  We consider the problem of leveraging prior experience to generate roadmaps
in sampling-based motion planning. A desirable roadmap is one that is sparse,
allowing for fast search, with nodes spread out at key locations such that a
low-cost feasible path exists. An increasingly popular approach is to learn a
distribution of nodes that would produce such a roadmap. State-of-the-art is to
train a conditional variational auto-encoder (CVAE) on the prior dataset with
the shortest paths as target input. While this is quite effective on many
problems, we show it can fail in the face of complex obstacle configurations or
mismatch between training and testing.
  We present an algorithm LEGO that addresses these issues by training the CVAE
with target samples that satisfy two important criteria. Firstly, these samples
belong only to bottleneck regions along near-optimal paths that are otherwise
difficult-to-sample with a uniform sampler. Secondly, these samples are spread
out across diverse regions to maximize the likelihood of a feasible path
existing. We formally define these properties and prove performance guarantees
for LEGO. We extensively evaluate LEGO on a range of planning problems,
including robot arm planning, and report significant gains over heuristics as
well as learned baselines.
