From 6ea40a753fada423daff3db2d69bc3e58bb99d51 Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Mon, 28 Sep 2015 18:18:39 -0400
Subject: LT is submodular only in expectation

---
 notes/extensions.tex | 7 +++----
 1 file changed, 3 insertions(+), 4 deletions(-)

(limited to 'notes/extensions.tex')

diff --git a/notes/extensions.tex b/notes/extensions.tex
index 78cb336..cb78e35 100644
--- a/notes/extensions.tex
+++ b/notes/extensions.tex
@@ -41,8 +41,7 @@ $$2\sigma(a+b) \geq \sigma(a+b+c) + \sigma(a)$$
 
 We see that for $a=.5$, $,b=.5$, and $c=1$, the equality is violated.
 Interestingly however is that if we let the scale parameter of the sigmoid go to
-infinity, we recover the LT model which is submodular. In fact, by slowly
-increasing the scale parameter, it becomes harder to find values of $a, b, c$
-which violate the inequality. We therefore have a parameterized function which
-is `close to submodular'.
+infinity, it is harder to violate the inequality. (TH) note that the LT model is
+NOT submodular for every fixed value of thresholds, but only in expectation over
+the random draw of these thresholds.
 \end{document}
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