Learning to Stop with Surprisingly Few Samples

TL;DR

This paper demonstrates that in a discounted optimal stopping problem, a minimal exploration phase—sometimes just a single sample—can enable near-optimal decisions, challenging the belief that extensive exploration is necessary.

Contribution

It shows that a short or even single-sample exploration phase is sufficient for near-optimal stopping decisions, even with unknown distributions.

Findings

Logarithmic exploration duration suffices for performance close to full information.

Single-sample exploration can achieve near-optimal results in heavy-tailed cases.

Proper tuning of exploration-exploitation balance is crucial for success.

Abstract

We consider a discounted infinite horizon optimal stopping problem. If the underlying distribution is known a priori, the solution of this problem is obtained via dynamic programming (DP) and is given by a well known threshold rule. When information on this distribution is lacking, a natural (though naive) approach is "explore-then-exploit," whereby the unknown distribution or its parameters are estimated over an initial exploration phase, and this estimate is then used in the DP to determine actions over the residual exploitation phase. We show: (i) with proper tuning, this approach leads to performance comparable to the full information DP solution; and (ii) despite common wisdom on the sensitivity of such "plug in" approaches in DP due to propagation of estimation errors, a surprisingly "short" (logarithmic in the horizon) exploration horizon suffices to obtain said performance. In cases where the underlying distribution is heavy-tailed, these observations are even more pronounced: a ${\it single \, sample}$ exploration phase suffices.