From 0d6c68c9ff5dbfd48ceb0f7f1da55d6bff6d0b1e Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Thu, 10 Jan 2019 15:49:54 -0500
Subject: Add Solar lectures 2 and 3 in Roughgarden

---
 roughgarden.tex | 141 +++++++++++++++++++++++++++++++++++++++++++++++++++-----
 1 file changed, 129 insertions(+), 12 deletions(-)

(limited to 'roughgarden.tex')

diff --git a/roughgarden.tex b/roughgarden.tex
index 29a61df..422ea35 100644
--- a/roughgarden.tex
+++ b/roughgarden.tex
@@ -2,10 +2,11 @@
 \usepackage[T1]{fontenc}
 \usepackage[utf8]{inputenc}
 \usepackage[hmargin=1.2in, vmargin=1.2in]{geometry}
+\usepackage{amsfonts}
 
-\title{Review of \emph{Complexity Theory, Game Theory and Economics} (Tim
-Roughgarden)}
-\author{Thibaut Horel}
+\title{Review of \emph{Complexity Theory, Game Theory and Economics} by Tim
+Roughgarden}
+%\author{Thibaut Horel}
 
 \begin{document}
 
@@ -16,11 +17,16 @@ Roughgarden)}
 
 \section*{General impression} Very nice selection of topics, striking
 a nice balance between covering classical results and recent papers. The level
-of exposition feels right for a general theory audience.
+of exposition feels right for a general theory audience. My feedback is fairly
+local and mostly consists in suggestions for improving the exposition and typo
+fixes.
 
 \section*{Solar lecture 1}
 
 \begin{itemize}
+	\item Section 1.1: for organisational purposes, it might make sens to move
+		the plan to the foreword, or a separated ``Introduction'' or
+		``Preface'' chapter.
 
 	\item Section 1.2.3: I suggest also introducing the concept of \emph{pure
 		strategy} when defining a \emph{mixed strategy} below the payoff
@@ -31,12 +37,12 @@ of exposition feels right for a general theory audience.
 	the first equation to make it clear what the maximization is over.
 		Similarly, add $w, y$ below the $\min$ in the second equation.
 
-\item Section 1.2.6: check oppositional/cooperative. Furthermore, I think the
-	content of this section should be moved to the beginning of Section 1.4 to
-		motivate the transition to bimatrix games. At the moment, it feels that
-		it is breaking the flow.
-
-\item Section 1.2.7: add quote by Jain.
+\item Section 1.2.6: I think ``competitive'' is more standard than
+	``oppositional'' in the contex of games. Similarly, ``collaborative''
+		instead of ``cooperative'' in footnote 8. Furthermore, I think the
+		content of this section should be moved to the beginning of Section 1.4
+		to motivate the transition to bimatrix games. At the moment, it feels
+		that it is breaking the flow.
 
 \item Section 1.3.1: the first paragraph is slightly confusing, because it is
 	said that players only know their own payoffs (and not those of the other
@@ -67,8 +73,10 @@ of exposition feels right for a general theory audience.
 \item Section 1.3.7: the first sentence is unnecessary, the proof seems
 	complete to me.
 
-\item Example 1.13: Is this the ``battle of the sexes'' game?
-
+\item Example 1.13: If I am not mistaken, this game is commonly known as the
+	``Battle of the sexes'' game. Even though the name is questionable, it
+		might be worth mentioning so that people can recognize it in other
+		contexts.
 \end{itemize}
 
 \paragraph{Typos.} \begin{itemize}
@@ -79,7 +87,116 @@ of exposition feels right for a general theory audience.
 
 \section*{Solar lecture 2}
 
+\begin{itemize}
+	\item Section 2.3.2 Equation (2.2), I would simply write the first term as
+		$x_1'^TAx_2$: this is easier to read and more consistent with the
+		previous lecture.
+
+	\item Lemma 2.5: I would significantly expand the proof of Sperner's lemma
+		because at the moment it is really hard to even get an intuition of why
+		it is true. After looking it up it seems that a complete proof can be
+		given in a small amount of space: 1. first highlight that there has to
+		be an odd number of bichromatic edges through which you enter the
+		triangle (this is the dimension 1 case) 2. explain how you
+		orient the edges and walk the graph 3. explain why you are guaranteed
+		to end up at a sink node (sometimes you will exit the triangle instead
+		of finding one, but this removes two bichromatic edges so you can
+		iterate by induction).
+
+	\item Section 2.5: this is a minor thing, but I found the roadmap hard to
+		read and refer to later. I suggest maybe a bullet list with each item
+		starting with ``Step $i$'' ($i\in[4]$) clearly identified in bold.
+		Maybe using notations like $\epsilon$-BFP$\leq \epsilon$-NE although
+		not fully accurate could help quickly follow the chain of reductions.
+		Maybe a picture as well would help visualizing the chain of reductions?
+
+	\item Section 2.6: I found the oracle model confusing: why is it useful to
+		consider instances where the $pred$ and $succ$ functions might not be
+		consistent with each other (i.e. where $succ(v) = w$ but $pred(w) \neq
+		v$)? It seems that when this problem is used in the next lecture, it is
+		only used for the promise version of the problem where $pred$ and
+		$succ$ are guaranteed to be consistent. If it is not needed, then one
+		could get rid of cases (iii) and (iv) in the framed box. If it is
+		indeed needed, then I think some explanation of why would be nice.
+		
+		On the same point, the formulation is confusing because it says that
+		(for example) $succ(v)$ is NULL if there is no successor, but later on
+		in the same paragraph, we understand that it is allowed for
+		$succ(v)=w$, even when $v$ has no successor as long as $pred(w)\neq v$.
+		I think it is important to be very clear and distinguish between the
+		``true'' successor (or absence thereof) according to the real graph $G$
+		and the successor from the perspective of the $succ$ function which is
+		allowed to be inconsistent with the graph.
+\end{itemize}
+
 \paragraph{Typos.} \begin{itemize}
+	\item last paragraph of Section 2.1: Theorem 1.8 $\rightarrow$ Theorem 2.1
+	\item Theorem 2.2: space missing before $\mathbb{R}^d$
+\end{itemize}
+
+\section*{Solar lecture 3}
+
+\begin{itemize}
+	\item Section 3.1.2 and Section 3.1.4. An intuition which was useful for me
+		when trying to understand these sections and is already implicitly
+		there but might be worth making more explicit:
+		\begin{itemize}
+			\item $G$ is used to define a ``gradient field'' $g$ on the entire
+				hypercube by extrapolation. The extrapolation is designed such
+				that the sink nodes of $G$ and source nodes (different from
+				$0^n$) correspond to stationary points of the gradient field.
+			\item the function $f$ is defined such that finding a fixed point
+				of $f$ is equivalent to finding a stationary point of the
+				gradient field (this might be a very personal bias but I always
+				think of $f$ as expressing a ``gradient ascent'' update:
+				$x_{k+1} = x_k + g(x_k)$, for which fixed points correspond to
+				stationary points).
+			\item by factoring out the parameter $\delta$ out of the definition
+				of $g$ and defining $f(x) = x + \delta \cdot g(x)$ we can now
+				interpret $\delta$ as the ``step size'' of gradient ascent (in
+				any case, factoring $\delta$ out would make things a bit easier
+				to read).
+			\item it also follows from this gradient ascent perspective: that
+				one can indeed solve the EOL problem by ``following'' the
+				gradient field. The reduction basically shows that one cannot
+				beat gradient ascent to find stationary points\footnote{This
+				also made me wonder whether lower bounds on the query
+				complexity of finding stationary points and the fact that
+				gradient descent is optimal under reasonable assumptions could
+				have somehow been leveraged somewhere in these reductions}.
+		\end{itemize}
+	
+	\item Section 3.1.3: I know it is out of scope of this section to give all
+		the details, but I wish a little bit more could have been said about
+		the ``soft'' classification of points to one of the three categories:
+		how is the ``intermediate regime'' defined and why is it useful? Given
+		that it is referred in Section 3.1.4, I feel knowing a little bit more
+		would help.
+
+	\item Section 3.1.4: I think reordering the cases and using a bit more math
+		instead of plain English could make the description of $g$ easier to
+		read:
+		\begin{itemize}
+			\item for directed edge $e = (u, v)\in G$ define $\overrightarrow{e}
+				= \frac{\sigma(v)-\sigma(u)}{\|\sigma(v)-\sigma(u)\|}$ also
+				denote by $\overrightarrow{d}$ a default direction.
+			\item if $x$ is close to $\sigma(e)$ with $e\in G$, then
+				$g(x)=\overrightarrow{e}$
+			\item if $x$ is close to $v$ with incoming edge $(u,v)$ and
+				outgoing edge $(v, w)$, $g(x)$ interpolates between
+				$\overrightarrow{uv}$ and $\overrightarrow{vw}$.
+			\item if $x$ is close to a source or sink node $v$ with adjacent
+				edge $e$, $g(x)$ interpolates between $0^n$ and
+				$\overrightarrow{e}$.
+			\item in all other cases $g(x)=\overrightarrow{d}$.
+		\end{itemize}
+	
+	\item Section 3.2.2: small typo $H_\epsilon$ is a set of $(1/\epsilon)^d$
+		points (and not $(1/\epsilon)^n$).
+\end{itemize}
+
+\paragraph{Typos.}\begin{itemize}
+	\item Section 3.1.3 footnote 3: details,, $\rightarrow$ details,
 \end{itemize}
 
 \end{document}
-- 
cgit v1.2.3-70-g09d2

