Optimal Approximation - Smoothness Tradeoffs for Soft-Max Functions

A soft-max function has two main efficiency measures: (1) approximation - which corresponds to how well it approximates the maximum function, (2) smoothness - which shows how sensitive it is to changes of its input. Our goal is to identify the optimal approximation-smoothness tradeoffs for different measures of approximation and smoothness. This leads to novel soft-max functions, each of which is optimal for a different application. The most commonly used soft-max function, called exponential mechanism, has optimal tradeoff between approximation measured in terms of expected additive approximation and smoothness measured with respect to Renyi Divergence. We introduce a soft-max function, called "piecewise linear soft-max", with optimal tradeoff between approximation, measured in terms of worst-case additive approximation and smoothness, measured with respect to $\ell_q$-norm. The worst-case approximation guarantee of the piecewise linear mechanism enforces sparsity in the output of our soft-max function, a property that is known to be important in Machine Learning applications [Martins et al. '16, Laha et al. '18] and is not satisfied by the exponential mechanism. Moreover, the $\ell_q$-smoothness is suitable for applications in Mechanism Design and Game Theory where the piecewise linear mechanism outperforms the exponential mechanism. Finally, we investigate another soft-max function, called power mechanism, with optimal tradeoff between expected multiplicative approximation and smoothness with respect to the Renyi Divergence, which provides improved theoretical and practical results in differentially private submodular optimization.