Decision-Oriented Learning Using Differentiable Submodular Maximization for Multi-Robot Coordination

TL;DR

This work addresses cost-aware decision making in multi-robot routing where the objective combines a monotone submodular performance term and context-dependent action costs, modeled as $g(S,w)=f(S)-\lambda c(S,w)$ with $c(S,w)=w^{\top}x$. It introduces a decision-oriented learning (DOL) framework that differentiates through a non-monotone submodular maximization solver by proposing the Differentiable Cost-Scaled Greedy (D-CSG) via the multilinear extension, enabling end-to-end training. The main contributions are formulating the decision-oriented loss $\ell_{DOL}$, developing D-CSG, and demonstrating that DOL yields better downstream decisions than the traditional two-stage approach, especially with limited training data. The approach offers a practical, interpretable method for improving cost prediction in energy-constrained multi-robot coordination and related planning problems.

Abstract

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.

Figures (7)

• Figure 1: The motivating case study of energy-aware multi-robot coordination. A team of robots needs to cover a task area and for every planning epoch, they need to trade off the area they will cover and the energy cost, which depends on the weather and terrain conditions.
• Figure 2: The proposed framework that incorporates the non-monotone submodular maximization into the learning process.
• Figure 3: Computational graph of the proposed differentiable algorithm. (a) The structure of the algorithm.
• Figure 4: Computational graph of the proposed differentiable algorithm. The internal structure of the differentiable cost-scaled greedy operation.
• Figure 5: A qualitative example to show how DOL is different from the two-stage approach. (a) Ground truth data and the optimal decision boundary. (b) Learned linear models using MSE loss. (c) Learned linear models using the DOL framework.

Theorems & Definitions (2)

• Definition 1: Submodularity
• Remark 1