  In this paper, we propose an accelerated quasi-Newton proximal extragradient
(A-QPNE) method for solving unconstrained smooth convex optimization problems.
With access only to the gradients of the objective, we prove that our method
can achieve a convergence rate of ${O}\bigl(\min\{\frac{1}{k^2},
\frac{\sqrt{d\log k}}{k^{2.5}}\}\bigr)$, where $d$ is the problem dimension and
$k$ is the number of iterations. In particular, in the regime where $k =
{O}(d)$, our method matches the optimal rate of ${O}(\frac{1}{k^2})$ by
Nesterov's accelerated gradient (NAG). Moreover, in the the regime where $k =
\Omega(d \log d)$, it outperforms NAG and converges at a faster rate of
${O}\bigl(\frac{\sqrt{d\log k}}{k^{2.5}}\bigr)$. To the best of our knowledge,
this result is the first to demonstrate a provable gain of a quasi-Newton-type
method over NAG in the convex setting. To achieve such results, we build our
method on a recent variant of the Monteiro-Svaiter acceleration framework and
adopt an online learning perspective to update the Hessian approximation
matrices, in which we relate the convergence rate of our method to the dynamic
regret of a specific online convex optimization problem in the space of
matrices.
