The cosmic-ray fluxes of electrons and positrons ( $e^{\pm}$) are measured with high precision by the space-borne particle spectrometers AMS-02. To infer a precise interpretation of the dominant production process for $e^{\pm}$ in our Galaxy, it is necessary to have a correct description of the secondary component, produced by the interaction of cosmic-ray proton and helium with the interstellar medium. We update the parametrization of the $e^{\pm}$ cross sections in order to obtain a new estimate of the lepton secondary component flux of the cosmic radiation. In the light of new cross section measurements performed at collider experiments of p+p→ $\pi^{\pm}$+X, we update the parametrization of the cross sections for these processes and then compute the $e^{\pm}$ ones from $\pi^{\pm}$ decay. We use for the first time in this field the $e^{\pm}$ spectrum obtained from the muon decay computed till the next to leading order. The peculiarity of this work is the experiment based approach, that we adopt in order to obtain a better shape determination and a significant reduction of uncertainty of the current secondary cosmic ray $e^{\pm}$ flux predictions.