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Let's say we're doing regression over $[0,1]^d$ -- either in the PAC sense with bounded-range agnostic noise or in the more classical-statistics sense with additive Gaussian noise. Suppose further that an additional hindrance stands between the data and the learner: The space $[0,1]^d$ is divided into a grid of cubes with side length $\eta$, and rather than seeing the full labeled sample $(X_i,Y_i)$, the learner only knows how many points $X$ fell into each of the grid cubes and the average of their $Y$ values in that cube.

Questions: 1. Terminology: Is there a name for this learning setting? "Summary" or "aggregate" statistics come to mind, but I want to be consistent with the known literature. 2. Prior results: Either in the PAC or the the additive-noise setting, is anything known? Efficient algorithms, minimax rates?

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The related work I know of is experimental, but a similar setup has been called Learning from Label Proportions: https://link.springer.com/article/10.1007/s13278-017-0478-6

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