Convergence in quadratic mean of averaged stochastic gradient algorithms without strong convexity nor bounded gradient

Online averaged stochastic gradient algorithms are more and more studied since (i) they can deal quickly with large sample taking values in high-dimensional spaces, (ii) they enable to treat data sequentially, (iii) they are known to be asymptotically efficient. In this paper, we focus on giving explicit bounds of the quadratic mean error of the estimates, and this, without supposing that the function we would like to minimize is strongly convex or admits a bounded gradient.