From b2ae282bb842e3d61052fc9332038cd19aa0772b Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Tue, 28 Jun 2022 14:11:10 -0400
Subject: SIOPT 2021 140246, 1st revision

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+\documentclass[10pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage[hmargin=1in, vmargin=1in]{geometry}
+\usepackage{amsmath,amsfonts}
+
+\title{\large Review of \emph{The stochastic Auxiliary Problem Principle in Banach Spaces}\\
+First Revision}
+\date{}
+
+\begin{document}
+
+\maketitle
+
+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.
+
+%\bibliographystyle{plain}
+%\bibliography{main}
+
+\end{document}
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