We present a generalization of Nesterov's accelerated gradient descent algorithm. Our algorithm (AGNES) provably achieves acceleration for smooth convex minimization tasks with noisy gradient estimates if the noise intensity is proportional to the magnitude of the gradient. Nesterov's accelerated gradient descent does not converge under this noise model if the constant of proportionality exceeds one. AGNES fixes this deficiency and provably achieves an accelerated convergence rate no matter how small the signal to noise ratio in the gradient estimate. Empirically, we demonstrate that this is an appropriate model for mini-batch gradients in overparameterized deep learning. Finally, we show that AGNES outperforms stochastic gradient descent with momentum and Nesterov's method in the training of CNNs.

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