An importance weight quantifies the relative importance of one example over
another, coming up in applications of boosting, asymmetric classification
costs, reductions, and active learning. The standard approach for dealing with
importance weights in gradient descent is via multiplication of the gradient.
We first demonstrate the problems of this approach when importance weights are
large, and argue in favor of more sophisticated ways for dealing with them. We
then develop an approach which enjoys an invariance property: that updating
twice with importance weight $h$ is equivalent to updating once with importance
weight $2h$. For many important losses this has a closed form update which
satisfies standard regret guarantees when all examples have $h=1$. We also
briefly discuss two other reasonable approaches for handling large importance
weights. Empirically, these approaches yield substantially superior prediction
with similar computational performance while reducing the sensitivity of the
algorithm to the exact setting of the learning rate. We apply these to online
active learning yielding an extraordinarily fast active learning algorithm that
works even in the presence of adversarial noise.

本文论述了在基于梯度的优化方法中，如何处理样本的重要性权重对优化过程的影响，提出了一种基于重要性权重的更新方法，并在在线主动学习中取得了显著的实验效果。