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Let $\mathcal{X}$ denotes the input space of dimension $n$, $\mathcal{Y}$ denotes the codomain.

In PAC learning with realizability assumption, we assume randomness over covariates $\mathcal{D}_{\mathcal{X}}$ and assume the existence of a function that maps those covariates to the codomain $\mathcal{Y}$.

The definition adds efficiency as a constraint on the runtime of the learning algorithm $Poly \Bigl(n, \frac{1}{\epsilon}, \frac{1}{\delta} \Bigl)$.

Is there a study that shows efficiency for free in the case of learning only on the region of $\mathcal{X}$ where the distribution mass is high/concentrated?

It seems like the answer is negative because that's the goal of active learning: sampling from high probability regions (informative samples), if that's true, how does active learning reduce time complexity in passive learning ? But also, is there a passive learning version where the training data only comes from high probability regions?

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