  We study a variant of an accelerated gradient algorithm of Nesterov and
Spokoiny. We call this algorithm the Nesterov--Spokoiny Acceleration (NSA). The
NSA algorithm simultaneously satisfies the following property: For a smooth
convex objective $f \in \mathscr{F}_{L}^{\infty,1} (\mathbb{R}^n) $, the
sequence $\{ \mathbf{x}_k \}_{k \in \mathbb{N}}$ governed by NSA satisfies $
\limsup\limits_{k \to \infty } k^2 ( f (\mathbf{x}_k ) - f^* ) = 0 $, where
$f^* > -\infty$ is the minimum of $f$.
