# The bias of the bootstrap estimator

From what I gather, a Bootstrap estimation of the generalization error for a ML procedure is optimistically biased, e.g.:

As far as I understand it, the source of the optimism is that with bootstrap we are testing on much of the data that we are training with.

I presume the bias depends on what we are estimating, so there is no universal formula for it (e.g. MSE of a particular regressor). Is this correct?

If so, are there still any known universal lower or upper theoretical bounds of the bias of the bootstrap estimator? Or does it fully depend on the generalization error that we are estimating and e.g. the underlying ML model?