We show that in pool-based active classification without assumptions on the underlying distribution, if the learner is given the power to abstain from some predictions by paying the price marginally smaller than the average loss $1/2$ of a random guess, exponential savings in the number of label requests are possible whenever they are possible in the corresponding realizable problem. We extend this result to provide a necessary and sufficient condition for exponential savings in pool-based active classification under the model misspecification.