  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 computed via an
iterative procedure, which provides an approximation of the exact proximity
operator. In that case, we obtain algorithms with two nested loops. We show
that the strategy that minimizes the computational cost to reach a solution
with a desired accuracy in finite time is to set the number of inner iterations
to a constant, which differs from the strategy indicated by a convergence rate
analysis. In the process, we also present a new procedure called SIP (that is
Speedy Inexact Proximal-gradient algorithm) that is both computationally
efficient and easy to implement. Our numerical experiments confirm the
theoretical findings and suggest that SIP can be a very competitive alternative
to the standard procedure.
