For smooth convex functions, relatively inexact gradient descent has a three-regime worst-case one-step rate, and the largest provably safe stepsize is 2/(1+delta).
Gradient directions and relative inexactness in optimization and machine learning
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In this paper, we investigate the influence of noise giving an estimate of the gradient having a acute angle with the original. Noise amplitude has a relative model. The work offers both theoretical calculations and theorems, as well as experimental results. Classic machine learning problems were chosen as experiments -- linear and logistic regression, computer vision and natural language processing.
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Worst-case convergence analysis of relatively inexact gradient descent on smooth convex functions
For smooth convex functions, relatively inexact gradient descent has a three-regime worst-case one-step rate, and the largest provably safe stepsize is 2/(1+delta).