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diff --git a/general.tex b/general.tex index 23145ad..595023d 100644 --- a/general.tex +++ b/general.tex @@ -13,7 +13,7 @@ This optimization, commonly known as \emph{ridge regression}, includes an additi Let $\entropy(\beta)$ be the entropy of $\beta$ under this distribution, and $\entropy(\beta\mid y_S)$ the entropy of $\beta$ conditioned on the experiment outcomes $Y_S$, for some $S\subseteq \mathcal{N}$. In this setting, a natural objective to select a set of experiments $S$ that maximizes her \emph{information gain}: $$ I(\beta;y_S) = \entropy(\beta)-\entropy(\beta\mid y_S). $$ -Assuming normal noise variables, the information gain is equal (upto a constant) to the following value function \cite{chaloner1995bayesian}: +Assuming normal noise variables, the information gain is equal (up to a constant) to the following value function \cite{chaloner1995bayesian}: \begin{align} V(S) = \frac{1}{2}\log\det(R + \T{X_S}X_S)\label{bayesianobjective} \end{align} @@ -21,7 +21,54 @@ Our objective \eqref{,,,} clearly follows from \eqref{bayesianobjective} by sett Moreover, our results can be extended to the general Bayesian case, by replacing $I_d$ with the positive semidefinite matrix $R$: -TODO: state theorem, discuss dependence on $\det R$. +\thibaut{Discussion about the value function below} + +When there is an $R$ in the value function, it seems to make more sense to +study the modified value function: +\begin{displaymath} + \tilde{V}(S) = \frac{1}{2}\log\det(R + \T{X_S}X_S) - \frac{1}{2}\log\det R +\end{displaymath} +For two reasons: +\begin{itemize} + \item $\tilde{V}(\emptyset) = 0$: the value function is normalized, I think + this assumption is needed somewhere in mechanism design. + \item $\tilde{V}(S) = \frac{1}{2}\log\det(I_d + R^{-1}\T{X_S}X_S)$, so we + can apply our result to get an $\alpha$ approximation ratio (see the value + of $\alpha$ below). If we take $V$ instead of $\tilde{V}$ then one can + write: + \begin{displaymath} + V(S) = \frac{1}{2}\log\det R + \tilde{V}(S) + \end{displaymath} + thus: + \begin{displaymath} + OPT(V) = \frac{1}{2}\log\det R + OPT(\tilde{V}) + \end{displaymath} + we can find $S^*$ such that $OPT(\tilde{V}) \leq \alpha \tilde{V}(S)$, so: + \begin{displaymath} + OPT(V) = \frac{1}{2}\log\det R + \alpha\tilde{V}(S) + \end{displaymath} + But this does not give an $\alpha$ approximation ratio for $V$, because + $\log\det R$ can be negative. This is only an \emph{asymptotic} + approximation ratio\ldots. +\end{itemize} + +\begin{theorem} + For the function $\tilde{V}$ defined above, there is a truthful, budget + feasible mechanism which achieves an approximation ratio of: + \begin{displaymath} + \frac{5e-1}{e-1}\frac{\log(1+\mu)}{\mu} + A + \end{displaymath} +where $\mu$ is the smallest eigenvalue of $R$ (and $A$ is a constant that +I will compute tomorrow, it should be roughly around 10). +\end{theorem} + +Note that everything becomes nice when $R \geq I_d$. In this case, the smallest +eigenvalue is larger than 1. Hence $\log\det R\geq 1$ and an approximation on +$\tilde{V}$ gives an approximation ration on $V$ (see discussion above). +Furthermore, we can bound $\log(1+\mu)/\mu$ by 1 and I think we fall back on +the approximation ratio of section 2. + +Can we motivate that $R\geq 1$ ? \subsection{Beyond Linear Models} TODO: Independent noise model. Captures models such as logistic regression, classification, etc. Arbitrary prior. Show that change in the entropy is submodular (cite Krause, Guestrin). |