Behavioral Shaping for Geometric Concepts

In a search problem, an agent uses the membership oracle of a target concept to find a positive example of the concept. In a shaped search problem the agent is aided by a sequence of increasingly restrictive concepts leading to the target concept (analogous to behavioral shaping). The concepts are given by membership oracles, and the agent has to find a positive example of the target concept while minimizing the total number of oracle queries. We show that for the concept class of intervals on the real line an agent using a bounded number of queries per oracle exists. In contrast, for the concept class of unions of two intervals on the real line no agent with a bounded number of queries per oracle exists. We then investigate the (amortized) number of queries per oracle needed for the shaped search problem over other concept classes. We explore the following methods to obtain efficient agents. For axis-parallel rectangles we use a bootstrapping technique to obtain gradually better approximations of the target concept. We show that given rectangles R ⊆ A ⊆ Rd one can obtain a rectangle A′ ⊇ R with vol(A′)/vol(R) ≤ 2, using only O(d · vol(A)/vol(R)) random samples from A. For ellipsoids of bounded eccentricity in Rd we analyze a deterministic ray-shooting process which uses a sequence of rays to get close to the centroid. Finally, we use algorithms for generating random points in convex bodies (Dyer et al., 1991; Kannan et al., 1997) to give a randomized agent for the concept class of convex bodies. In the final section, we explore connections between our bootstrapping method and active learning. Specifically, we use the bootstrapping technique for axis-parallel rectangles to active learn axis-parallel rectangles under the uniform distribution in O(d ln(1/ε)) labeled samples.