Suppose we are given a (possibly large) dataset $X \subset \mathcal{X}$ for some complete, separable space $\mathcal{X}$, and a space $\mathcal{Y}$ for which completeness and separability are not guaranteed. Let $f: \mathcal{X} \times \mathcal{Y} \to \mathbb{R}$, and assume $f(\cdot) \in [0, b]$. Suppose we are interested in solving
$$ \min\limits_{y \in \mathcal{Y}} f(X, y) $$
Instead, suppose we use the following batched version:
$$ \min\limits_{y \in \mathcal{Y}} \max\limits_{\substack{x_i \subset X \\\\ \cup_i x_i = X}} f(x_i, y) $$
That is, we form a partition of $X$ and compute the upper bounds for each subset in the partition. We then minimize these upper bounds. Is there a way to bound the difference between these two?
Specifically, I wonder if the standard covering number guarantee from Duchi, Example 4.3.9, or a variant thereof, could be relevant here. Since $f(\cdot) \in [0, b]$, it is also in $[-b, b]$, which seems to be a condition for 4.3.9. The key difference is that we are not interested in bounding $\sup \lvert P_n f - Pf \rvert$.
