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We seek to understand fundamental tradeoffs between the accuracy of priorinformation that a learner has on a given problem and its learning performance.We introduce the notion of prioritized risk, which differs from traditionalnotions of minimax and Bayes risk by allowing us to study such fundamentaltradeoffs in settings where reality does not necessarily conform to thelearner's prior. We present a general reduction-based approach for extendingclassical minimax lower-bound techniques in order to lower bound theprioritized risk for statistical estimation problems. We also introduce a novelgeneralization of Fano's inequality (which may be of independent interest) forlower bounding the prioritized risk in more general settings involvingunbounded losses. We illustrate the ability of our framework to provideinsights into tradeoffs between prior information and learning performance forproblems in estimation, regression, and reinforcement learning.

Fundamental Tradeoffs in Learning with Prior Information