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We investigate three related and important problems connected to machinelearning: approximating a submodular function everywhere, learning a submodularfunction (in a PAC-like setting [53]), and constrained minimization ofsubmodular functions. We show that the complexity of all three problems dependson the 'curvature' of the submodular function, and provide lower and upperbounds that refine and improve previous results [3, 16, 18, 52]. Our prooftechniques are fairly generic. We either use a black-box transformation of thefunction (for approximation and learning), or a transformation of algorithms touse an appropriate surrogate function (for minimization). Curiously, curvaturehas been known to influence approximations for submodular maximization [7, 55],but its effect on minimization, approximation and learning has hitherto beenopen. We complete this picture, and also support our theoretical claims byempirical results.

Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions