Accelerated extra-gradient descent: a novel accelerated first-order method
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Citation (published version)Lorenzo Orecchia, Jelena Diakonikolas. 2018. "Accelerated Extra-Gradient Descent: A Novel Accelerated First-Order Method." 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). https://doi.org/10.4230/LIPIcs.ITCS.2018.23
We provide a novel accelerated first-order method that achieves the asymptotically optimal convergence rate for smooth functions in the first-order oracle model. To this day, Nesterov’s Accelerated Gradient Descent (agd) and variations thereof were the only methods achieving acceleration in this standard blackbox model. In contrast, our algorithm is significantly different from agd, as it relies on a predictor-corrector approach similar to that used by Mirror-Prox  and ExtraGradient Descent  in the solution of convex-concave saddle point problems. For this reason, we dub our algorithm Accelerated Extra-Gradient Descent (axgd). Its construction is motivated by the discretization of an accelerated continuous-time dynamics  using the classical method of implicit Euler discretization. Our analysis explicitly shows the effects of discretization through a conceptually novel primal-dual viewpoint. Moreover, we show that the method is quite general: it attains optimal convergence rates for other classes of objectives (e.g., those with generalized smoothness properties or that are non-smooth and Lipschitz-continuous) using the appropriate choices of step lengths. Finally, we present experiments showing that our algorithm matches the performance of Nesterov’s method, while appearing more robust to noise in some cases.
RightsCopyright © Jelena Diakonikolas and Lorenzo Orecchia 2018; licensed under Creative Commons Attribution License (CC-BY)