise model under some frequently used assumptions. Finally, simulations are performed to test the robustness of BS-SA against impulsive noise and to evaluate the accuracy of theoretical expressions derived for statistical performance.

A continuous-time neurodynamic approach and its discretization for distributed convex optimization over multi-agent systems

Distributed optimization problem (DOP) over multi-agent systems, which can be described by minimizing the sum of agents’ local objective functions, has recently attracted widespread attention owing to its applications in diverse domains. In this paper, inspired by penalty method and subgradient descent method, a continuous-time neurodynamic approach is proposed for solving a DOP with inequality and set constraints. The state of continuous-time neurodynamic approach exists globally and converges to an optimal solution of the considered DOP. Comparisons reveal that the proposed neurodynamic approach can not only resolve more general convex DOPs, but also has lower dimension of solution space. Additionally, the discretization of the neurodynamic approach is also introduced for the convenience of implementation in practice. The iteration sequence of discrete-time method is also convergent to an optimal solution of DOP from any initial point. The effectiveness of the neurodynamic approach is verified by simulation examples and an application in L1-norm minimization problem in the end.

Accelerating Sequential Minimal Optimization via Stochastic Subgradient Descent

Sequential minimal optimization (SMO) is one of the most popular methods for solving a variety of support vector machines (SVMs). The shrinking and caching techniques are commonly used to accelerate SMO. An interesting phenomenon of SMO is that most of the computational time is wasted on the first half of iterations for building a good solution closing to the optimal. However, as we all know, the stochastic subgradient descent (SSGD) method is extremely fast for building a good solution. In this paper, we propose a generalized framework of accelerating SMO through SSGD for a variety of SVMs of binary classification, regression, ordinal regression, and so on. We also provide a deep insight about why SSGD can accelerate SMO. Experimental results on a variety of datasets and learning applications confirm that our method can effectively speed up SMO.

Asynchronous Lagrangian scenario decomposition

We present a distributed asynchronous algorithm for solving two-stage stochastic mixed-integer programs (SMIP) using scenario decomposition, aimed at industrial-scale instances of the stochastic unit commitment (SUC) problem. The algorithm is motivated by large differences in run times observed among scenario subproblems of SUC instances, which can result in inefficient use of distributed computing resources by synchronous parallel algorithms. Our algorithm performs dual iterations asynchronously using a block-coordinate subgradient descent method which allows performing block-coordinate updates using delayed information, while candidate primal solutions are recovered from the solutions of scenario subproblems using heuristics. We present a high performance computing implementation of the asynchronous algorithm, detailing the operations performed by each parallel process and the communication mechanisms among them. We conduct numerical experiments using SUC instances of the Western Electricity Coordinating Council system with up to 1000 scenarios and of the Central Western European system with up to 120 scenarios. We also conduct numerical experiments on generic SMIP instances from the SIPLIB library (DCAP and SSLP). The results demonstrate the general applicability of the proposed algorithm and its ability to solve industrial-scale SUC instances within operationally acceptable time frames. Moreover, we find that an equivalent synchronous parallel algorithm would leave cores idle up to 80.4% of the time on our realistic test instances, an observation which underscores the need for designing asynchronous optimization schemes in order to fully exploit distributed computing on real world applications.

A Discrete-time Distributed Algorithm for Minimum l1-Norm Solution of an Under-determined Linear Equation Set

This paper proposes a discrete-time, distributed algorithm for multi-agent networks to achieve the minimum l1-norm solution to a group of linear equations known to possess a family of solutions. We assume each agent in the network knows only one equation and can communicate with only its neighbors. The algorithm is developed based on a combination of the projection-consensus idea and the sub-gradient descent method. Given the underlying network graph to be directed and strongly connected, we prove that the algorithm enables all agents to achieve a common minimum l1-norm solution. The major difficulty to be dealt with is the non-smooth nature of the norm and the lack of strict convexity of the associated relevant performance index.

Multilayered Attack–Defense Model on Networks

The classical Germeier’s attack–defense game is generalized for the case of defense in depth (multilayered defense) that has a network structure. The generalization is based on the work by Hohzaki and Tanaka. In distinction from this work, the defense in each possible direction of motion between the network nodes given by directed arcs may have multiple layers, which leads in the general case to convex minimax problems that can be solved using the subgradient descent method. In particular, the proposed model generalizes the classical attack–defense model for the multilayered defense without the simplifying assumption that the effectiveness of defense is independent of the defense layer.

Traffic‐aware resource allocation scheme for mMTC in dynamic TDD systems

The asymmetry traffic between downlink (DL) and uplink (UL) in massive machine-type communication (mMTC) systems is so prominent that it makes the traditionally fixed frame protocols insufficient to handle. Meanwhile, the dynamic time-division duplexing (D-TDD) is a promising and attractive technology since its number of time slots for the DL and UL can be asymmetric and adjusted dynamically. In the cellular and mMTC co-existing network, to balance the discrepancy of the UL/DL ratio and alleviate the interference as well, in this study, the authors design a D-TDD-based transmission frame structure to first fulfil the basic transmission requirements of the human-type communication (HTC) users with a low power almost blank subframe. Herein, the stochastic geometry methods are adopted to calculate spectral efficiencies of the HTC user equipment. Then, focusing on the worst queue state in mMTC, they devise the slot allocation problem with the min–max objective of the UL/DL queues and utilise the sub-gradient descent (SGD) method for a solution. Simulation results show that the proposed traffic aware sub-frame configuration is more appropriate for the dynamical asymmetry environment. Meanwhile, the adopted dynamic step size SGD algorithm can achieve a trade-off between the worst-case queue and the network throughput.

Network Resource Allocation via Stochastic Subgradient Descent: Convergence Rate

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. Toward 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.

A Distributed Traffic Control Strategy Based on Cell-Transmission Model

This paper presents a distributed traffic control strategy based on cell-transmission model (CTM). In view of the drawbacks of the CTM in describing the evolution of traffic flow on the intersections under the control signals, we modify the CTM and propose a simplified model. A series of rules is designed with the help of the subgradient descent method to update the status of intersections. Unlike the conventional method, the traffic signal timing plan is found in a fully distributed manner. The evolution of traffic flow for several kinds of typical traffic states is studied through numerical simulation, and the validity and benefits of our method are evaluated against the most popular mixed-integer linear programing.

New computational guarantees for solving convex optimization problems with first order methods, via a function growth condition measure

Motivated by recent work of Renegar (A framework for applying subgradient methods to conic optimization problems, arXiv:1503.02611v2 , 2015) we present new computational methods and associated computational guarantees for solving convex optimization problems using first-order methods. Our problem of interest is the general convex optimization problem $$f^* = \min _{x \in Q} f(x)$$ , where we presume knowledge of a strict lower bound $$f_{\mathrm{slb}}< f^*$$ . [Indeed, $$f_{\mathrm{slb}}$$ is naturally known when optimizing many loss functions in statistics and machine learning (least-squares, logistic loss, exponential loss, total variation loss, etc.) as well as in Renegar’s transformed version of the standard conic optimization problem arXiv:1503.02611v2 , 2015; in all these cases one has $$f_{\mathrm{slb}}= 0 < f^*$$ .] We introduce a new functional measure called the growth constant G for $$f(\cdot )$$ , that measures how quickly the level sets of $$f(\cdot )$$ grow relative to the function value, and that plays a fundamental role in the complexity analysis. When $$f(\cdot )$$ is non-smooth, we present new computational guarantees for the Subgradient Descent Method and for smoothing methods, that can improve existing computational guarantees in several ways, most notably when the initial iterate $$x^0$$ is far from the optimal solution set. When $$f(\cdot )$$ is smooth, we present a scheme for periodically restarting the Accelerated Gradient Method that can also improve existing computational guarantees when $$x^0$$ is far from the optimal solution set, and in the presence of added structure we present a scheme using parametrically increased smoothing that further improves the associated computational guarantees.

Multi-level defense system models: overcoming by means of attacks with several phase constraints

The Germeyer “defense–attack” model is modified to allow for an echeloned defence system in a given direction. The model is a special case of the terminal-type discrete optimal control problem, and it can be solved using the subgradient-descent method. The posed control problem refines the resource level-distribution problem for a given direction, counting more general constraints with allowance for the action abilities of defense tools at various borders.

Stochastic subgradient descent method for large-scale robust chance-constrained support vector machines

Robust chance-constrained Support Vector Machines (SVM) with second-order moment information can be reformulated into equivalent and tractable Semidefinite Programming (SDP) and Second Order Cone Programming (SOCP) models. However, practical applications involve processing large-scale data sets. For the reformulated SDP and SOCP models, existed solvers by primal-dual interior method do not have enough computational efficiency. This paper studies the stochastic subgradient descent method and algorithms to solve robust chance-constrained SVM on large-scale data sets. Numerical experiments are performed to show the efficiency of the proposed approaches. The result of this paper breaks the computational limitation and expands the application of robust chance-constrained SVM.

L1-norm latent SVM for compact features in object detection

The deformable part model is one of the most effective methods for object detection. However, it simultaneously computes the scores for a holistic filter and several part filters in a relatively high-dimensional feature space, which causes the problem of low computational efficiency. This paper proposes an approach to select compact and effective features by learning a sparse deformable part model using L1-norm latent SVM. A stochastic truncated sub-gradient descent method is presented to solve the L1-norm latent SVM problem. Convergence of the algorithm is proved. Extensive experiments are conducted on the INRIA and PASCAL VOC 2007 datasets. A highly compact feature in our method can reach the state-of-the-art performance. The feature dimensionality is reduced to 12% of the original one in the INRIA dataset and less than 30% in most categories of PASCAL VOC 2007 dataset. Compared with the features used in L2-norm latent SVM, the average precisions (AP) have almost no drop using the reduced feature. With our method, the speed of the detection score computation is faster than that of the L2-norm latent SVM method by 3 times. When the cascade strategy is applied, it can be further speeded up by about an order of magnitude.

An efficient primal dual prox method for non-smooth optimization

We study the non-smooth optimization problems in machine learning, where both the loss function and the regularizer are non-smooth functions. Previous studies on efficient empirical loss minimization assume either a smooth loss function or a strongly convex regularizer, making them unsuitable for non-smooth optimization. We develop a simple yet efficient method for a family of non-smooth optimization problems where the dual form of the loss function is bilinear in primal and dual variables. We cast a non-smooth optimization problem into a minimax optimization problem, and develop a primal dual prox method that solves the minimax optimization problem at a rate of $$O(1/T)$$O(1/T) assuming that the proximal step can be efficiently solved, significantly faster than a standard subgradient descent method that has an $$O(1/\sqrt{T})$$O(1/T) convergence rate. Our empirical studies verify the efficiency of the proposed method for various non-smooth optimization problems that arise ubiquitously in machine learning by comparing it to the state-of-the-art first order methods.

Learning image-to-class distance metric for image classification

Image-To-Class (I2C) distance is a novel distance used for image classification and has successfully handled datasets with large intra-class variances. However, it uses Euclidean distance for measuring the distance between local features in different classes, which may not be the optimal distance metric in real image classification problems. In this article, we propose a distance metric learning method to improve the performance of I2C distance by learning per-class Mahalanobis metrics in a large margin framework. Our I2C distance is adaptive to different classes by combining with the learned metric for each class. These multiple per-class metrics are learned simultaneously by forming a convex optimization problem with the constraints that the I2C distance from each training image to its belonging class should be less than the distances to other classes by a large margin. A subgradient descent method is applied to efficiently solve this optimization problem. For efficiency and scalability to large-scale problems, we also show how to simplify the method to learn a diagonal matrix for each class. We show in experiments that our learned Mahalanobis I2C distance can significantly outperform the original Euclidean I2C distance as well as other distance metric learning methods in several prevalent image datasets, and our simplified diagonal matrices can preserve the performance but significantly speed up the metric learning procedure for large-scale datasets. We also show in experiment that our method is able to correct the class imbalance problem, which usually leads the NN-based methods toward classes containing more training images.

Towards Scalable Summarization of Consumer Videos Via Sparse Dictionary Selection

The rapid growth of consumer videos requires an effective and efficient content summarization method to provide a user-friendly way to manage and browse the huge amount of video data. Compared with most previous methods that focus on sports and news videos, the summarization of personal videos is more challenging because of its unconstrained content and the lack of any pre-imposed video structures. We formulate video summarization as a novel dictionary selection problem using sparsity consistency, where a dictionary of key frames is selected such that the original video can be best reconstructed from this representative dictionary. An efficient global optimization algorithm is introduced to solve the dictionary selection model with the convergence rates as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) (where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> is the iteration counter), in contrast to traditional sub-gradient descent methods of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (1/√ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> ). Our method provides a scalable solution for both key frame extraction and video skim generation, because one can select an arbitrary number of key frames to represent the original videos. Experiments on a human labeled benchmark dataset and comparisons to the state-of-the-art methods demonstrate the advantages of our algorithm.

Ergodic Mirror Descent

We generalize stochastic subgradient descent methods to situations in which we do not receive independent samples from the distribution over which we optimize, instead receiving samples coupled over time. We show that as long as the source of randomness is suitably ergodic---it converges quickly enough to a stationary distribution---the method enjoys strong convergence guarantees, both in expectation and with high probability. This result has implications for stochastic optimization in high-dimensional spaces, peer-to-peer distributed optimization schemes, decision problems with dependent data, and stochastic optimization problems over combinatorial spaces.