  We present a differentiable, decision-oriented learning framework for cost
prediction in a class of multi-robot decision-making problems, in which the
robots need to trade off the task performance with the costs of taking actions
when they select actions to take. Specifically, we consider the cases where the
task performance is measured by a known monotone submodular function (e.g.,
coverage, mutual information), and the cost of actions depends on the context
(e.g., wind and terrain conditions). We need to learn a function that maps the
context to the costs. Classically, we treat such a learning problem and the
downstream decision-making problem as two decoupled problems, i.e., we first
learn to predict the cost function without considering the downstream
decision-making problem, and then use the learned function for predicting the
cost and using it in the decision-making problem. However, the loss function
used in learning a prediction function may not be aligned with the downstream
decision-making. We propose a decision-oriented learning framework that
incorporates the downstream task performance in the prediction phase via a
differentiable optimization layer. The main computational challenge in such a
framework is to make the combinatorial optimization, i.e., non-monotone
submodular maximization, differentiable. This function is not naturally
differentiable. We propose the Differentiable Cost Scaled Greedy algorithm
(D-CSG), which is a continuous and differentiable relaxation of CSG. We
demonstrate the efficacy of the proposed framework through numerical
simulations. The results show that the proposed framework can result in better
performance than the traditional two-stage approach.
