Fundamental Tradeoffs in Learning with Prior Information
This work addresses a foundational problem in machine learning theory for researchers and practitioners, offering a novel framework to analyze prior information tradeoffs, though it is incremental in extending existing lower-bound techniques.
The paper tackles the problem of understanding tradeoffs between the accuracy of a learner's prior information and its learning performance, introducing prioritized risk to study scenarios where reality may not align with the prior. It presents a reduction-based approach and a generalization of Fano's inequality to lower bound prioritized risk, applying the framework to estimation, regression, and reinforcement learning problems.
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.