  Stochastic gradient descent (SGD) is one of the most widely used algorithms
for large scale optimization problems. While classical theoretical analysis of
SGD for convex problems studies (suffix) \emph{averages} of iterates and
obtains information theoretically optimal bounds on suboptimality, the
\emph{last point} of SGD is, by far, the most preferred choice in practice. The
best known results for last point of SGD \cite{shamir2013stochastic} however,
are suboptimal compared to information theoretic lower bounds by a $\log T$
factor, where $T$ is the number of iterations. \cite{harvey2018tight} shows
that in fact, this additional $\log T$ factor is tight for standard step size
sequences of $\OTheta{\frac{1}{\sqrt{t}}}$ and $\OTheta{\frac{1}{t}}$ for
non-strongly convex and strongly convex settings, respectively. Similarly, even
for subgradient descent (GD) when applied to non-smooth, convex functions, the
best known step-size sequences still lead to $O(\log T)$-suboptimal convergence
rates (on the final iterate). The main contribution of this work is to design
new step size sequences that enjoy information theoretically optimal bounds on
the suboptimality of \emph{last point} of SGD as well as GD. We achieve this by
designing a modification scheme, that converts one sequence of step sizes to
another so that the last point of SGD/GD with modified sequence has the same
suboptimality guarantees as the average of SGD/GD with original sequence. We
also show that our result holds with high-probability. We validate our results
through simulations which demonstrate that the new step size sequence indeed
improves the final iterate significantly compared to the standard step size
sequences.
