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\title[LATIN 2014]{Budget Feasible Mechanism\\for Experimental
Design}
\author[Thibaut Horel]{\textbf{Thibaut Horel}, Stratis Ioannidis, and S. Muthukrishnan}

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\newcommand{\ie}{\emph{i.e.}}
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\newcommand{\etal}{\emph{et al.}}
\newcommand{\reals}{\ensuremath{\mathbb{R}}}

\begin{document}

\maketitle

\begin{frame}{Motivation}
   \begin{center}
        \includegraphics<1>[scale=0.4]{stg-8.pdf}
        \includegraphics<2>[scale=0.4]{stg-7.pdf}
        \includegraphics<3>[scale=0.4]{stg-6.pdf}
        \includegraphics<4>[scale=0.4]{stg-5.pdf}
        \includegraphics<5>[scale=0.4]{stg-4.pdf}
        \includegraphics<6>[scale=0.4]{stg-3.pdf}
        \includegraphics<7>[scale=0.4]{stg-2.pdf}
        \includegraphics<8>[scale=0.4]{stg-1.pdf}
   \end{center}

\end{frame} 

\begin{frame}{Applications}
    \begin{itemize}
	\item<1-> Applications
            \begin{itemize}
		\item Medicine/Sociology
		\item Online surveys
		\item Data markets
	    \end{itemize}
\vspace*{1cm}
        \item<2-> Challenges
	     \begin{itemize}
	     	\item Which users are \alert{the most valuable}?
            \item What if users are \alert{strategic}?
        \end{itemize}
    \end{itemize}
\end{frame}

\begin{frame}{Contributions}
\pause
    \begin{itemize}
    \item Formulating the problem in the Experimental Design framework
            \pause
        \item Experimental design with \alert{strategic agents}?\pause
            \begin{itemize}
                \item Budget Feasible Mechanisms [Singer, 2010]
            \end{itemize}
    \end{itemize}
	\pause
	\vspace*{1cm}

    For Linear Regression:
	    \begin{itemize}
            \item deterministic, poly-time, truthful, budget
                feasible, 12.9-approximate mechanism. 
                \pause
            \item Lower bound of 2.
            \pause
            \item Prior results:
            \pause
            \begin{itemize}
            \item deterministic exponential-time mechanism [Singer, 2010; Chen \etal, 2011]
            \pause
            \item universally truthful (randomized) mechanism [Chen \etal, 2011; Singer, 2012]
            \end{itemize}
	    \end{itemize}

\pause
\vspace*{1cm}

    Generalization to more general statistical learning tasks.
\end{frame}

\begin{frame}{Problem}
   \begin{center}
        \includegraphics<1>[scale=0.4]{st2.pdf}
        \includegraphics<2>[scale=0.4]{st3.pdf}
        \includegraphics<3>[scale=0.4]{st4.pdf}
        \includegraphics<4>[scale=0.4]{st5.pdf}
        \includegraphics<5>[scale=0.4]{st6.pdf}
        \includegraphics<6>[scale=0.4]{st6a.pdf}
        \includegraphics<7>[scale=0.4]{st6b.pdf}
        \includegraphics<8>[scale=0.4]{st6c.pdf}
        \includegraphics<9>[scale=0.4]{st6d.pdf}
        \includegraphics<10->[scale=0.4]{st6e.pdf}
    \end{center}
	\vspace*{-2em}
    \begin{center}
        \begin{overprint}
        \onslide<1>\begin{center}$N$ ``experiment subjects'', $[N]\equiv
        \{1,\ldots,N\}$ \end{center}
            \onslide<2>
            \begin{center}
                $x_i\in \reals^d$: public features (\eg, age, gender, height, \etc)
            \end{center}
            \onslide<3>
            \begin{center}
                $y_i\in \reals$: private data (\eg, survey answer, medical test outcome, movie rating\ldots)
            \end{center}
            \onslide<4>
                \textbf{Gaussian Linear Model.}  There exists $\beta\in \reals^d$ s.t.\vspace*{-1em}
    \begin{displaymath}
        y_i = \beta^T x_i + \varepsilon_i,\quad
        \varepsilon_i\sim\mathcal{N}(0,\sigma^2), \quad i\in [N]
    \end{displaymath}
	    \onslide<5>
	    \begin{center}
		Experimenter {\tt E} wishes to learn $\beta$. 
	    \end{center}
       \onslide<6>
       \begin{center}
	Each subject $i\in [N]$ has a cost $c_i\in \reals_+$
	\end{center}
            \onslide<7>
             \begin{center}
                \E\ has a budget $B$.
            \end{center}
	    \onslide<8-9>
	    \begin{center}
		\E\ pays subjects\visible<9>{; $y_i$ is revealed upon payment.}
	    \end{center} 
	    \onslide<10>
	    \begin{center}
            {\tt E}  estimates ${\beta}$  through \emph{ridge regression}.
	    \end{center}
	    \onslide<11>
	    \begin{center}
            Goal: Determine who to pay how much so that $\hat{\beta}$ is as
            accurate as possible.
        \end{center}
        \onslide<12>
        \begin{center}
            Users are \alert{strategic} and may lie about costs $c_i$\\users cannot manipulate $x_i$ or $y_i$.
        \end{center}

        \end{overprint}
    \end{center}
\end{frame}

\begin{frame}{Value of data}

Let $S\subset [N]$ be the set of selected users.

\E\ has a \emph{prior} on $\beta$:
$$\beta \sim \mathcal{N}(0,\sigma^2 R^{-1}). $$

\pause
Information Gain: reduction of uncertainty on $\beta$
\begin{align*}
    I(\beta;S)&=H(\beta)-H(\beta\mid y_i,i\in S)\\
    \pause
    & = \frac{1}{2}\log\det \Big(R+\sum_{i\in S}x_ix_i^T\Big)-\frac{1}{2}\log\det R
\end{align*}
\end{frame}

\begin{frame}{Experimental Design}
\begin{block}{\textsc{Experimental Design Problem (EDP)} }
\vspace*{-0.4cm}		
\begin{align*}
			\text{Maximize}\qquad &V(S) = \log \det (I+\sum_{i\in S}x_ix_i^T) \\
			\text{subj. to}\qquad &\textstyle\sum_{i\in S}c_i\leq B
\end{align*}
\end{block}

\vspace*{1cm}
\pause
\begin{itemize}
\item EDP is NP-hard 
    \pause
\item $V$ is submodular, monotone, non-negative, and $V(\emptyset)=0$
    \pause
\item $\frac{e}{e-1}$-approximable (Sviridenko 2004, Krause and Guestrin 2005)
\end{itemize}
\end{frame}

\begin{frame}{Experimental Design with Strategic Agents}
\begin{block}{Value function}
\vspace*{-0.4cm}		
\begin{align*}
			\text{Maximize}\qquad &V(S) = \log \det (I+\sum_{i\in S}x_ix_i^T) \\
			\text{subj. to}\qquad &\textstyle\sum_{i\in S}c_i\leq B
\end{align*}
\end{block}

\vspace*{1cm}
\pause
Budgeted auction mechanism design problem. \pause

\vspace*{0.5cm}

Payments and allocation rule must be:
\begin{itemize}
\item normalized, truthful and individually rational
\pause
\item computable in polynomial time, give a good approximation ratio
\end{itemize}
\end{frame}


\begin{frame}{Our Main Results}
   \begin{theorem}
        There exists a deterministic,  budget feasible, individually rational and truthful mechanism for
        EDP which runs in polynomial time. Its
        approximation ratio is:
        \begin{displaymath}
        \frac{10e-3 + \sqrt{64e^2-24e + 9}}{2(e-1)}
        \simeq 12.98
        \end{displaymath}
    \end{theorem}
    
    \pause
    \vspace*{0.5cm}

   \begin{theorem}
        There is no 2-approximate, budget feasible, individually rational and truthful mechanism for
        EDP.     \end{theorem}

        \pause
        \vspace*{0.5cm}

    \alert{Bonus:} two technical approximation results.
\end{frame}

\begin{frame}{Blueprint for Deterministic, Poly-time Algorithm}
Azar and Gamzu 2008, Singer 2010, Chen \etal\ 2011:
    \begin{block}{Allocation rule}
        \begin{itemize}
        \item Find $i^* = \arg\max_i V(\{i\})$
        \item Compute $S_G$ greedily
        \item Return:
            \begin{itemize}
                \item $\{i^*\}$ if $V(\{i^*\}) \geq \alert<2>{OPT_{-i^*}}$ 
                \item $S_G$ otherwise
            \end{itemize}
        \end{itemize}
    \end{block}
    \vspace{0.5cm}
    \pause
    \pause
    \begin{columns}[c]
        \begin{column}{0.53\textwidth}
Replace $OPT_{-i^*}$ with a \alert{relaxation $L^*$}:\pause
            \begin{itemize}
		   \item computable in polynomial time
                    \pause
                \item close to $OPT_{-i^*}$
		    \pause
		\item monotone in costs $c$
            \end{itemize}
 	\end{column}
	\pause
        \begin{column}{0.45\textwidth}
   	\begin{itemize}
                \item \textsc{Knapsack} (Chen \etal, 2011)
                \item \textsc{Coverage} (Singer, 2012)
            \end{itemize}
	\end{column}
       \end{columns}
\end{frame}

\begin{frame}{A relaxation for \textsc{EDP}}
\alt<1>{
    \begin{block}{\textsc{EDP} }
        \vspace*{-0.4cm}		
        \begin{align*}
            \text{Maximize}\qquad &V(S) = \log \det\big(I+\sum_{i\in S}x_ix_i^T\big) \\
            \text{subj. to}\qquad &\textstyle\sum_{i\in S}c_i\leq B
        \end{align*}
    \end{block}
}
{
    \begin{block}{Convex Relaxation}
        \vspace*{-0.4cm}		
        \begin{align*}
            \text{Maximize}\qquad &\alert<2>{L(\lambda)} = \log \det \big(I+\sum_{i\in [N]}\alert<2>{\lambda_i}x_ix_i^T\big) \\
            \text{subj. to}\qquad &\textstyle\sum_{i\in [N]}\alert<2>{\lambda_i}c_i\leq B, \lambda_i\in [0,1]
        \end{align*}
    \end{block}
}
\pause
\vspace{0.5cm}
$L(\lambda^*)$ is:
\begin{itemize}
    \item Monotone in $c$
        \pause
    \item Poly-time: $L$ is concave
\end{itemize}
\end{frame}

\begin{frame}{First technical result: integrality gap}
        \begin{align*}
            \text{Maximize}\qquad &L(\lambda) = \log \det \big(I+\sum_{i\in [N]}\lambda_ix_ix_i^T\big) \\
            \text{subj. to}\qquad &\textstyle\sum_{i\in [N]}\lambda_ic_i\leq B, \lambda_i\in [0,1]
        \end{align*}

        \pause
        \vspace{0.5cm}

        \begin{block}{Integrality gap}
            For $\lambda^*$ the optimal solution:
            \begin{displaymath}
                OPT\leq  L(\lambda^*) \leq 2 OPT + 2V(\{i^*\})
            \end{displaymath}
        \end{block}

        \pause
        \vspace{0.5cm}
        
        \alert{Proof:}
        \begin{itemize}
            \item relate $L$ to the multilinear extension of $V$
            \item use \alert{pipage rounding} [Ageev and Sviridenko, 2004]
        \end{itemize}

\end{frame}

\begin{frame}{Second technical result: monotone approximation}
    \begin{block}{EDP-relaxed}
        \begin{align*}
            \text{Maximize}\qquad &L(\lambda) = \log \det \big(I+\sum_{i\in [N]}\lambda_ix_ix_i^T\big) \\
            \text{subj. to}\qquad &\textstyle\sum_{i\in [N]}\lambda_ic_i\leq B, \lambda_i\in [\only<1-2>{0}\only<3->{\alert<3>{\alpha}},1]
        \end{align*}
        \end{block}

        \vspace*{1cm}
        \pause

    Convex optimization only gives $\delta$-approximation $\alert{\Rightarrow}$ breaks monotonicity.

    \vspace*{1cm}
    \pause

    Shrink feasible set:
    \begin{itemize}
    \pause
        \item $L$ is ``sufficiently'' monotone
        \pause
        \item a $\delta$-approximation of its optimum is $\varepsilon(\delta,\alpha)$-monotone in the costs
        \pause
        \item tune $\delta$, $\alpha$ $\Rightarrow$ $\varepsilon$-truthful mechanism
    \end{itemize}
    
    
\end{frame}


%\begin{frame}{Towards More General ML Tasks}
%   \begin{center}
%        \includegraphics<1>[scale=0.4]{st2.pdf}
%        \includegraphics<2>[scale=0.4]{st3.pdf}
%        \includegraphics<3>[scale=0.4]{st4.pdf}
%        \includegraphics<4-5>[scale=0.4]{st25.pdf}
%        \includegraphics<6->[scale=0.4]{st26.pdf}
%    \end{center}
%
%    \begin{overprint}
%        \onslide<1>
%		\begin{center}$N$ experiment subjects\end{center}
%       	\onslide<2>
%            \begin{center}
%                $x_i\in \Omega$: public features
%            \end{center}
%        \onslide<3>
%            \begin{center}
%                $y_i\in \reals$: private data 
%            \end{center}
%        \onslide<4-5>
%             \begin{center}
%                \textbf{Hypothesis.}  There exists $h:\Omega\to \reals$ s.t.
% %   \begin{displaymath}
%        $y_i =  h(x_i) + \varepsilon_i,$ %\quad
%	where $\varepsilon_i$ are independent.
%	\visible<5>{Examples: Logistic Regression, LDA, SVMs, \etc}
%  % \end{displaymath}
%		\end{center}
%%$$y_i = \beta^Tx_i + \varepsilon_i, i=1,\ldots, N$$ 
%            %\end{center}
%	    \onslide<6>
%	    \begin{center}
%		Experimenter {\tt E} wishes to learn $h$, and has a prior distribution over $$\mathcal{H}=\{\text{mappings from }\Omega\text{ to }\reals\}.$$ 
%	    \end{center}
%    \onslide<7->\begin{center}
%\vspace*{-0.5cm}
%%    \begin{block}
%%{Value function: Information gain}
%        \begin{displaymath}
%            V(S) = H(h) - H(h\mid y_i,i\in S),\quad S\subseteq[N]
%        \end{displaymath}
% %   \end{block}
%	\visible<8>{$V$ is submodular! [Krause and Guestrin 2009]}
%    \end{center}
%	\end{overprint}
%
%\end{frame}

\begin{frame}{Conclusion}
    \begin{itemize}
        \item Experimental design + Budget Feasible Mechanisms
        \vspace{1cm}
        \pause
        \item Deterministic mechanism for other ML Tasks? General submodular functions?
            \vspace{1cm}
            \pause
        \item  Approximation ratio $\simeq \alert{13}$. Lower bound: \alert{$2$}
    \end{itemize}
\end{frame}
\begin{frame}{}
\vspace{\stretch{1}}
\begin{center}
{\Huge Thank you!}
\end{center}
\vspace{\stretch{1}}
\end{frame}

\begin{frame}{Non-Homotropic Prior}
Recall that:
\begin{align*}
V(S)&=H(\beta)-H(\beta\mid y_i,i\in S)\\
& = \frac{1}{2}\log\det (R^{-1}+\sum_{i\in S}x_ix_i^T)
\end{align*}
What if $R\neq I$?
\begin{theorem}
 There exists a truthful, poly-time, and budget feasible mechanism for the objective
 function $V$ above. Furthermore, 
 the algorithm computes a set $S^*$ such that:
 \begin{displaymath}
    OPT \leq 
    \frac{5e-1}{e-1}\frac{2}{\mu\log(1+1/\mu)}V(S^*) + 5.1 
\end{displaymath}
where $\mu$ is the largest eigenvalue of $R$.
\end{theorem}
\end{frame}

\end{document}
