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Register a new userFast Stochastic Variance Reduced Gradient Method with Momentum Acceleration for Machine Learning
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Fanhua Shang
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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Recently, research on accelerated stochastic gradient descent methods (e.g., SVRG) has made exciting progress (e.g., linear convergence for strongly convex problems). However, the best-known methods (e.g., Katyusha) requires at least two auxiliary variables and two momentum parameters. In this paper, we propose a fast stochastic variance reduction gradient (FSVRG) method, in which we design a novel update rule with the Nesterovs momentum and incorporate the technique of growing epoch size. FSVRG has only one auxiliary variable and one momentum weight, and thus it is much simpler and has much lower per-iteration complexity. We prove that FSVRG achieves linear convergence for strongly convex problems and the optimal $mathcal{O}(1/T^2)$ convergence rate for non-strongly convex problems, where $T$ is the number of outer-iterations. We also extend FSVRG to directly solve the problems with non-smooth component functions, such as SVM. Finally, we empirically study the performance of FSVRG for solving various machine learning problems such as logistic regression, ridge regression, Lasso and SVM. Our results show that FSVRG outperforms the state-of-the-art stochastic methods, including Katyusha.
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Stochastic gradient methods (SGMs) have been extensively used for solving stochastic problems or large-scale machine learning problems. Recent works employ various techniques to improve the convergence rate of SGMs for both convex and nonconvex cases. Most of them require a large number of samples in some or all iterations of the improved SGMs. In this paper, we propose a new SGM, named PStorm, for solving nonconvex nonsmooth stochastic problems. With a momentum-based variance reduction technique, PStorm can achieve the optimal complexity result $O(varepsilon^{-3})$ to produce a stochastic $varepsilon$-stationary solution, if a mean-squared smoothness condition holds and $Theta(varepsilon^{-1})$ samples are available for the initial update. Different from existing optimal methods, PStorm can still achieve a near-optimal complexity result $tilde{O}(varepsilon^{-3})$ by using only one or $O(1)$ samples in every update. With this property, PStorm can be applied to online learning problems that favor real-time decisions based on one or $O(1)$ new observations. In addition, for large-scale machine learning problems, PStorm can generalize better by small-batch training than other optimal methods that require large-batch training and the vanilla SGM, as we demonstrate on training a sparse fully-connected neural network and a sparse convolutional neural network.
In recent years, stochastic variance reduction algorithms have attracted considerable attention for minimizing the average of a large but finite number of loss functions. This paper proposes a novel Riemannian extension of the Euclidean stochastic variance reduced gradient (R-SVRG) algorithm to a manifold search space. The key challenges of averaging, adding, and subtracting multiple gradients are addressed with retraction and vector transport. For the proposed algorithm, we present a global convergence analysis with a decaying step size as well as a local convergence rate analysis with a fixed step size under some natural assumptions. In addition, the proposed algorithm is applied to the computation problem of the Riemannian centroid on the symmetric positive definite (SPD) manifold as well as the principal component analysis and low-rank matrix completion problems on the Grassmann manifold. The results show that the proposed algorithm outperforms the standard Riemannian stochastic gradient descent algorithm in each case.
Machine Learning algorithms based on Brain-inspired Hyperdimensional (HD) computing imitate cognition by exploiting statistical properties of high-dimensional vector spaces. It is a promising solution for achieving high energy-efficiency in different machine learning tasks, such as classification, semi-supervised learning and clustering. A weakness of existing HD computing-based ML algorithms is the fact that they have to be binarized for achieving very high energy-efficiency. At the same time, binarized models reach lower classification accuracies. To solve the problem of the trade-off between energy-efficiency and classification accuracy, we propose the QubitHD algorithm. It stochastically binarizes HD-based algorithms, while maintaining comparable classification accuracies to their non-binarized counterparts. The FPGA implementation of QubitHD provides a 65% improvement in terms of energy-efficiency, and a 95% improvement in terms of the training time, as compared to state-of-the-art HD-based ML algorithms. It also outperforms state-of-the-art low-cost classifiers (like Binarized Neural Networks) in terms of speed and energy-efficiency by an order of magnitude during training and inference.
Distributionally robust supervised learning (DRSL) is emerging as a key paradigm for building reliable machine learning systems for real-world applications -- reflecting the need for classifiers and predictive models that are robust to the distribution shifts that arise from phenomena such as selection bias or nonstationarity. Existing algorithms for solving Wasserstein DRSL -- one of the most popular DRSL frameworks based around robustness to perturbations in the Wasserstein distance -- involve solving complex subproblems or fail to make use of stochastic gradients, limiting their use in large-scale machine learning problems. We revisit Wasserstein DRSL through the lens of min-max optimization and derive scalable and efficiently implementable stochastic extra-gradient algorithms which provably achieve faster convergence rates than existing approaches. We demonstrate their effectiveness on synthetic and real data when compared to existing DRSL approaches. Key to our results is the use of variance reduction and random reshuffling to accelerate stochastic min-max optimization, the analysis of which may be of independent interest.
Federated Averaging (FedAvg, also known as Local-SGD) (McMahan et al., 2017) is a classical federated learning algorithm in which clients run multiple local SGD steps before communicating their update to an orchestrating server. We propose a new federated learning algorithm, FedPAGE, able to further reduce the communication complexity by utilizing the recent optimal PAGE method (Li et al., 2021) instead of plain SGD in FedAvg. We show that FedPAGE uses much fewer communication rounds than previous local methods for both federated convex and nonconvex optimization. Concretely, 1) in the convex setting, the number of communication rounds of FedPAGE is $O(frac{N^{3/4}}{Sepsilon})$, improving the best-known result $O(frac{N}{Sepsilon})$ of SCAFFOLD (Karimireddy et al.,2020) by a factor of $N^{1/4}$, where $N$ is the total number of clients (usually is very large in federated learning), $S$ is the sampled subset of clients in each communication round, and $epsilon$ is the target error; 2) in the nonconvex setting, the number of communication rounds of FedPAGE is $O(frac{sqrt{N}+S}{Sepsilon^2})$, improving the best-known result $O(frac{N^{2/3}}{S^{2/3}epsilon^2})$ of SCAFFOLD (Karimireddy et al.,2020) by a factor of $N^{1/6}S^{1/3}$, if the sampled clients $Sleq sqrt{N}$. Note that in both settings, the communication cost for each round is the same for both FedPAGE and SCAFFOLD. As a result, FedPAGE achieves new state-of-the-art results in terms of communication complexity for both federated convex and nonconvex optimization.
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