What happens when error cannot be computed (is infinite) with leave-one-out CV?

[tadworthington]

I thought of this while working on the homework for the class. Let's say I have three points: (-1,0), (1,0), and (1,1). I want to use a linear model (h(x) = mx + b) to do the fitting, and I use LOO to check my cross validation error. The problem becomes apparent right away:

Leave out (-1,0), and fit (1,0), (1,1).  Fitting gives a vertical line, x = 1.

Of course, I am now unable to compute the squared error for the point (-1,0) that was left out - the error will be infinite.

Is the solution that I can't choose a vertical line (x = k, for some k) when fitting the data?

[tadworthington]

I think my question didn't make sense. Of course I can't get a vertical line when producing a hypothesis of the form h(x) = mx + b.

Too bad there is no delete on the forum

[magdon]

As posed, the LOO error is indeed not defined (infinite) however, you question is interesting when your last data point is (say) .

By choosing appropriately small, you can make the LOO error arbitrarily large.

However, there is no problem with that; remember that your LOO error is an estimate of your when learning with N-1 points. If your distribution can generate the two points and with high probability (which is verified by the very existence of this data set) then indeed, the out-of-sample error you should expect when learning from 2 data points is very large.

Originally Posted by tadworthington

I thought of this while working on the homework for the class. Let's say I have three points: (-1,0), (1,0), and (1,1). I want to use a linear model (h(x) = mx + b) to do the fitting, and I use LOO to check my cross validation error. The problem becomes apparent right away:

Leave out (-1,0), and fit (1,0), (1,1).  Fitting gives a vertical line, x = 1.

Of course, I am now unable to compute the squared error for the point (-1,0) that was left out - the error will be infinite.

Is the solution that I can't choose a vertical line (x = k, for some k) when fitting the data?