Recall the definition: a set function f(S) is submodular, if for any subsets S S0 Z, and i Z S0, f(S {i}) f(S) f(S0 {i}) f(S0). For experiments in section 5.2, all checkpoints are instances of resnet-50. They are trained by a batch size of 128, and an initial learning rate of 0.1. We run for 200 epochs, with learning rate decay at the 60th, 120th and 160th epoch. A typical validation accuracy from these checkpoint (on its own task) is about 83% (reasonably good).

The idea is to first develop a prediction model without concern for the downstream risk profile or robustness guarantee, and then utilize calibration (or recalibration) methods to quantify the uncertainty of the prediction.

Finding models that satisfy these twoconditions ischallenging and current methods tendtotackle only one ofthe two. Exponential and implicit generative models have typically strong approximation properties (see e.g.

PSD studied probability have14] Notation.