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In this paper, we investigate the trade-off between convergence rate andcomputational cost when minimizing a composite functional withproximal-gradient methods, which are popular optimisation tools in machinelearning. We consider the case when the proximity operator is computed via aniterative procedure, which provides an approximation of the exact proximityoperator. In that case, we obtain algorithms with two nested loops. We showthat the strategy that minimizes the computational cost to reach a solutionwith a desired accuracy in finite time is to set the number of inner iterationsto a constant, which differs from the strategy indicated by a convergence rateanalysis. In the process, we also present a new procedure called SIP (that isSpeedy Inexact Proximal-gradient algorithm) that is both computationallyefficient and easy to implement. Our numerical experiments confirm thetheoretical findings and suggest that SIP can be a very competitive alternativeto the standard procedure.

Optimal Computational Trade-Off of Inexact Proximal Methods