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We consider a class of discrete optimization problems that aim to maximize a submodular objective function subject to a distributed partition matroid constraint. More precisely, we consider a networked scenario in which multiple agents choose actions from local strategy sets with the goal of maximizing a submodular objective function defined over the set of all possible actions. Given this distributed setting, we develop Constraint-Distributed Continuous Greedy (CDCG), a message passing algorithm that converges to the tight (1-1/e) approximation factor of the optimum global solution using only local computation and communication. It is known that a sequential greedy algorithm can only achieve a 1/2 multiplicative approximation of the optimal solution for this class of problems in the distributed setting. Our framework relies on lifting the discrete problem to a continuous domain and developing a consensus algorithm that achieves the tight (1-1/e) approximation guarantee of the global discrete solution once a proper rounding scheme is applied. We also offer empirical results from a multi-agent area coverage problem to show that the proposed method significantly outperforms the state-of-the-art sequential greedy method.
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Learning to make decisions via submodular regularization
Many sequential decision making tasks can be viewed as combinatorial optimiza- tion problems over a large number of actions. When the cost of evaluating an ac- tion is high, even a greedy algorithm, which iteratively picks the best action given the history, is prohibitive to run. In this paper, we aim to learn a greedy heuris- tic for sequentially selecting actions as a surrogate for invoking the expensive oracle when evaluating an action. In particular, we focus on a class of combinato- rial problems that can be solved via submodular maximization (either directly on the objective function or via submodular surrogates). We introduce a data-driven optimization framework based on the submodular-norm loss, a novel loss func- tion that encourages the resulting objective to exhibit diminishing returns. Our framework outputs a surrogate objective that is efficient to train, approximately submodular, and can be made permutation-invariant. The latter two properties al- low us to prove strong approximation guarantees for the learned greedy heuristic. Furthermore, our model is easily integrated with modern deep imitation learning pipelines for sequential prediction tasks. We demonstrate the performance of our algorithm on a variety of batched and sequential optimization tasks, including set cover, active learning, and data-driven protein engineering.
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- Award ID(s):
- 1645832
- PAR ID:
- 10330320
- Date Published:
- Journal Name:
- International Conference on Learning Representations
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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