The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multi-agent co-ordination, estimation in sensor networks, and large-scale optimization in machine learning. We develop and analyze distributed algorithms based on dual averaging of subgradients, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our method of analysis allows for a clear separation between the convergence of the optimization algorithm itself and the effects of communication constraints arising from the network structure. In particular, we show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network. The sharpness of this prediction is confirmed both by theoretical lower bounds and simulations for various networks. Our approach includes both the cases of deterministic optimization and communication, as well as problems with stochastic optimization and/or communication. graphs being a prime example-than on poorly connected networks (e.g., chains, trees or single cycles). Johansson et al.[JRJ09] analyze a low communication peer-to-peer protocol that attains rates dependent on network structure. However, in their algorithm only one node has a current parameter value, while all nodes in our algorithm maintain good estimates of the optimum at all time. This is important in online, streaming, and control problems where nodes are expected to act or answer queries in real time. In additional comparison to previous work, our analysis gives network scaling terms that are often substantially sharper. Our development yields an algorithm with convergence rate that scales inversely in the spectral gap of the network. By exploiting known results on spectral gaps for graphs with n nodes, we show that (disregarding logarithmic factors) our algorithm obtains an ǫ-optimal solution in O(n 2 /ǫ 2 ) iterations for a single cycle or path, O(n/ǫ 2 ) iterations for a two-dimensional grid, and O(1/ǫ 2 ) iterations for a bounded degree expander graph. Moreover, simulation results show an excellent agreement with these theoretical predictions.Our analysis covers several settings for distributed minimization. We begin by studying fixed communication protocols, which are of interest in a variety of areas such as cluster computing or sensor networks with a fixed hardware-dependent protocol. We also investigate randomized communication protocols as well as randomized network failures, which are often essential to handle gracefully in wireless sensor networks and large clusters with potential node failures. Randomized communication also provides an interesting tradeoff between communication savings and convergence rates. In this setting, we obtain much sharper results than previous work by studying the spectral properties of the expected transi...
We analyze the convergence of gradient-based optimization algorithms that base their updates on delayed stochastic gradient information. The main application of our results is to the development of gradient-based distributed optimization algorithms where a master node performs parameter updates while worker nodes compute stochastic gradients based on local information in parallel, which may give rise to delays due to asynchrony. We take motivation from statistical problems where the size of the data is so large that it cannot fit on one computer; with the advent of huge datasets in biology, astronomy, and the internet, such problems are now common. Our main contribution is to show that for smooth stochastic problems, the delays are asymptotically negligible and we can achieve order-optimal convergence results. In application to distributed optimization, we develop procedures that overcome communication bottlenecks and synchronization requirements. We show n-node architectures whose optimization error in stochastic problems-in spite of asynchronous delays-scales asymptotically as O(1/ √ nT ) after T iterations. This rate is known to be optimal for a distributed system with n nodes even in the absence of delays. We additionally complement our theoretical results with numerical experiments on a statistical machine learning task.
Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardness of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining an understanding of these complexity-theoretic issues is important. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We improve upon known results and obtain tight minimax complexity estimates for various function classes.
Many statistical M -estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient and composite gradient methods for solving such problems, working within a high-dimensional framework that allows the data dimension d to grow with (and possibly exceed) the sample size n. This high-dimensional structure precludes the usual global assumptions-namely, strong convexity and smoothness conditions-that underlie much of classical optimization analysis. We define appropriately restricted versions of these conditions, and show that they are satisfied with high probability for various statistical models. Under these conditions, our theory guarantees that projected gradient descent has a globally geometric rate of convergence up to the statistical precision of the model, meaning the typical distance between the true unknown parameter θ * and an optimal solution θ. This result is substantially sharper than previous convergence results, which yielded sublinear convergence, or linear convergence only up to the noise level. Our analysis applies to a wide range of Mestimators and statistical models, including sparse linear regression using Lasso (ℓ 1 -regularized regression); group Lasso for block sparsity; log-linear models with regularization; low-rank matrix recovery using nuclear norm regularization; and matrix decomposition. Overall, our analysis reveals interesting connections between statistical precision and computational efficiency in high-dimensional estimation.
We analyze a class of estimators based on convex relaxation for solving high-dimensional matrix decomposition problems. The observations are noisy realizations of a linear transformation X of the sum of an (approximately) low rank matrix Θ ⋆ with a second matrix Γ ⋆ endowed with a complementary form of low-dimensional structure; this set-up includes many statistical models of interest, including forms of factor analysis, multi-task regression with shared structure, and robust covariance estimation. We derive a general theorem that gives upper bounds on the Frobenius norm error for an estimate of the pair (Θ ⋆ , Γ ⋆ ) obtained by solving a convex optimization problem that combines the nuclear norm with a general decomposable regularizer. Our results are based on imposing a "spikiness" condition that is related to but milder than singular vector incoherence. We specialize our general result to two cases that have been studied in past work: low rank plus an entrywise sparse matrix, and low rank plus a columnwise sparse matrix. For both models, our theory yields non-asymptotic Frobenius error bounds for both deterministic and stochastic noise matrices, and applies to matrices Θ ⋆ that can be exactly or approximately low rank, and matrices Γ ⋆ that can be exactly or approximately sparse. Moreover, for the case of stochastic noise matrices and the identity observation operator, we establish matching lower bounds on the minimax error, showing that our results cannot be improved beyond constant factors. The sharpness of our theoretical predictions is confirmed by numerical simulations.
We consider the problem of sparse coding, where each sample consists of a sparse linear combination of a set of dictionary atoms, and the task is to learn both the dictionary elements and the mixing coefficients. Alternating minimization is a popular heuristic for sparse coding, where the dictionary and the coefficients are estimated in alternate steps, keeping the other fixed. Typically, the coefficients are estimated via ℓ 1 minimization, keeping the dictionary fixed, and the dictionary is estimated through least squares, keeping the coefficients fixed. In this paper, we establish local linear convergence for this variant of alternating minimization and establish that the basin of attraction for the global optimum (corresponding to the true dictionary and the coefficients) is O 1/s 2 , where s is the sparsity level in each sample and the dictionary satisfies RIP. Combined with the recent results of approximate dictionary estimation, this yields provable guarantees for exact recovery of both the dictionary elements and the coefficients, when the dictionary elements are incoherent.The problem of sparse coding consists of unsupervised learning of the dictionary and the coefficient matrices. Thus, given only unlabeled data, we aim to learn the set of dictionary atoms or basis functions that provide a good fit to the observed data. Sparse coding is applied in a variety of domains. Sparse coding of natural images has yielded dictionary atoms which resemble the receptive fields of neurons in the visual cortex [26,27], and has also yielded localized dictionary elements on speech and video data [19,25].An important strength of sparse coding is that it can incorporate overcomplete dictionaries, where the number of dictionary atoms r can exceed the observed dimensionality d. It has been argued that having overcomplete representation provides greater flexibility is modeling and more robustness to noise , which is crucial for encoding complex signals present in images, speech and video. It has been shown that the performance of most machine learning methods employed downstream is critically dependent on the choice of data representations, and overcomplete representations are the key to obtaining state-of-art prediction results .On the downside, the problem of learning sparse codes is computationally challenging, and is in general, NP-hard . In practice, heuristics are employed based on alternating minimization. At a high level, this consists of alternating steps, where the dictionary is kept fixed and the coefficients are updated and vice versa. Such alternating minimization methods have enjoyed empirical success in a number of settings [18,10,2,20,35]. In this paper, we carry out a theoretical analysis of the alternating minimization procedure for sparse coding.
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