Learning to Stop with Surprisingly Few Samples
This addresses the problem of optimal stopping under uncertainty for decision-makers, offering an incremental improvement by reducing exploration requirements compared to common wisdom.
The paper tackles the discounted infinite horizon optimal stopping problem when the underlying distribution is unknown, showing that an explore-then-exploit approach with proper tuning achieves performance comparable to the full information dynamic programming solution, requiring only a logarithmic exploration horizon in the horizon length or even a single sample for heavy-tailed distributions.
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.