Title: Convergence Analysis of Nesterov's Accelerated Gradient Descent under Relaxed Assumptions
Primary area: optimization
Keywords: Nesterov's Accelerated Gradient, Convergence Theory, Convex Optimization, Stochastic Optimization, Affine Variance

Abstract: We study convergence rates of Nesterov's Accelerated Gradient Descent (NAG) method for convex optimization in both deterministic and stochastic settings.
We focus on a more general smoothness condition raised from several machine learning problems empirically and theoretically.
We show the accelerated convergence rate of order $\mathcal{O}\left(1/T^2\right)$ in terms of the function value gap, given access to exact gradients of objective functions, matching the optimal rate for standard smooth convex optimization in \citep{nesterov1983method}.
Under the relaxed affine-variance noise assumption for stochastic optimization, we establish the high-probability convergence rate of order $\tilde{\mathcal{O}}\left(\sqrt{\log\left(1/\delta\right)/T}\right)$ and this rate could improve to $\tilde{\mathcal{O}}\left(\log\left(1/\delta\right)/T^2\right)$ when the noise parameters are sufficiently small.
Here, $T$ denotes the total number of iterations and $\delta$ is the probability margin.
Up to logarithm factors, our probabilistic convergence rate reaches the same order of the expected rate obtained in \citep{ghadimi2016accelerated} where the assumptions of  bounded variance noise and Lipschitz smoothness are required.