  In this paper, we study the finite-sum convex optimization problem focusing
on the general convex case. Recently, the study of variance reduced (VR)
methods and their accelerated variants has made exciting progress. However, the
step size used in the existing VR algorithms typically depends on the
smoothness parameter, which is often unknown and requires tuning in practice.
To address this problem, we propose two novel adaptive VR algorithms: Adaptive
Variance Reduced Accelerated Extra-Gradient (AdaVRAE) and Adaptive Variance
Reduced Accelerated Gradient (AdaVRAG). Our algorithms do not require knowledge
of the smoothness parameter. AdaVRAE uses $\mathcal{O}\left(n\log\log
n+\sqrt{\frac{n\beta}{\epsilon}}\right)$ gradient evaluations and AdaVRAG uses
$\mathcal{O}\left(n\log\log n+\sqrt{\frac{n\beta\log\beta}{\epsilon}}\right)$
gradient evaluations to attain an $\mathcal{O}(\epsilon)$-suboptimal solution,
where $n$ is the number of functions in the finite sum and $\beta$ is the
smoothness parameter. This result matches the best-known convergence rate of
non-adaptive VR methods and it improves upon the convergence of the state of
the art adaptive VR method, AdaSVRG. We demonstrate the superior performance of
our algorithms compared with previous methods in experiments on real-world
datasets.
