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\title{\large Review of \emph{The stochastic Auxiliary Problem Principle in Banach Spaces}\\
First Revision}
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I would like to thank the authors for their careful revision. I believe that my comments and suggestions have been adequately addressed. However, I have a few additional (minor) concerns about some of the new content (in blue) that was added due to Reviewer 2's concerns:
\begin{itemize}
\item page 22 line 887: I don't see why equation (4.13) implies that the sequence $\{\mathbb{E}(J(U_k) - J(u^\sharp))\}$ is bounded. Could it be that the authors simply meant “finite” instead of “bounded”? It seems that finiteness would suffice where boundedness is used (line 940 on page 23). I think the author should either write “finite” below (4.13) if it is indeed sufficient, or explain why boundedness holds and why it is required in line 940 on page 23.
\item page 24 equation (4.15): I believe $\beta_{k-1}$ should be $\beta_k$
\item page 24 equation (4.17): similarly, $\beta_{k-1}$ should be $\beta_k$
\item page 24 line 967: it is not obvious that the existence of the constant $M'$ is guaranteed by Corollary 4.3. Indeed, one could apply Corollary A.3 to equation (4.16) for fixed $n$ and $i$ to obtain the following proposition: \emph{for all $j\in \mathbb{N}$ there exists $M_j$ such that for all $k\geq j$, $\mathbb{E}(\ell_{U_j}(U_k))\leq M_j$}. But the constant $M_j$ could depend on $j$ \emph{a priori} and it seems that an additional argument is required to justify that the sequence $\{M_j\}$ is bounded by a universal constant $M'$ as claimed by the authors.
\end{itemize}
Overall, my positive impression of the paper remains and I recommend publication in SIOPT once the above points are clarified.
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