Figure 1: Offline Model-Based Optimization (MBO): The goal of offline MBO is to optimize an unknown objective function with respect to, provided access to only as static, previously-collected dataset of designs. Machine learning methods have shown tremendous promise on prediction problems: predicting the efficacy of a drug, predicting how a protein will fold, or predicting the strength of a composite material. But can we use machine learning for design? Conventionally, such problems have been tackled with black-box optimization procedures that repeatedly query an objective function. For instance, if designing a drug, the algorithm will iteratively modify the drug, test it, then modify it again.

We tackle some fundamental problems in probability theory on corrupted random processes on the integer line. We analyze when a biased random walk is expected to reach its bottommost point and when intervals of integer points can be detected under a natural model of noise. We apply these results to problems in learning thresholds and intervals under a new model for learning under adversarial design.