  Estimation procedures based on recursive algorithms are interesting and
powerful techniques that are able to deal rapidly with (very) large samples of
high dimensional data. The collected data may be contaminated by noise so that
robust location indicators, such as the geometric median, may be preferred to
the mean. In this context, an estimator of the geometric median based on a fast
and efficient averaged non linear stochastic gradient algorithm has been
developed by Cardot, C\'enac and Zitt (2013). This work aims at studying more
precisely the non asymptotic behavior of this algorithm by giving non
asymptotic confidence balls. This new result is based on the derivation of
improved $L^2$ rates of convergence as well as an exponential inequality for
the martingale terms of the recursive non linear Robbins-Monro algorithm.
