On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes
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Stochastic gradient descent is the method of choice for large scale optimization of machine learning objective functions. Yet, its performance is greatly variable and heavily depends on the choice of the stepsizes. This has motivated a large body of research on adaptive stepsizes. However, there is currently a gap in our theoretical understanding of these methods, especially in the non-convex setting. In this paper, we start closing this gap: we theoretically analyze the use of adaptive stepsizes, like the ones in AdaGrad, in the non-convex setting. We show sufficient conditions for almost sure convergence to a stationary point when the adaptive stepsizes are used, proving the first guarantee for AdaGrad in the non-convex setting. Moreover, we show explicit rates of convergence that automatically interpolates between O(1/T) and O(1/√(T)) depending on the noise of the stochastic gradients, in both the convex and non-convex setting.
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