K. Antonakopoulos, T. Pethick, A. Kavis, P. Mertikopoulos, and V. Cevher. In NeurIPS '21: Proceedings of the 35th International Conference on Neural Information Processing Systems, 2021.
We examine a flexible algorithmic framework for solving monotone variational inequalities in the presence of randomness and uncertainty. The proposed template encompasses a wide range of popular first-order methods, including dual averaging, dual extrapolation and optimistic gradient algorithms – both adaptive and non-adaptive. Our first result is that the algorithm achieves the optimal rates of convergence for cocoercive problems when the profile of the randomness is known to the optimizer: $\mathcal{O}(1/\sqrt{T})$ for absolute noise profiles, and $\mathcal{O}(1/T)$ for relative ones. Subsequently, we drop all prior knowledge requirements (the absolute/relative variance of the randomness affecting the problem, the operator’s cocoercivity constant, etc.), and we analyze an adaptive instance of the method that gracefully interpolates between the above rates – i.e. it achieves $\mathcal{O}(1/\sqrt{T})$ and $\mathcal{O}(1/T)$ in the absolute and relative cases, respectively. To our knowledge, this is the first universality result of its kind in the literature and, somewhat surprisingly, it shows that an extra-gradient proxy step is not required to achieve optimal rates.