Optimal Stopping Problems in Low-Dimensional Feature Spaces: Lossless Conditions and Approximations

Optimal control problems can be solved by dy-namic programming. However, this method suffers from the curse of dimensionality. To resolve this, simplified versions of the original problem are often constructed in lower-dimensional feature spaces, leading to approximate policies. Yet, the connections between the original and the approximate policy and costs are rarely formalized. This paper addresses this challenge for optimal stopping problems. We start by providing conditions for lossless feature representations. This means that from an optimal policy obtained in feature space, an optimal policy in the original space can be constructed. Then, we search for modified versions of the original problem that (i) admit a lossless feature representation of far lower dimension; and (ii) provide upper and lower bounds on the optimal cost of the original problem. We can then use policies obtained in feature space using these modified problems to provide approximate policies for the original problem that are guaranteed to perform better than or equal to this aforementioned cost upper bound. We apply our tools in a high-dimensional precision farming intervention problem, where our tools allow for a dramatic decrease in complexity with only a small increase in the cost.