Elicitation Complexity of Statistical Properties

A property, or statistical functional, is said to be elicitable if it minimizes expected loss for some loss function. The study of which properties are elicitable sheds light on the capabilities and limits of empirical risk minimization. While several recent papers have asked which properties are elicitable, we instead advocate for a more nuanced question: how many dimensions are required to indirectly elicit a given property? This number is called the elicitation complexity of the property. We lay the foundation for a general theory of elicitation complexity, including several basic results about how elicitation complexity behaves, and the complexity of standard properties of interest. Building on this foundation, we establish several upper and lower bounds for the broad class of Bayes risks. We apply these results by proving tight complexity bounds, with respect to identifiable properties, for variance, financial risk measures, entropy, norms, and new properties of interest. We then show how some of these bounds can extend to other practical classes of properties, and conclude with a discussion of open directions.