In this work, we approach the problem of multi-task submodular optimization
with the perspective of local distributional robustness, within the
neighborhood of a reference distribution which assigns an importance score to
each task. We initially propose to introduce a regularization term which makes
use of the relative entropy to the standard multi-task objective. We then
demonstrate through duality that this novel formulation itself is equivalent to
the maximization of a submodular function, which may be efficiently carried out
through standard greedy selection methods. This approach bridges the existing
gap in the optimization of performance-robustness trade-offs in multi-task
subset selection. To numerically validate our theoretical results, we test the
proposed method in two different setting, one involving the selection of
satellites in low Earth orbit constellations in the context of a sensor
selection problem, and the other involving an image summarization task using
neural networks. Our method is compared with two other algorithms focused on
optimizing the performance of the worst-case task, and on directly optimizing
the performance on the reference distribution itself. We conclude that our
novel formulation produces a solution that is locally distributional robust,
and computationally inexpensive.

在本研究中，我们通过引入基于本地分布鲁棒性的参考分布邻域，提出了一种多任务子模块优化问题的方法。我们首先建议引入一个正则化项，利用相对熵与标准多任务目标之间的关系。然后通过对偶性证明这个新的公式本身等同于最大化一个子模块函数，这可以通过标准的贪婪选择方法高效实现。这种方法填补了多任务子集选择中性能 - 鲁棒性权衡优化的现有差距。为了数值验证我们的理论结果，我们在两种不同的情况下进行了测试，一种涉及对地球低轨道星座中的卫星进行选择以解决传感器选择问题，另一种涉及使用神经网络进行图像摘要任务。我们将我们的方法与两种其他算法进行了比较，这两种算法侧重于最坏情况下任务的性能优化和直接优化参考分布本身的性能。我们得出结论，我们的新公式提供了一种局部分布鲁棒且计算费用低廉的解决方案。