  This paper is devoted to the estimators of the mean that provide strong
non-asymptotic guarantees under minimal assumptions on the underlying
distribution. The main ideas behind proposed techniques are based on bridging
the notions of symmetry and robustness. We show that existing methods, such as
median-of-means and Catoni's estimators, can often be viewed as special cases
of our construction. The main contribution of the paper is the proof of uniform
bounds for the deviations of the stochastic process defined by proposed
estimators. Moreover, we extend our results to the case of adversarial
contamination where a constant fraction of the observations is arbitrarily
corrupted. Finally, we apply our methods to the problem of robust multivariate
mean estimation and show that obtained inequalities achieve optimal dependence
on the proportion of corrupted samples.
