Learning with non-modular losses is an important problem when sets of predictions are made simultaneously. The main tools for constructing convex surrogate loss functions for set prediction are margin rescaling and slack rescaling. In this work, we show that these strategies lead to tight convex surrogates iff the underlying loss function is increasing in the number of incorrect predictions. However, gradient or cutting-plane computation for these functions is NP-hard for non-supermodular loss functions. We propose instead a novel surrogate loss function for submodular losses, the Lovász hinge, which leads to O(p logp) complexity with O(p) oracle accesses to the loss function to compute a gradient or cutting-plane. We prove that the Lovász hinge is convex and yields an extension. As a result, we have developed the first tractable convex surrogates in the literature for submodular losses. We demonstrate the utility of this novel convex surrogate through several set prediction tasks, including on the PASCAL VOC and Microsoft COCO datasets.

中文翻译：

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Lovász铰链：亚模损耗的新型凸代理。

当同时进行多组预测时，非模块损耗的学习是一个重要的问题。构造用于集预测的凸代理损失函数的主要工具是余量重定标和松弛重定标。在这项工作中，我们表明，如果潜在的损失函数在不正确的预测中不断增加，则这些策略会导致紧密的凸替代。但是，对于非超模量损耗函数，这些函数的梯度或切平面计算是NP-hard的。相反，我们提出了一种用于子模损耗的新型代理损耗函数Lovász铰链，它导致O（p logp）复杂性，而O（p）oracle访问损耗函数以计算梯度或切割平面。我们证明Lovász铰链是凸的并产生延伸。因此，我们已经针对亚模损耗开发了文献中的第一个可处理的凸代理。我们通过包括PASCAL VOC和Microsoft COCO数据集在内的几个集合预测任务，演示了这种新颖的凸替代产品的实用性。