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This paper investigates the stochastic optimization problem focusing on developing scalable parallel algorithms for deep learning tasks. Our solution involves a reformation of the objective function for stochastic optimization in neural network models, along with a novel parallel computing strategy, coined the weighted aggregating stochastic gradient descent (WASGD). Following a theoretical analysis on the characteristics of the new objective function, WASGD introduces a decentralized weighted aggregating scheme based on the performance of local workers. Without any center variable, the new method automatically gauges the importance of local workers and accepts them by their contributions. Furthermore, we have developed an enhanced version of the method, WASGD+, by (1) implementing a designed sample order and (2) upgrading the weight evaluation function. To validate the new method, we benchmark our pipeline against several popular algorithms including the state-of-the-art deep neural network classifier training techniques (e.g., elastic averaging SGD). Comprehensive validation studies have been conducted on four classic datasets: CIFAR-100, CIFAR-10, Fashion-MNIST, and MNIST. Subsequent results have firmly validated the superiority of the WASGD scheme in accelerating the training of deep architecture. Better still, the enhanced version, WASGD+, is shown to be a significant improvement over its prototype. [ABSTRACT FROM AUTHOR]

We consider a class of nonconvex regularized optimization problems, which appear frequently in machine learning and data processing. Due to the structure of the problems, the iteratively reweighted algorithm was developed and applied to the consensus optimization. In this paper, we propose the acceleration of this scheme by adding an inertial term in each iteration. The proposed algorithms inherit the advantages of classical decentralized algorithms: they can be implemented over a connected network, in which the agents communicate with their neighbors and perform local computations. We also employ the diminishing stepsizes technique for the iteratively reweighted algorithm and consider its acceleration. In specific cases, our algorithms reduce to existing decentralized schemes and also indicate novel ones. Mathematically, we prove the convergence for both algorithms with several assumptions on the objective functions. With Kurdyka-Łojasiewicz property, convergence rates can be derived for constant stepsize case. Numerical results demonstrate the efficiency of the algorithms. [ABSTRACT FROM AUTHOR]

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]

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]

In real-world applications, it is important for machine learning algorithms to be robust against data outliers or corruptions. In this paper, we focus on improving the robustness of a large class of learning algorithms that are formulated as low-rank semi-definite programming (SDP) problems. Traditional formulations use the square loss, which is notorious for being sensitive to outliers. We propose to replace this with more robust noise models, including the $\ell _1$ ℓ 1 -loss and other nonconvex losses. However, the resultant optimization problem becomes difficult as the objective is no longer convex or smooth. To alleviate this problem, we design an efficient algorithm based on majorization-minimization. The crux is on constructing a good optimization surrogate, and we show that this surrogate can be efficiently obtained by the alternating direction method of multipliers (ADMM). By properly monitoring ADMM's convergence, the proposed algorithm is empirically efficient and also theoretically guaranteed to converge to a critical point. Extensive experiments are performed on four machine learning applications using both synthetic and real-world data sets. Results show that the proposed algorithm is not only fast but also has better performance than the state-of-the-arts. [ABSTRACT FROM AUTHOR]

Due to the growth of modern dataset size and the desire to harness computing power of multiple machines, there is a recent surge of interest in the design of distributed machine learning algorithms. However, distributed algorithms are sensitive to Byzantine attackers who can send falsified data to prevent the convergence of algorithms or lead the algorithms to converge to value of the attackers’ choice. Some recent work proposed interesting algorithms that can deal with the scenario when up to half of the workers are compromised. In this paper, we propose a novel algorithm that can deal with an arbitrary number of Byzantine attackers. The main idea is to ask the parameter server to randomly select a small clean dataset and compute noisy gradient using this small dataset. This noisy gradient will then be used as a ground truth to filter out information sent by compromised workers. We show that the proposed algorithm converges to the neighborhood of the population minimizer regardless the number of Byzantine attackers. We further provide numerical examples to show that the proposed algorithm can benefit from the presence of good workers and achieve better performance than existing algorithms. [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]

Dual Coordinate Descent (DCD) is one of the most popular optimization methods. The parallelization of DCD is difficult, as DCD is sequential in nature. As such, simultaneously running multiple DCD threads on batches of data elements causes result inaccuracy and slow convergence, due to the concurrent updates of multiple coordinates. Some parallelization methods adopt separable approximate functions that depend on the degree of parallelism. Such dependencies result in both poor scalability and slow-convergence. To address these challenges, in this paper we present a new parallel primal-dual algorithm for DCD, called PPD. In PPD, the block data distribution is utilized to obtain a new approximate function that is independent from the parallelism. Moreover, PPD is designed with a novel primal-dual acceleration scheme to approach the optimal solution closely and quickly. We demonstrate the advantages of PPD in terms of scalability and efficiency through experiments. [ABSTRACT FROM AUTHOR]

With growth in the number of smart devices and advancements in their hardware, in recent years, data-driven machine learning techniques have drawn significant attention. However, due to privacy and communication issues, it is not possible to collect this data at a centralized location. Federated learning is a machine learning setting where the centralized location trains a learning model over remote devices. Federated learning algorithms cannot be employed in the real world scenarios unless they consider unreliable and resource-constrained nature of the wireless medium. In this paper, we propose a federated learning algorithm that is suitable for cellular wireless networks. We prove its convergence, and provide a sub-optimal scheduling policy that improves the convergence rate. We also study the effect of local computation steps and communication steps on the convergence of the proposed algorithm. We prove, in practice, federated learning algorithms may solve a different problem than the one that they have been employed for if the unreliability of wireless channels is neglected. Finally, through numerous experiments on real and synthetic datasets, we demonstrate the convergence of our proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we present an algorithm for the sparse signal recovery problem that incorporates damped Gaussian generalized approximate message passing (GGAMP) into expectation-maximization-based sparse Bayesian learning (SBL). In particular, GGAMP is used to implement the E-step in SBL in place of matrix inversion, leveraging the fact that GGAMP is guaranteed to converge with appropriate damping. The resulting GGAMP-SBL algorithm is much more robust to arbitrary measurement matrix A than the standard damped GAMP algorithm while being much lower complexity than the standard SBL algorithm. We then extend the approach from the single measurement vector case to the temporally correlated multiple measurement vector case, leading to the GGAMP-TSBL algorithm. We verify the robustness and computational advantages of the proposed algorithms through numerical experiments. [ABSTRACT FROM AUTHOR]

The emergence of big data has caused a dramatic shift in the operating regime for optimization algorithms. The performance bottleneck, which used to be computations, is now often communications. Several gradient compression techniques have been proposed to reduce the communication load at the price of a loss in solution accuracy. Recently, it has been shown how compression errors can be compensated for in the optimization algorithm to improve the solution accuracy. Even though convergence guarantees for error-compensated algorithms have been established, there is very limited theoretical support for quantifying the observed improvements in solution accuracy. In this paper, we show that Hessian-aided error compensation, unlike other existing schemes, avoids accumulation of compression errors on quadratic problems. We also present strong convergence guarantees of Hessian-based error compensation for stochastic gradient descent. Our numerical experiments highlight the benefits of Hessian-based error compensation, and demonstrate that similar convergence improvements are attained when only a diagonal Hessian approximation is used. [ABSTRACT FROM AUTHOR]

Learning Automata (LA) can be reckoned to be the founding algorithms on which the field of Reinforcement Learning has been built. Among the families of LA, Estimator Algorithms (EAs) are certainly the fastest, and of these, the family of discretized algorithms are proven to converge even faster than their continuous counterparts. However, it has recently been reported that the previous proofs for 휖-optimality for all the reported algorithms for the past three decades have been flawed. We applaud the researchers who discovered this flaw, and who further proceeded to rectify the proof for the Continuous Pursuit Algorithm (CPA). The latter proof examines the monotonicity property of the probability of selecting the optimal action, and requires the learning parameter to be continuously changing. In this paper, we provide a new method to prove the 휖-optimality of the Discretized Pursuit Algorithm (DPA) which does not require this constraint, by virtue of the fact that the DPA has, in and of itself, absorbing barriers to which the LA can jump in a discretized manner. Unlike the proof given (Zhang et al., Appl Intell 41:974-985, ) for an absorbing version of the CPA, which utilizes the single-action Hoeffding's inequality, the current proof invokes what we shall refer to as the 'multi-action' version of the Hoeffding's inequality. We believe that our proof is both unique and pioneering. It can also form the basis for formally showing the 휖-optimality of the other EAs that possess absorbing states. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel sparse signal recovery algorithm called the trainable iterative soft thresholding algorithm (TISTA). The proposed algorithm consists of two estimation units: a linear estimation unit and a minimum mean squared error (MMSE) estimator based shrinkage unit. The error variance required in the MMSE shrinkage unit is precisely estimated from a tentative estimate of the original signal. The remarkable feature of the proposed scheme is that TISTA includes adjustable variables that control step size and the error variance for the MMSE shrinkage function. The variables are adjusted by standard deep learning techniques. The number of trainable variables of TISTA is nearly equal to the number of iteration rounds and is much smaller than that of known learnable sparse signal recovery algorithms. This feature leads to highly stable and fast training processes of TISTA. Computer experiments show that TISTA is applicable to various classes of sensing matrices, such as Gaussian matrices, binary matrices, and matrices with large condition numbers. Numerical results also demonstrate that, in many cases, TISTA provides significantly faster convergence than approximate message passing (AMP) and the learned iterative shrinkage thresholding algorithm and also outperforms orthogonal AMP in the NMSE performance. [ABSTRACT FROM AUTHOR]

This paper examines the mean-square error performance of diffusion stochastic algorithms under a generalized coordinate-descent scheme. In this setting, the adaptation step by each agent is limited to a random subset of the coordinates of its stochastic gradient vector. The selection of coordinates varies randomly from iteration to iteration and from agent to agent across the network. Such schemes are useful in reducing computational complexity at each iteration in power-intensive large data applications. They are also useful in modeling situations where some partial gradient information may be missing at random. Interestingly, the results show that the steady-state performance of the learning strategy is not always degraded, while the convergence rate suffers some degradation. The results provide yet another indication of the resilience and robustness of adaptive distributed strategies. [ABSTRACT FROM AUTHOR]

Stochastic approximation techniques play an important role in solving many problems encountered in machine learning or adaptive signal processing. In these contexts, the statistics of the data are often unknown a priori or their direct computation is too intensive, and they have thus to be estimated online from the observed signals. For batch optimization of an objective function being the sum of a data fidelity term and a penalization (e.g., a sparsity promoting function), Majorize-Minimize (MM) methods have recently attracted much interest since they are fast, highly flexible, and effective in ensuring convergence. The goal of this paper is to show how these methods can be successfully extended to the case when the data fidelity term corresponds to a least squares criterion and the cost function is replaced by a sequence of stochastic approximations of it. In this context, we propose an online version of an MM subspace algorithm and we study its convergence by using suitable probabilistic tools. Simulation results illustrate the good practical performance of the proposed algorithm associated with a memory gradient subspace, when applied to both nonadaptive and adaptive filter identification problems. [ABSTRACT FROM PUBLISHER]

Wireless distributed machine learning is envisaged to facilitate advanced learning services and applications in wireless networks consisting of devices with limited computing capability. Distributed machine learning algorithms are more vulnerable in wireless systems since information exchange in learning is limited by wireless resources and channel conditions. Moreover, their performance can be significantly degraded by attacks of the Byzantine devices, and information distorted by channel fading can be treated as Byzantine attacks. Consequently, protection of wireless distributed machine learning from Byzantine devices is paramount. Leveraging over-the-air computation, we put forth a novel wireless distributed stochastic gradient descent system which is resilient to Byzantine attacks. The proposed learning system is underpinned by two novel and distinct features which enable more accurate and faster distributed machine learning resilient to Byzantine attacks: collecting training data in the PS to obtain its own training results and grouping the distributed devices. We derive upper bounds of the mean square error of the global parameter when the proposed algorithms are used in the cases with and without Byzantine devices, and prove the convergence of the proposed algorithms with the derived bounds. The effectiveness of the algorithms is validated by showing the accuracy and convergence speed. [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]

The $k$ -vectors algorithm for learning regression functions proposed here is akin to the well-known $k$ -means algorithm. Both algorithms partition the feature space, but unlike the $k$ -means algorithm, the $k$ -vectors algorithm aims to reconstruct the response rather than the feature. The partitioning rule of the algorithm is based on maximizing the correlation (inner product) of the feature vector with a set of $k$ vectors, and generates polyhedral cells, similar to the ones generated by the nearest-neighbor rule of the $k$ -means algorithm. Similarly to $k$ -means, the learning algorithm alternates between two types of steps. In the first type of steps, $k$ labels are determined via a centroid-type rule (in the response space), which uses a surrogate hinge-type loss function to the mean squared error loss function. In the second type of steps, the $k$ vectors which determine the partition are updated according to a multiclass classification rule, in the spirit of support vector machines. It is proved that both steps of the algorithm only require solving convex optimization problems, and that the algorithm is empirically consistent - as the length of the training sequence increases to infinity, fixed-points of the empirical version of the algorithm tend to fixed points of the population version of the algorithm. Learnability of the predictor class posit by the algorithm is also established. [ABSTRACT FROM AUTHOR]

The notion of computation has changed the world more than any previous expressions of knowledge. However, as know-how in its particular algorithmic embodiment, computation is closed to meaning. Therefore, computer-based data processing can only mimic life's creative aspects, without being creative itself. AI's current record of accomplishments shows that it automates tasks associated with intelligence, without being intelligent itself. Mistaking the abstract (computation) for the concrete (computer) has led to the religion of "everything is an output of computation"—even the humankind that conceived the computer. The hypostatized role of computers explains the increased dependence on them. The convergence machine called deep learning is only the most recent form through which the deterministic theology of the machine claims more than what it actually is: extremely effective data processing. A proper understanding of complexity, as well as the need to distinguish between the reactive nature of the artificial and the anticipatory nature of the living are suggested as practical responses to the challenges posed by machine theology. [ABSTRACT FROM AUTHOR]

The multiplicative update (MU) algorithm has been extensively used to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizers. However, theoretical convergence guarantees have only been derived for a few special divergences without regularization. In this work, we provide a conceptually simple, self-contained, and unified proof for the convergence of the MU algorithm applied on NMF with a wide range of divergences and regularizers. Our main result shows the sequence of iterates (i.e., pairs of basis and coefficient matrices) produced by the MU algorithm converges to the set of stationary points of the nonconvex NMF optimization problem. Our proof strategy has the potential to open up new avenues for analyzing similar problems in machine learning and signal processing. [ABSTRACT FROM PUBLISHER]

We present a matrix-factorization algorithm that scales to input matrices with both huge number of rows and columns. Learned factors may be sparse or dense and/or nonnegative, which makes our algorithm suitable for dictionary learning, sparse component analysis, and nonnegative matrix factorization. Our algorithm streams matrix columns while subsampling them to iteratively learn the matrix factors. At each iteration, the row dimension of a new sample is reduced by subsampling, resulting in lower time complexity compared to a simple streaming algorithm. Our method comes with convergence guarantees to reach a stationary point of the matrix-factorization problem. We demonstrate its efficiency on massive functional magnetic resonance imaging data (2 TB), and on patches extracted from hyperspectral images (103 GB). For both problems, which involve different penalties on rows and columns, we obtain significant speed-ups compared to state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

The analysis of crosscut stability is an indispensable task in underground mining activities. Crosscut instabilities usually cause geological disasters and delay of the project. On site, mining engineers analyze and predict the crosscut condition by monitoring its convergence and stress; however, stress monitoring is time-consuming and expensive. In this study, we propose an improved extreme learning machine (ELM) algorithm to predict crosscut's stress based on convergence data, for the first time in literature. The performance of the proposed technique is validated using a crosscut response by means of the FLAC3D finite difference program. It is found that the improved ELM algorithm performs higher generalization performance compared to traditional ELM, as it eliminates the random selection for input weights. Furthermore, a crosscut construction project in an underground mine, Yanqianshan iron mine, located in Liaoning Province (China), is selected as the case study. The accuracy and efficiency of the improved ELM algorithm has been demonstrated by comparing predicted stress data to measured data on site. Additionally, a comparison is conducted between the improved ELM algorithm and other commonly used artificial neural network algorithms. [ABSTRACT FROM AUTHOR]