american options - Foresight bias in least square monte carlo

Foresight bias means we tend to over estimate the American option value. This we observe in other areas of statistics - e.g. in sample test almost always gives better prediction than out of sample tests in linear regression models.

My question is: is it known (theoretically or empirically) such bias is reduced by increasing sample size in least square monte carlo algorithm?

I have done some simulation myself. From my simulation, at least for the models I have worked on, the answer seems to be yes.

Presumably this is simply because bigger sample tends to mean the sampling distribution of the estimator has smaller variance blah blah blah?

Is there any theoretical results or empirical studies on this?

Edit: foresight bias is also known as look-ahead bias