Ranking methods or models based on their performance is of prime importance but is tricky because performance is fundamentally multidimensional. In the case of classification, precision and recall are scores with probabilistic interpretations that are both important to consider and complementary. The rankings induced by these two scores are often in partial contradiction. In practice, therefore, it is extremely useful to establish a compromise between the two views to obtain a single, global ranking. Over the last fifty years or so,it has been proposed to take a weighted harmonic mean, known as the F-score, F-measure, or $F_β$. Generally speaking, by averaging basic scores, we obtain a score that is intermediate in terms of values. However, there is no guarantee that these scores lead to meaningful rankings and no guarantee that the rankings are good tradeoffs between these base scores. Given the ubiquity of $F_β$ scores in the literature, some clarification is in order. Concretely: (1) We establish that $F_β$-induced rankings are meaningful and define a shortest path between precision- and recall-induced rankings. (2) We frame the problem of finding a tradeoff between two scores as an optimization problem expressed with Kendall rank correlations. We show that $F_1$ and its skew-insensitive version are far from being optimal in that regard. (3) We provide theoretical tools and a closed-form expression to find the optimal value for $β$ for any distribution or set of performances, and we illustrate their use on six case studies.

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.

Artificial Neural-Network-based (ANN-based) lossy compressors have recently obtained striking results on several sources. Their success may be ascribed to an ability to identify the structure of low-dimensional manifolds in high-dimensional ambient spaces. Indeed, prior work has shown that ANN-based compressors can achieve the optimal entropy-distortion curve for some such sources. In contrast, we determine the optimal entropy-distortion tradeoffs for two low-dimensional manifolds with circular structure and show that state-of-the-art ANN-based compressors fail to optimally compress them.