  We propose dynamic sampled stochastic approximation (SA) methods for
stochastic optimization with a heavy-tailed distribution (with finite 2nd
moment). The objective is the sum of a smooth convex function with a convex
regularizer. Typically, it is assumed an oracle with an upper bound $\sigma^2$
on its variance (OUBV). Differently, we assume an oracle with
\emph{multiplicative noise}. This rarely addressed setup is more aggressive but
realistic, where the variance may not be bounded. Our methods achieve optimal
iteration complexity and (near) optimal oracle complexity. For the smooth
convex class, we use an accelerated SA method a la FISTA which achieves, given
tolerance $\epsilon>0$, the optimal iteration complexity of
$\mathcal{O}(\epsilon^{-\frac{1}{2}})$ with a near-optimal oracle complexity of
$\mathcal{O}(\epsilon^{-2})[\ln(\epsilon^{-\frac{1}{2}})]^2$. This improves
upon Ghadimi and Lan [\emph{Math. Program.}, 156:59-99, 2016] where it is
assumed an OUBV. For the strongly convex class, our method achieves optimal
iteration complexity of $\mathcal{O}(\ln(\epsilon^{-1}))$ and optimal oracle
complexity of $\mathcal{O}(\epsilon^{-1})$. This improves upon Byrd et al.
[\emph{Math. Program.}, 134:127-155, 2012] where it is assumed an OUBV. In
terms of variance, our bounds are local: they depend on variances
$\sigma(x^*)^2$ at solutions $x^*$ and the per unit distance multiplicative
variance $\sigma^2_L$. For the smooth convex class, there exist policies such
that our bounds resemble those obtained if it was assumed an OUBV with
$\sigma^2:=\sigma(x^*)^2$. For the strongly convex class such property is
obtained exactly if the condition number is estimated or in the limit for
better conditioned problems or for larger initial batch sizes. In any case, if
it is assumed an OUBV, our bounds are thus much sharper since typically
$\max\{\sigma(x^*)^2,\sigma_L^2\}\ll\sigma^2$.
