Popular asymmetric loss functions for bounding or constructing loss functions to guarantee bounding?

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I'm working on a open-ended project for my Masters using the Online Encyclopedia of Integer Sequences and attempting to upper bound future sequence values instead of predicting them. I've managed to get reasonable results using RNNs and Linear-Exponential (LINEX) loss but was wondering what other popular asymmetric loss functions for bounding there are or what general principles I should consider in constructing a loss function if my aim is to ensure all predicted bounds are deterministically (and not probabilistically) sufficient, if possible?


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