what do you think of https://arxiv.org/abs/1608.05343 ?
A reasonable candidate for replacing backprop in deep learning: ADMM (alternating direction method of multipliers).
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how do you decompose the loss?
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You could also think about using Chambolle and Pock's primal dual https://hal.archives-ouvertes.fr/hal-00490826/document …
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Tom Goldstein's lab at UMD has been trying to get this to work for a while now. Heard from a friend who's been somewhat involved.
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Here's a recent paper on the topic https://arxiv.org/pdf/1605.02026v1.pdf …
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New conversation
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