  Efficient sampling from constraint manifolds, and thereby generating a
diverse set of solutions for feasibility problems, is a fundamental challenge.
We consider the case where a problem is factored, that is, the underlying
nonlinear program is decomposed into differentiable equality and inequality
constraints, each of which depends only on some variables. Such problems are at
the core of efficient and robust sequential robot manipulation planning. Naive
sequential conditional sampling of individual variables, as well as fully joint
sampling of all variables at once (e.g., leveraging optimization methods), can
be highly inefficient and non-robust. We propose a novel framework to learn how
to break the overall problem into smaller sequential sampling problems.
Specifically, we leverage Monte-Carlo Tree Search to learn assignment orders
for the variable-subsets, in order to minimize the computation time to generate
feasible full samples. This strategy allows us to efficiently compute a set of
diverse valid robot configurations for mode-switches within sequential
manipulation tasks, which are waypoints for subsequent trajectory optimization
or sampling-based motion planning algorithms. We show that the learning method
quickly converges to the best sampling strategy for a given problem, and
outperforms user-defined orderings or fully joint optimization, while providing
a higher sample diversity.
