Recently, there has been growing interest in developing optimization methods for solving large-scale machine learning problems. Most of these problems boil down to the problem of minimizing an average of a finite set of smooth and strongly convex functions where the number of functions @math is large. Gradient descent method (GD) is successful in minimizing convex problems at a fast linear rate; however, it is not applicable to the considered large-scale optimization setting because of the high computational complexity. Incremental methods resolve this drawback of gradient methods by replacing the required gradient for the descent direction with an incremental gradient approximation. They operate by evaluating one gradient per iteration and executing the average of the @math available gradients as a gradient approximate. Although, incremental methods reduce the computational cost of GD, their convergence rates do not justify their advantage relative to GD in terms of the total number of gradient evaluations until convergence. In this paper, we introduce a Double Incremental Aggregated Gradient method (DIAG) that computes the gradient of only one function at each iteration, which is chosen based on a cyclic scheme, and uses the aggregated average gradient of all the functions to approximate the full gradient. The iterates of the proposed DIAG method uses averages of both iterates and gradients in oppose to classic incremental methods that utilize gradient averages but do not utilize iterate averages. We prove that not only the proposed DIAG method converges linearly to the optimal solution, but also its linear convergence factor justifies the advantage of incremental methods on GD. In particular, we prove that the worst case performance of DIAG is better than the worst case performance of GD.
One may use a cyclic order instead of stochastic selection of functions in SGD which leads to the update of incremental gradient descent method (IGD) as in @cite_26 @cite_17 . Similar to the case for SGD, the sequence of iterates generated by the IGD method converges to the optimal argument at a sublinear rate of the order @math when the stepsize is diminishing. SGD and IGD are able to reduce the computational complexity of GD by requiring only one gradient evaluation per iteration; however, they both suffer from slow (sublinear) convergence rates.
@cite_26 @cite_17
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We consider stochastic strongly convex optimization with a complex inequality constraint. This complex inequality constraint may lead to computationally expensive projections in algorithmic iterations of the stochastic gradient descent (SGD) methods. To reduce the computation costs pertaining to the projections, we propose an Epoch-Projection Stochastic Gradient Descent (Epro-SGD) method. The proposed Epro-SGD method consists of a sequence of epochs; it applies SGD to an augmented objective function at each iteration within the epoch, and then performs a projection at the end of each epoch. Given a strongly convex optimization and for a total number of @math iterations, Epro-SGD requires only @math projections, and meanwhile attains an optimal convergence rate of @math , both in expectation and with a high probability. To exploit the structure of the optimization problem, we propose a proximal variant of Epro-SGD, namely Epro-ORDA, based on the optimal regularized dual averaging method. We apply the proposed methods on real-world applications; the empirical results demonstrate the effectiveness of our methods. Stochastic gradient descent (SGD) is a simple and popular method to solve stochastic optimization problems which arise in machine learning. For strongly convex problems, its convergence rate was known to be O( (T) T), by running SGD for T iterations and returning the average point. However, recent results showed that using a different algorithm, one can get an optimal O(1 T) rate. This might lead one to believe that standard SGD is suboptimal, and maybe should even be replaced as a method of choice. In this paper, we investigate the optimality of SGD in a stochastic setting. We show that for smooth problems, the algorithm attains the optimal O(1 T) rate. However, for non-smooth problems, the convergence rate with averaging might really be ( (T) T), and this is not just an artifact of the analysis. On the flip side, we show that a simple modification of the averaging step suffices to recover the O(1 T) rate, and no other change of the algorithm is necessary. We also present experimental results which support our findings, and point out open problems. Stochastic gradient descent (SGD) on a low-rank factorization is commonly employed to speed up matrix problems including matrix completion, subspace tracking, and SDP relaxation. In this paper, we exhibit a step size scheme for SGD on a low-rank least-squares problem, and we prove that, under broad sampling conditions, our method converges globally from a random starting point within @math steps with constant probability for constant-rank problems. Our modification of SGD relates it to stochastic power iteration. We also show experiments to illustrate the runtime and convergence of the algorithm. Stochastic gradient descent (SGD) holds as a classical method to build large scale machine learning models over big data. A stochastic gradient is typically calculated from a limited number of samples (known as mini-batch), so it potentially incurs a high variance and causes the estimated parameters bounce around the optimal solution. To improve the stability of stochastic gradient, recent years have witnessed the proposal of several semi-stochastic gradient descent algorithms, which distinguish themselves from standard SGD by incorporating global information into gradient computation. In this paper we contribute a novel stratified semi-stochastic gradient descent (S3GD) algorithm to this nascent research area, accelerating the optimization of a large family of composite convex functions. Though theoretically converging faster, prior semi-stochastic algorithms are found to suffer from high iteration complexity, which makes them even slower than SGD in practice on many datasets. In our proposed S3GD, the semi-stochastic gradient is calculated based on efficient manifold propagation, which can be numerically accomplished by sparse matrix multiplications. This way S3GD is able to generate a highly-accurate estimate of the exact gradient from each mini-batch with largely-reduced computational complexity. Theoretic analysis reveals that the proposed S3GD elegantly balances the geometric algorithmic convergence rate against the space and time complexities during the optimization. The efficacy of S3GD is also experimentally corroborated on several large-scale benchmark datasets.