  Averaging scheme has attracted extensive attention in deep learning as well
as traditional machine learning. It achieves theoretically optimal convergence
and also improves the empirical model performance. However, there is still a
lack of sufficient convergence analysis for strongly convex optimization.
Typically, the convergence about the last iterate of gradient descent methods,
which is referred to as individual convergence, fails to attain its optimality
due to the existence of logarithmic factor. In order to remove this factor, we
first develop gradient descent averaging (GDA), which is a general
projection-based dual averaging algorithm in the strongly convex setting. We
further present primal-dual averaging for strongly convex cases (SC-PDA), where
primal and dual averaging schemes are simultaneously utilized. We prove that
GDA yields the optimal convergence rate in terms of output averaging, while
SC-PDA derives the optimal individual convergence. Several experiments on SVMs
and deep learning models validate the correctness of theoretical analysis and
effectiveness of algorithms.
