Conditional variance of the absolute sum of zero-mean i.i.d. random variables Why does the following conditional variance formula hold, and does it hold in general?
$$\text{Var} \left( \left \vert \sum^n_{i=1} U_i \right \vert \, \middle | \,D_n \right) = n \text{Var}(U_i \vert D_n).$$
Here $U_i$ are zero-mean independent and identically distributed random variables, and nothing more about their distribution is stated. Further information on $U_i$ and $D_n$ is provided below.
I am comfortable with the standard result that for i.i.d. random variables $X_i$,
$$\text{Var} \left( \sum^n_{i=1} X_i \right) = \sum^n_{i=1} \text{Var}(X_i) = n \text{Var}(X_i),$$
but the presence of the absolute values in the formula of interest has thrown me.

Further information on $U_i$ and $D_n$.
In response to the helpful comments and answer by James Martin and Ben, it is clear that the equation does not hold in general, and that additional context on the $U_i$ and $D_n$ is required. Here is an extract of the proof where the equation is used:




Where $D_n = {(X_1, Y_1,), \dots, (X_n, Y_n)}$ and $D_n' = (X_1', Y_1'), \dots (X_n', Y_n')$ are two sets of i.i.d. random variables sampled from unknown joint distribution $D_n, D_n' \overset{i.i.d}{\sim} P(X, Y)$. The conditional expectation in $U_i$ is such that $\mathbb{E}_{D_n' | D_n}[\cdot \vert D_n]$. $f: \mathbb{R}^d \rightarrow \{0, 1 \}$ is a classifier function , while $\tilde{f}$ is a random classifier function, dependent on $D_n$.
The argument comes from the proof of the Vapnik-Chervonenkis inequality in an expository note by Robert Nowak (2009), following the strategy of Devroye,  Györfi and Lugosi (1996) in their proof of Glivenko-Cantelli theorem.

This is a new answer looking at the later question supplemented by your additional information.  This new answer should be seen as augmenting my previous general observations in the other answer.
With the new information added to the question, you now have a specific distribution for the values under consideration.  Suppose we define the parameter $\theta = \mathbb{P}(\tilde{f}(X_i') \neq Y_i'|D_n)$ and note that this quantity is fixed for all $i$.$^\dagger$  From the information specified in the excerpt, we have the conditional distribution:
$$U_1,...,U_n | D_n \sim \text{IID Bern}(\theta) - \theta,$$
which gives $\sum_{i=1}^n U_i | D_n \sim \text{Bin}(n, \theta) - n \theta$.
Consequently, we get:
$$\begin{align}
\mathbb{E} \Bigg( \bigg| \sum_{i=1}^n U_i \bigg| \Bigg| D_n \Bigg)
&= \sum_{k=0}^n |k-n\theta| \cdot \text{Bin}(k|n, \theta) \\[6pt]
&= \sum_{k=0}^{\lfloor n \theta \rfloor} (n\theta-k) \cdot \text{Bin}(k|n, \theta) + \sum_{k=\lfloor n \theta \rfloor+1}^n (k-n\theta) \cdot \text{Bin}(k|n, \theta), \\[6pt]
\end{align}$$
and:
$$\begin{align}
\mathbb{V} \Bigg( \bigg| \sum_{i=1}^n U_i \bigg| \Bigg| D_n \Bigg)
&= \mathbb{E} \Bigg( \bigg( \sum_{i=1}^n U_i \bigg)^2 \Bigg| D_n \Bigg) - \mathbb{E} \Bigg( \bigg| \sum_{i=1}^n U_i \bigg| \Bigg| D_n \Bigg) ^2\\[6pt]
&= n \theta (1-\theta + n\theta) \\[6pt]
&\quad + \Bigg( \sum_{k=0}^{\lfloor n \theta \rfloor} (n\theta-k) \cdot \text{Bin}(k|n, \theta) + \sum_{k=\lfloor n \theta \rfloor+1}^n (k-n\theta) \cdot \text{Bin}(k|n, \theta) \Bigg)^2. \\[6pt]
\end{align}$$
The latter quantity is not generally equivalent to:
$$\mathbb{V} \Bigg( \sum_{i=1}^n U_i \Bigg| D_n \Bigg)
= n \mathbb{V}(U_i|D_n)
= n \theta (1-\theta),$$
so this step in the paper appears to be wrong to me.  Most likely, the author meant to assert the inequality $\mathbb{V}(| \sum U_i | |D_n) \geqslant n \mathbb{V}(U_i|D_n)$ here rather than asserting equivalence (which is wrong).  The overall proof still works if you use the inequality relation instead, so this is just one of those cases where a step in the proof is written incorrectly, but the proof itself still works.

$^\dagger$ The information specifies that the values $U_1,...,U_n | D_n$ are IID, which can only be the case if the quantity $\theta$ is fixed over all $i=1,...,n$.
