  A major challenge for building statistical models in the big data era is that
the available data volume far exceeds the computational capability. A common
approach for solving this problem is to employ a subsampled dataset that can be
handled by available computational resources. In this paper, we propose a
general subsampling scheme for large-scale multi-class logistic regression and
examine the variance of the resulting estimator. We show that asymptotically,
the proposed method always achieves a smaller variance than that of the uniform
random sampling. Moreover, when the classes are conditionally imbalanced,
significant improvement over uniform sampling can be achieved. Empirical
performance of the proposed method is compared to other methods on both
simulated and real-world datasets, and these results match and confirm our
theoretical analysis.
