Fast Rates for Online Prediction with Abstention

TL;DR

This paper demonstrates that allowing a small abstention cost in online binary sequence prediction enables regret bounds independent of the time horizon, with precise dependence on abstention cost and number of experts.

Contribution

It provides the first exact characterization of regret bounds with abstention, showing a significant improvement over traditional rates without abstention.

Findings

Regret bounds are independent of the time horizon when abstention cost is below 0.5.

Matching upper and lower bounds of order (log N)/(1-2c) are established.

Extensions include regret bounds for changing abstention costs over time.

Abstract

In the setting of sequential prediction of individual $\{0, 1\}$-sequences with expert advice, we show that by allowing the learner to abstain from the prediction by paying a cost marginally smaller than $\frac 12$ (say, $0.49$), it is possible to achieve expected regret bounds that are independent of the time horizon $T$. We exactly characterize the dependence on the abstention cost $c$ and the number of experts $N$ by providing matching upper and lower bounds of order $\frac{\log N}{1-2c}$, which is to be contrasted with the best possible rate of $\sqrt{T\log N}$ that is available without the option to abstain. We also discuss various extensions of our model, including a setting where the sequence of abstention costs can change arbitrarily over time, where we show regret bounds interpolating between the slow and the fast rates mentioned above, under some natural assumptions on the sequence of abstention costs.