# Comparative Learning: A Sample Complexity Theory for Two Hypothesis Classes

In many learning theory problems, a central role is played by a hypothesis class: we might assume that the data is labeled according to a hypothesis in the class (usually referred to as the realizable setting), or we might evaluate the learned model by comparing it with the best hypothesis in the class (the agnostic setting). Taking a step beyond these classic setups that involve only a single hypothesis class, we introduce comparative learning as a combination of the realizable and agnostic settings in PAC learning: given two binary hypothesis classes S and B, we assume that the data is labeled according to a hypothesis in the source class S and require the learned model to achieve an accuracy comparable to the best hypothesis in the benchmark class B. Even when both S and B have infinite VC dimensions, comparative learning can still have a small sample complexity. We show that the sample complexity of comparative learning is characterized by the mutual VC dimension 𝖵𝖢(S,B) which we define to be the maximum size of a subset shattered by both S and B. We also show a similar result in the online setting, where we give a regret characterization in terms of the mutual Littlestone dimension 𝖫𝖽𝗂𝗆(S,B). These results also hold for partial hypotheses. We additionally show that the insights necessary to characterize the sample complexity of comparative learning can be applied to characterize the sample complexity of realizable multiaccuracy and multicalibration using the mutual fat-shattering dimension, an analogue of the mutual VC dimension for real-valued hypotheses. This not only solves an open problem proposed by Hu, Peale, Reingold (2022), but also leads to independently interesting results extending classic ones about regression, boosting, and covering number to our two-hypothesis-class setting.

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