We seek to understand fundamental tradeoffs between the accuracy of prior information that a learner has on a given problem and its learning performance. We introduce the notion of prioritized risk, which differs from traditional notions of minimax and Bayes risk by allowing us to study such fundamental tradeoffs in settings where reality does not necessarily conform to the learner's prior. We present a general reduction-based approach for extending classical minimax lower-bound techniques in order to lower bound the prioritized risk for statistical estimation problems. We also introduce a novel generalization of Fano's inequality (which may be of independent interest) for lower bounding the prioritized risk in more general settings involving unbounded losses. We illustrate the ability of our framework to provide insights into tradeoffs between prior information and learning performance for problems in estimation, regression, and reinforcement learning.

翻译：我们旨在理解学习者在给定问题上拥有的先验信息准确性与其学习性能之间的基本权衡。我们引入了“优先风险”的概念，该概念不同于传统的最小最大风险和贝叶斯风险，允许我们在现实不一定符合学习者先验假设的情况下研究此类基本权衡。我们提出了一种通用的基于归约的方法，用于扩展经典的最小最大下界技术，从而为统计估计问题中的优先风险设定下界。我们还引入了范诺不等式的一种新颖推广（可能具有独立意义），用于在涉及无界损失的更一般情境中为优先风险设定下界。我们通过实例展示了该框架在为估计、回归和强化学习问题中先验信息与学习性能之间的权衡提供深刻洞见方面的能力。