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In the context of finite sums minimization, variance reduction techniques are widely used to improve the performance of state-of-the-art stochastic gradient methods. Their practical impact is clear, as well as their theoretical properties. Stochastic proximal point algorithms have been studied as an alternative to stochastic gradient algorithms since they are more stable with respect to the choice of the step size. However, their variance-reduced versions are not as well studied as the gradient ones. In this work, we propose the first unified study of variance reduction techniques for stochastic proximal point algorithms. We introduce a generic stochastic proximal-based algorithm that can be specified to give the proximal version of SVRG, SAGA, and some of their variants. For this algorithm, in the smooth setting, we provide several convergence rates for the iterates and the objective function values, which are faster than those of the vanilla stochastic proximal point algorithm. More specifically, for convex functions, we prove a sublinear convergence rate of O(1/k). In addition, under the Polyak-łojasiewicz condition, we obtain linear convergence rates. Finally, our numerical experiments demonstrate the advantages of the proximal variance reduction methods over their gradient counterparts in terms of the stability with respect to the choice of the step size in most cases, especially for difficult problems. [ABSTRACT FROM AUTHOR]

Machine learning (ML) is evolving rapidly and has made many theoretical breakthroughs while widely applied in various fields. ML allows systems the ability to access data and use it to enable computers to execute cognitive processes such as learning and improving from previous experiences and solving complicated issues. Many first-order stochastic optimization methods have been used to solve the optimization model of ML. These algorithms adopt Barzilai-Borwein (BB) step size instead of fixed or diminishing step size to improve performance. However, the BB step size format involves fractional calculation, which inevitably leads to a zero denominator, especially when the objective function is non-convex. The BB technique will be violated if the denominator is near 0 or even negative. To improve the computation of the step size, a Positive Defined Stabilized Barzilai-Borwein (PDSBB) approach is introduced in this paper. Integrating PDSBB with the stochastic variance reduced gradient (SVRG) approach, a new method SVRG-PDSBB is proposed. Numerical experiments have shown that the new algorithm has stabilized the performance of the new step size, which successfully avoiding zero denominators and effectively solving the common problems in machine learning. The convergence of SVRG-PDSBB is theoretically and numerically proven, and the effectiveness of the new algorithm is shown by comparison with other algorithms. [ABSTRACT FROM AUTHOR]

The machine learning model converges slowly and has unstable training since large variance by random using a sample estimate gradient in SGD. To this end, we propose a noise reduction method for Stochastic Variance Reduction gradient (SVRG), called N-SVRG, which uses small batches samples instead of all samples for the average gradient calculation, while performing an incremental update of the average gradient. In each round of iteration, a small batch of samples is randomly selected for the average gradient calculation, while the average gradient is updated by rounding of the past model gradients during internal iterations. By suitably reducing the batch size B, the memory storage as well as the number of iterations can be reduced. The experiments are compared with the state-of-the-art Mini-Batch SGD, AdaGrad, RMSProp, SVRG and SCSG, and it is demonstrated that N-SVRG outperforms SVRG and SASG, and is on par with SCSG. Finally, by exploring the relationship between the small values of different parameters n. B and k and the effectiveness of the algorithm, we prove that ourN-SVRG algorithm has some stability and can achieve sufficient accuracy even in the case of small batch size. The advantages and disadvantages of various methods are experimentally compared, and the stability of N-SVRG is explored by parameter settings. [ABSTRACT FROM AUTHOR]