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# A Light Touch for Heavily Constrained SGD

; 29th Annual Conference on Learning Theory, PMLR 49:729-771, 2016.

#### Abstract

Minimizing empirical risk subject to a set of constraints can be a useful strategy for learning restricted classes of functions, such as monotonic functions, submodular functions, classifiers that guarantee a certain class label for some subset of examples, etc. However, these restrictions may result in a very large number of constraints. Projected stochastic gradient descent (SGD) is often the default choice for large-scale optimization in machine learning, but requires a projection after each update. For heavily-constrained objectives, we propose an efficient extension of SGD that stays close to the feasible region while only applying constraints probabilistically at each iteration. Theoretical analysis shows a compelling trade-off between per-iteration work and the number of iterations needed on problems with a large number of constraints.