Accelerated Primal-Dual Methods for Convex-Strongly-Concave Saddle Point Problems

In this work, we aim to investigate Primal-Dual (PD) methods for convex-strongly-concave saddle point problems (SPP). In many cases, the computation of the proximal oracle over the primal-only function is inefficient. Hence, we use its first-order linear approximation in the proximal step resulting in a Linearized PD (LPD) method. Even when the coupling term is bilinear, we observe that LPD has a suboptimal dependence on the Lipschitz constant of the primal-only function. In contrast, LPD has optimal convergence for the strongly-convex concave case. This observation induces us to present our accelerated linearized primal-dual (ALPD) algorithm to solve convex strongly-concave SPP. ALPD is a single-loop algorithm that combines features of Nesterov's accelerated gradient descent (AGD) and LPD. We show that when the coupling term is semi-linear (which contains bilinear as a specific case), ALPD obtains the optimal dependence on the Lipschitz constant of primal-only function. Hence, it is an optimal algorithm. When the coupling term has a general nonlinear form, the ALPD algorithm has suboptimal dependence on the Lipschitz constant of the primal part of the coupling term. To improve this dependence, we present an inexact APD algorithm. This algorithm performs AGD iterations in the inner loop to find an approximate solution to a proximal subproblem of APD. We show that inexact APD maintains optimal number of gradients evaluations (gradient complexity) of primal-only and dual parts of the problem. It also significantly improves the gradient-complexity of the primal coupling term.