Optimal Computational Trade-Off of Inexact Proximal Methods

In this paper, we investigate the trade-off between convergence rate and computational cost when minimizing a composite functional with proximal-gradient methods, which are popular optimisation tools in machine learning. We consider the case when the proximity operator is approximated via an iterative procedure, which yields algorithms with two nested loops. We show that the strategy minimizing the computational cost to reach a desired accuracy in finite time is to keep the number of inner iterations constant, which differs from the strategy indicated by a convergence rate analysis.