从真 Proper Scoring Rules 到最大-最小最优预测聚合
From Proper Scoring Rules to Max-Min Optimal Forecast Aggregation
AI总结：
本文建立适当评分规则与预测聚合之间的对应关系，提出关于评分规则的 QA 池化，证明其最大-最小最优性、凹性、低遗憾在线学习能力及公理化刻画。
摘要：
本文在两个看似无关的预测问题之间建立了紧密联系：激励相容的预测信息征集与预测聚合。适当评分规则（proper scoring rules）是解决前一问题的著名方案。对于每个这样的规则 $s$，我们给出一种相应的聚合方法，将专家预测和专家权重映射为一个“共识预测”，我们称之为关于 $s$ 的*拟算术（QA）池化*。我们从多个方面论证这种对应关系：关于两个研究最充分的评分规则（二次评分规则和对数评分规则）的 QA 池化，对应于两种研究最充分的预测聚合方法（线性方法和对数方法）。给定用于支付的评分规则 $s$，一个预测者智能体若将任务分包给若干专家，并按专家权重成比例地向其支付报酬，则该智能体使用关于 $s$ 的 QA 池化来聚合专家报告是最优的；这意味着该策略会在所有可能结果上最大化其最坏情形利润。使用 QA 池化的聚合者所得分数关于专家权重是凹的。因此，可以利用在线梯度下降从重复实验中学习合适的专家权重，并实现低遗憾。所有 QA 池化方法构成的类别还可由一组自然公理化条件刻画，这推广了 Kolmogorov 关于拟算术均值的经典工作。
英文摘要：
This paper forges a strong connection between two seemingly unrelated forecasting problems: incentive-compatible forecast elicitation and forecast aggregation. Proper scoring rules are the well-known solution to the former problem. To each such rule $s$ we associate a corresponding method of aggregation, mapping expert forecasts and expert weights to a "consensus forecast," which we call *quasi-arithmetic (QA) pooling* with respect to $s$. We justify this correspondence in several ways: - QA pooling with respect to the two most well-studied scoring rules (quadratic and logarithmic) corresponds to the two most well-studied forecast aggregation methods (linear and logarithmic). - Given a scoring rule $s$ used for payment, a forecaster agent who sub-contracts several experts, paying them in proportion to their weights, is best off aggregating the experts' reports using QA pooling with respect to $s$, meaning this strategy maximizes its worst-case profit (over the possible outcomes). - The score of an aggregator who uses QA pooling is concave in the experts' weights. As a consequence, online gradient descent can be used to learn appropriate expert weights from repeated experiments with low regret. - The class of all QA pooling methods is characterized by a natural set of axioms (generalizing classical work by Kolmogorov on quasi-arithmetic means).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
