From 431085fd4bb4902922e1a4539e4ba4386537522f Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Mon, 5 Nov 2012 17:52:37 +0100
Subject: General

---
 general.tex | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

(limited to 'general.tex')

diff --git a/general.tex b/general.tex
index 46337d6..3815c83 100644
--- a/general.tex
+++ b/general.tex
@@ -45,7 +45,7 @@ analysis of the approximation ratio, we get the following result which extends
 Theorem~\ref{thm:main}:
 
 \begin{theorem}
- There exists a truthful and budget feasible mechanism for the objective
+ There exists a truthful, individually rational and budget feasible mechanism for the objective
  function $\tilde{V}$ given by \eqref{eq:normalized}. Furthermore, for any $\varepsilon
  > 0$, in time $O(\text{poly}(|\mathcal{N}|, d, \log\log \varepsilon^{-1}))$,
  the algorithm computes a set $S^*$ such that:
@@ -64,8 +64,8 @@ Selecting experiments that maximize the information gain in the Bayesian setup l
 where $h\in \mathcal{H}$ for some subset $\mathcal{H}$ of all possible mappings $h:\Omega\to\reals$, called the \emph{hypothesis space}, and $\varepsilon_i$ are random variables in $\reals$, not necessarily identically distributed, that are independent \emph{conditioned on $h$}. This model is quite broad, and captures many learning tasks, such as:
 \begin{enumerate}
 \item\textbf{Generalized Linear Regression.} $\Omega=\reals^d$,  $\mathcal{H}$ is the set of linear maps $\{h(x) = \T{\beta}x \text{ s.t. } \beta\in \reals^d\}$, and $\varepsilon_i$ are independent zero-mean normal variables, where  $\expt{\varepsilon_i^2}=\sigma_i$. 
-\item\textbf{Logistic Regression.} $\Omega=\reals^d$,  $\mathcal{H}$ is the set of maps $\{h(x) = \frac{e^{\T{\beta} x}}{1+e^{\T{\beta} x}} \text{ s.t. } \beta\in\reals^d\}$, and $\varepsilon_i$ are independent conditioned on $h$ such that $$\varepsilon_i=\begin{cases} 1- h(x_i),& \text{w.~prob.}~h(x_i)\\-h(x_i),&\text{w.~prob.}~1-h(x_i)\end{cases}$$
-\item\textbf{Learning Binary Functions with Bernoulli Noise.} $\Omega = \{0,1\}^d$, and $\mathcal{H}$ is some subset of $2^{\Omega\times\{0,1\}}$, and $$\varepsilon_i =\begin{cases}0, &\text{w.~prob.}~p\\\bar{h}(x_i)-h(x_i), \text{w.~prob.}~1-p\end{cases}$$
+\item\textbf{Logistic Regression.} $\Omega=\reals^d$,  $\mathcal{H}$ is the set of maps $\{h(x) = \frac{e^{\T{\beta} x}}{1+e^{\T{\beta} x}} \text{ s.t. } \beta\in\reals^d\}$, and $\varepsilon_i$ are independent conditioned on $h$ such that $$\varepsilon_i=\begin{cases} 1- h(x_i),& \text{w.~prob.}\;h(x_i)\\-h(x_i),&\text{w.~prob.}\;1-h(x_i)\end{cases}$$
+\item\textbf{Learning Binary Functions with Bernoulli Noise.} $\Omega = \{0,1\}^d$, and $\mathcal{H}$ is some subset of $2^{\Omega}$, and $$\varepsilon_i =\begin{cases}0, &\text{w.~prob.}\;p\\\bar{h}(x_i)-h(x_i), &\text{w.~prob.}\;1-p\end{cases}$$
 \end{enumerate}
 
 In this setup, assume that the experimenter has a prior distribution on the hypothesis $h\in \mathcal{H}$. Then, the information gain objective can be written again as the mutual information between $\beta$ and $y_S$.
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