One sided approximation degree Given a boolean function $f:\{0,1\}^n\rightarrow\{0,1\}$, let $P_{i,\epsilon}$ be minimum multivariate polynomial such that $P_{i,\epsilon}=i\iff f=i$,  $P_{i,\epsilon}\in(i-\epsilon,i+\epsilon)\iff f=1-i$ where $\epsilon\in(0,1)$ at each $i\in\{0,1\}$.

Is atleast for one of $i\in\{0,1\}$,  degrees of $P_{i,\epsilon},P_{i,\epsilon/2}$ polynomially always related? They should be, however I cannot find a reference.

Straight forward interpolation arguments seems to fail as $\epsilon$ gets closer to 1 (but not exactly $1$).
I don't think they are polynomially related. Consider the OR function on n bits. There is a polynomial $p(x) = \frac{1}{n} (x_1 + x_2 + \ldots + x_n)$, which approximates it with one sided error $\epsilon = 1- \frac{1}{n}$ because on the all zero input the polynomial is zero, and on any other input its value is between $\frac{1}{n}$ and 1.

However any polynomial approximating the OR function to one sided error $\epsilon/2$ (which is approximately $1/2$) is a bounded-error approximation of OR which is known to require degree $\Omega(\sqrt{n})$.