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.
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