# What is the trade off between having a larger validation set versus a smaller one?

• Suppose I am comparing several models, e,g, $$\{ax\}$$, $$\{ax+b\}$$ and $$\{ax^2 + bx + c\}$$, $$\{ax^3\}$$ on data set $$\mathcal{D} = \{x_i,y_i\}_{i = 1}^N$$

I partition $$\mathcal{D}$$ into training set ($$N-K$$ points) and validation set ($$K$$ points).

I first train my models on the training set to obtain hypothesis

e.g., $$h_1 = a^\star x$$, $$h_2 = a^\star x + b^\star$$, $$h_3 = a^\star x^2 + b^\star x + c^\star, \ldots$$

Then I run my hypothesis $$h_i$$ on the validation set to obtain the error, and choose the hypothesis with the smallest validation error.

What is the tradeoff between having a larger $$K$$ versus a smaller $$K$$?

I considered the extreme cases when $$K = 0$$ and $$K = N$$.

For $$K = 0$$, I am picking my hypothesis directly from the training set. Hence I might have more overfitting, and worse out of sample performance

For $$K = N$$, I pick each hypothesis from my training set at random, then sending all these random hypothesis through the validation set to obtain the minimum error. Hence I might miss out on the best hypothesis?

I can't seem to put this comparison into words. Can anyone help