  The Area Under the ROC Curve (AUC) is a widely used performance measure for
imbalanced classification arising from many application domains where
high-dimensional sparse data is abundant. In such cases, each $d$ dimensional
sample has only $k$ non-zero features with $k \ll d$, and data arrives
sequentially in a streaming form. Current online AUC optimization algorithms
have high per-iteration cost $\mathcal{O}(d)$ and usually produce non-sparse
solutions in general, and hence are not suitable for handling the data
challenge mentioned above.
  In this paper, we aim to directly optimize the AUC score for high-dimensional
sparse datasets under online learning setting and propose a new algorithm,
\textsc{FTRL-AUC}. Our proposed algorithm can process data in an online fashion
with a much cheaper per-iteration cost $\mathcal{O}(k)$, making it amenable for
high-dimensional sparse streaming data analysis. Our new algorithmic design
critically depends on a novel reformulation of the U-statistics AUC objective
function as the empirical saddle point reformulation, and the innovative
introduction of the "lazy update" rule so that the per-iteration complexity is
dramatically reduced from $\mathcal{O}(d)$ to $\mathcal{O}(k)$. Furthermore,
\textsc{FTRL-AUC} can inherently capture sparsity more effectively by applying
a generalized Follow-The-Regularized-Leader (FTRL) framework.
  Experiments on real-world datasets demonstrate that \textsc{FTRL-AUC}
significantly improves both run time and model sparsity while achieving
competitive AUC scores compared with the state-of-the-art methods. Comparison
with the online learning method for logistic loss demonstrates that
\textsc{FTRL-AUC} achieves higher AUC scores especially when datasets are
imbalanced.
