  The $\tau$ lepton anomalous magnetic moment: $a_\tau = \frac{g_{\tau}-2}{2}$
was measured, so far, with a precision of only several percents despite its
highly sensitivity to physics beyond the Standard Model such as compositeness
or Supersymmetry. A new study is presented to improve the sensitivity of the
$a_\tau $ measurement with photon-photon interactions from ultra-peripheral
lead-lead collisions at LHC. The theoretical approach used in this work is
based on an effective Lagrangian and on a photon flux implemented in the
MadGraph5 Monte Carlo simulation. Using a multivariate analysis to discriminate
the signal from the background processes, a sensitivity to the anomalous
magnetic moment $\rm{a_{\tau}}$ = 0 $_{+0.011} ^{-0.019}$ is obtained at 95\%
CL with a dataset corresponding to an integrated luminosity of 2 nb$^{-1}$ of
lead-lead collisions and assuming a conservative 10\% systematic uncertainty.
The present results are compared with previous calculations and available
measurements.
