Stochastic dual coordinate ascent (SDCA) is an effective technique for solving regularized loss minimization problems in machine learning. This paper considers an extension of SDCA under the mini-batch setting that is often used in practice. Our main contribution is to introduce an accelerated mini-batch version of SDCA and prove a fast convergence rate for this method. We discuss an implementation of our method over a parallel computing system, and compare the results to both the vanilla stochastic dual coordinate ascent and to the accelerated deterministic gradient descent method of nesterov2007gradient .
@cite_4: Online prediction methods are typically presented as serial algorithms running on a single processor. However, in the age of web-scale prediction problems, it is increasingly common to encounter situations where a single processor cannot keep up with the high rate at which inputs arrive. In this work, we present the distributed mini-batch algorithm, a method of converting many serial gradient-based online prediction algorithms into distributed algorithms. We prove a regret bound for this method that is asymptotically optimal for smooth convex loss functions and stochastic inputs. Moreover, our analysis explicitly takes into account communication latencies between nodes in the distributed environment. We show how our method can be used to solve the closely-related distributed stochastic optimization problem, achieving an asymptotically linear speed-up over multiple processors. Finally, we demonstrate the merits of our approach on a web-scale online prediction problem.
@cite_1: 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. We develop and analyze distributed algorithms based on dual averaging of subgradients, and provide sharp bounds on their convergence rates as a function of the network size and topology. Our analysis clearly separates the convergence of the optimization algorithm itself from the effects of communication constraints arising from the network structure. 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.
@cite_17: 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 @math -node architectures whose optimization error in stochastic problems---in spite of asynchronous delays---scales asymptotically as @math after @math iterations. This rate is known to be optimal for a distributed system with @math nodes even in the absence of delays. We additionally complement our theoretical results with numerical experiments on a statistical machine learning task.
@cite_6: We describe and analyze a simple and effective iterative algorithm for solving the optimization problem cast by Support Vector Machines (SVM). Our method alternates between stochastic gradient descent steps and projection steps. We prove that the number of iterations required to obtain a solution of accuracy e is O(1 e). In contrast, previous analyses of stochastic gradient descent methods require Ω (1 e2) iterations. As in previously devised SVM solvers, the number of iterations also scales linearly with 1 λ, where λ is the regularization parameter of SVM. For a linear kernel, the total run-time of our method is O (d (λe)), where d is a bound on the number of non-zero features in each example. Since the run-time does not depend directly on the size of the training set, the resulting algorithm is especially suited for learning from large datasets. Our approach can seamlessly be adapted to employ non-linear kernels while working solely on the primal objective function. We demonstrate the efficiency and applicability of our approach by conducting experiments on large text classification problems, comparing our solver to existing state-of-the-art SVM solvers. For example, it takes less than 5 seconds for our solver to converge when solving a text classification problem from Reuters Corpus Volume 1 (RCV1) with 800,000 training examples.
@cite_5: Mini-batch algorithms have been proposed as a way to speed-up stochastic convex optimization problems. We study how such algorithms can be improved using accelerated gradient methods. We provide a novel analysis, which shows how standard gradient methods may sometimes be insufficient to obtain a significant speed-up and propose a novel accelerated gradient algorithm, which deals with this deficiency, enjoys a uniformly superior guarantee and works well in practice.
@cite_10: In this paper we propose a new approach for constructing efficient schemes for non-smooth convex optimization. It is based on a special smoothing technique, which can be applied to functions with explicit max-structure. Our approach can be considered as an alternative to black-box minimization. From the viewpoint of efficiency estimates, we manage to improve the traditional bounds on the number of iterations of the gradient schemes from ** keeping basically the complexity of each iteration unchanged.
@cite_11: We address the issue of using mini-batches in stochastic optimization of SVMs. We show that the same quantity, the spectral norm of the data, controls the parallelization speedup obtained for both primal stochastic subgradient descent (SGD) and stochastic dual coordinate ascent (SCDA) methods and use it to derive novel variants of mini-batched SDCA. Our guarantees for both methods are expressed in terms of the original nonsmooth primal problem based on the hinge-loss.
Using mini-batches in stochastic learning has received a lot of attention in recent years. E.g. @cite_6 reported experiments showing that applying small mini-batches in Stochastic Gradient Descent (SGD) decreases the required number of iterations. @cite_4 and @cite_17 gave an analysis of SGD with mini-batches for smooth loss functions. @cite_5 studied SGD and accelerated versions of SGD with mini-batches and @cite_11 studied SDCA with mini-batches for SVMs. @cite_1 studied dual averaging in distributed networks as a function of spectral properties of the underlying graph. However, all of these methods have a polynomial dependence on @math , while we consider the strongly convex and smooth case in which a @math rate is achievable. It should be noted that one can use our results for Lipschitz functions as well by smoothing the loss function (see @cite_10 ). By doing so, we can interpolate between the @math rate of non-accelerated method and the @math rate of accelerated gradient.