Review for NeurIPS paper: Optimal Approximation - Smoothness Tradeoffs for Soft-Max Functions

Summary and Contributions: A soft-max function has two main efficiency measures, approximation and smoothness. Authors goal is to identify the optimal approximation-smoothness tradeoffs for different measures of approximation and smoothness. They introduce a soft-max function, called piece-wise linear soft-max, with optimal tradeoff between approximation measured in terms of worst-case additive approximation, and smoothness measured with respect to l -norm. The worst-case approximation guarantee of the piece-wise linear mechanism enforces sparsity in the output of our soft-max function, a property that is known to be important in Machine Learning applications and is not satisfied by the exponential mechanism. Finally, they investigate another soft-max function, called power mechanism, with optimal tradeoff between expected multiplicative approximation and smoothness with respect to the Rényi Divergence, which provides improved theoretical and practical results in differentially private submodular optimization.