Momentum methods such as Polyak's heavy ball (HB) method, Nesterov's accelerated gradient (AG) as well as accelerated projected gradient (APG) method have been commonly used in machine learning practice, but their performance is quite sensitive to noise in the gradients. We study these methods under a firstorder stochastic oracle model where noisy estimates of the gradients are available. For strongly convex problems, we show that the distribution of the iterates of AG converges with the accelerated linear rate to a ball of radius " centered at a unique invariant distribution in the 1Wasserstein metric where is the condition number as long as the noise variance is smaller than an explicit upper bound we can provide. Our analysis also certifies linear convergence rates as a function of the stepsize, momentum parameter and the noise variance; recovering the accelerated rates in the noiseless case and quantifying the level of noise that can be tolerated to achieve a given performance. To the best of our knowledge, these are the first linear convergence results for stochastic momentum methods under the stochastic oracle model. We also develop finer results for the special case of quadratic objectives, extend our results to the APG method andmore »
Variance Amplification of Accelerated FirstOrder Algorithms for Strongly Convex Quadratic Optimization Problems
We study performance of accelerated firstorder optimization algorithms in the presence of additive white stochastic disturbances. For strongly convex quadratic problems, we explicitly evaluate the steadystate variance of the optimization variable in terms of the eigenvalues of the Hessian of the objective function. We demonstrate that, as the condition number increases, variance amplification of both Nesterov's accelerated method and the heavyball method by Polyak is significantly larger than that of the standard gradient descent. In the context of distributed computation over networks, we examine the role of network topology and spatial dimension on the performance of these firstorder algorithms. For ddimensional tori, we establish explicit asymptotic dependence for the variance amplification on the network size and the corresponding condition number. Our results demonstrate detrimental influence of acceleration on amplification of stochastic disturbances and suggest that metrics other than convergence rate have to be considered when evaluating performance of optimization algorithms.
 Award ID(s):
 1809833
 Publication Date:
 NSFPAR ID:
 10128661
 Journal Name:
 2018 IEEE Conference on Decision and Control (CDC)
 Page Range or eLocationID:
 5753 to 5758
 Sponsoring Org:
 National Science Foundation
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