Stochastic Subgradient Research Articles

Network utility maximization in uplink multiuser wireless LANs

Multiuser Multiple-Input Multiple-Output (MU-MIMO) enables several simultaneous transmission from the stations toward a multiple antennas access point of a WLAN and thereby boosts the network throughput. However, this throughput enhancement greatly depends on which stations are transmitting concurrently, which in turn necessitates the design of an efficient scheduling algorithm. In this paper, we model uplink MU-MIMO scheduling as a network utility maximization problem that is a non-convex problem due to the wideband channel aware transmission rate function and scheduling constraints. Relying on the zero duality gap, a stochastic learning algorithm is designed to solve the dual of that problem. Determination of stochastic subgradient in the stochastic learning algorithm involves an NP- complete problem, hence imperialist competitive algorithm is exploited to solve this problem. The simulation results substantiate significant throughput improvement and airtime fairness superiority of this scheme compared to the existing ones. The results also show the proposed scheme is not able to guarantee the optimal throughput due to employing imperialist competitive algorithm.

Stochastic Subgradient Algorithms for Strongly Convex Optimization Over Distributed Networks

We study diffusion and consensus based optimization of a sum of unknown convex objective functions over distributed networks. The only access to these functions is through stochastic gradient oracles, each of which is only available at a different node; and a limited number of gradient oracle calls is allowed at each node. In this framework, we introduce a convex optimization algorithm based on stochastic subgradient descent (SSD) updates. We use a carefully designed time-dependent weighted averaging of the SSD iterates, which yields a convergence rate of $O\left(\frac{N\sqrt{N}}{(1-\sigma)T}\right)$ after $T$ gradient updates for each node on a network of $N$ nodes, where $0\leq \sigma denotes the second largest singular value of the communication matrix. This rate of convergence matches the performance lower bound up to constant terms. Similar to the SSD algorithm, the computational complexity of the proposed algorithm also scales linearly with the dimensionality of the data. Furthermore, the communication load of the proposed method is the same as the communication load of the SSD algorithm. Thus, the proposed algorithm is highly efficient in terms of complexity and communication load. We illustrate the merits of the algorithm with respect to the state-of-art methods over benchmark real life data sets.

Nonsmooth analysis and subgradient methods for averaging in dynamic time warping spaces

Time series averaging in dynamic time warping (DTW) spaces has been successfully applied to improve pattern recognition systems. This article proposes and analyzes subgradient methods for the problem of finding a sample mean in DTW spaces. The class of subgradient methods generalizes existing sample mean algorithms such as DTW Barycenter Averaging (DBA). We show that DBA is a majorize-minimize algorithm that converges to necessary conditions of optimality after finitely many iterations. Empirical results show that for increasing sample sizes the proposed stochastic subgradient (SSG) algorithm is more stable and finds better solutions in shorter time than the DBA algorithm on average. Therefore, SSG is useful in online settings and for non-small sample sizes. The theoretical and empirical results open new paths for devising sample mean algorithms: nonsmooth optimization methods and modified variants of pairwise averaging methods.

Affine Invariant Description and Large-Margin Dimensionality Reduction for Target Detection in Optical Remote Sensing Images

A novel target detection method based on affine invariant interest point detection, feature encoding, and large-margin dimensionality reduction (LDR) is proposed for optical remote sensing images. First, four types of interest point detectors are introduced, and their performance in extracting low-level affine invariant descriptors using affine shape estimation is compared. Such a description can deal with significant affine transformations, including viewpoints. Second, feature encoding, which extends bag-of-words (BOW) by encoding high-order statistics, is selected to generate mid-level representation. Finally, LDR based on the large-margin constraint and stochastic subgradient is introduced to make the high-dimensional mid-level representation applicable for target detection. The experiments on aircraft and vehicle detections illustrate the effectiveness of the affine invariant description and LDR (compared with principal component analysis) in improving the detection performance. The experiments also demonstrate the effectiveness of the proposed method compared with popular approaches including Gabor, HOG, LBP, BOW, and R-CNN.

Distributed Constrained Stochastic Subgradient Algorithms Based on Random Projection and Asynchronous Broadcast over Networks

We consider a distributed constrained optimization problem over a time-varying network, where each agent only knows its own cost functions and its constraint set. However, the local constraint set may not be known in advance or consists of huge number of components in some applications. To deal with such cases, we propose a distributed stochastic subgradient algorithm over time-varying networks, where the estimate of each agent projects onto its constraint set by using random projection technique and the implement of information exchange between agents by employing asynchronous broadcast communication protocol. We show that our proposed algorithm is convergent with probability 1 by choosing suitable learning rate. For constant learning rate, we obtain an error bound, which is defined as the expected distance between the estimates of agent and the optimal solution. We also establish an asymptotic upper bound between the global objective function value at the average of the estimates and the optimal value.

Uncertainty-safe large scale support vector machines

The issue of large scale binary classification when data is subject to random perturbations is addressed. The proposed model integrates a learning framework that adjusts its robustness to noise during learning. The method avoids over-conservative situations that can be encountered with worst-case robust support vector machine formulations. The algorithm could be seen as a technique to learn the support-set of the noise distribution during the training process. This is achieved by introducing optimization variables that control the magnitude of the noise perturbations that should be taken into account. The magnitude is tuned by optimizing a generalization error. Only rough estimates of perturbations bounds are required. Additionally, a stochastic bi-level optimization technique is proposed to solve the resulting formulation. The algorithm performs very cheap stochastic subgradient moves and is therefore well suited to large datasets. Encouraging experimental results show that the technique outperforms robust second order cone programming formulations.

Approximate Dual Averaging Method for Multiagent Saddle-Point Problems with Stochastic Subgradients

This paper considers the problem of solving the saddle-point problem over a network, which consists of multiple interacting agents. The global objective function of the problem is a combination of local convex-concave functions, each of which is only available to one agent. Our main focus is on the case where the projection steps are calculated approximately and the subgradients are corrupted by some stochastic noises. We propose an approximate version of the standard dual averaging method and show that the standard convergence rate is preserved, provided that the projection errors decrease at some appropriate rate and the noises are zero-mean and have bounded variance.

On Stochastic Subgradient Mirror-Descent Algorithm with Weighted Averaging

This paper considers stochastic subgradient mirror-descent method for solving constrained convex minimization problems. In particular, a stochastic subgradient mirror-descent method with weighted iterate-averaging is investigated and its per-iterate convergence rate is analyzed. The novel part of the approach is in the choice of weights that are used to construct the averages. Through the use of these weighted averages, we show that the known optimal rates can be obtained with simpler algorithms than those currently existing in the literature. Specifically, by suitably choosing the stepsize values, one can obtain the rate of the order $1/k$ for strongly convex functions, and the rate $1/\sqrt{k}$ for general convex functions (not necessarily differentiable). Furthermore, for the latter case, it is shown that a stochastic subgradient mirror-descent with iterate averaging converges (along a subsequence) to an optimal solution, almost surely, even with the stepsize of the form $1/\sqrt{1+k}$, which was not p...

Mixed-Timescale Precoding and Cache Control in Cached MIMO Interference Network

Consider media streaming in MIMO interference networks whereby multiple base stations (BS) simultaneously deliver media to their associated users using fixed data rates. The performance is fundamentally limited by the cross-link interference. We propose a cache-induced opportunistic cooperative MIMO (CoMP) for interference mitigation. By caching a portion of the media files, the BSs opportunistically employ CoMP to transform the cross-link interference into spatial multiplexing gain. We study a mixed-timescale optimization of MIMO precoding and cache control to minimize the transmit power under the rate constraint. The cache control is to create more CoMP opportunities and is adaptive to the long-term popularity of the media files. The precoding is to guarantee the rate requirement and is adaptive to the channel state information and cache state at the BSs. The joint stochastic optimization problem is decomposed into a short-term precoding and a long-term cache control problem. We propose a precoding algorithm which converges to a stationary point of the short-term problem. Based on this, we exploit the hidden convexity of the long-term problem and propose a low complexity and robust solution using stochastic subgradient. The solution has significant gains over various baselines and does not require explicit knowledge of the media popularity.

Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization

We consider a distributed multi-agent network system where the goal is to minimize a sum of convex objective functions of the agents subject to a common convex constraint set. Each agent maintains an iterate sequence and communicates the iterates to its neighbors. Then, each agent combines weighted averages of the received iterates with its own iterate, and adjusts the iterate by using subgradient information (known with stochastic errors) of its own function and by projecting onto the constraint set. The goal of this paper is to explore the effects of stochastic subgradient errors on the convergence of the algorithm. We first consider the behavior of the algorithm in mean, and then the convergence with probability 1 and in mean square. We consider general stochastic errors that have uniformly bounded second moments and obtain bounds on the limiting performance of the algorithm in mean for diminishing and non-diminishing stepsizes. When the means of the errors diminish, we prove that there is mean consensus between the agents and mean convergence to the optimum function value for diminishing stepsizes. When the mean errors diminish sufficiently fast, we strengthen the results to consensus and convergence of the iterates to an optimal solution with probability 1 and in mean square.

Telecommunication Network Capacity Design for Uncertain Demand

The expansion of telecommunication services has increased the number of users sharing network resources. When a given service is highly demanded, some demands may be unmet due to the limited capacity of the network links. Moreover, for such demands, telecommunication operators should pay penalty costs. To avoid rejecting demands, we can install more capacities in the existing network. In this paper we report experiments on the network capacity design for uncertain demand in telecommunication networks with integer link capacities. We use Poisson demands with bandwidths given by normal or log-normal distribution functions. The expectation function is evaluated using a predetermined set of realizations of the random parameter. We model this problem as a two-stage mixed integer program, which is solved using a stochastic subgradient procedure, the Barahona's volume approach and the Benders decomposition.

A method of stochastic subgradients with complete feedback stepsize rule for convex stochastic approximation problems

A stochastic subgradient algorithm for solving convex stochastic approximation problems is considered. In the algorithm, the stepsize coefficients are controlled on-line on the basis of information gathered in the course of computations according to a new, complete feedback rule derived from the concept of regularized improvement function. Convergence with probability 1 of the method is established.

Probability Inequalities for Sums of Independent Random Variables

Probability Inequalities for Sums of Independent Random Variables