\documentclass{acm_proc_article-sp}
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\usepackage{algorithm}
\usepackage{algpseudocode}
\newtheorem{lemma}{Lemma}
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\newtheorem{prop}{Proposition}
\newtheorem{theorem}{Theorem}
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\newcommand{\mse}{\mathop{\mathrm{MSE}}}
\DeclareMathOperator{\trace}{tr}
\DeclareMathOperator*{\argmax}{arg\,max}
\title{Budgeted Auctions for Experiment Design}
\begin{document}

\maketitle

\section{Intro}

\begin{itemize}
    \item already existing field of experiment design: survey-like setup, what
    are the best points to include in your experiment? Measure of the
    usefulness of the data: variance-reduction or entropy-reduction.
    \item nowadays, there is also a big focus on purchasing data: paid surveys,
    mechanical turk, etc. that add economic aspects to the problem of
    experiment design
    \item recent advances (Singer, Chen) in the field of budgeted mechanisms
    \item we study ridge regression, very widely used in statistical learning,
    and treat it as a problem of budgeted experiment design
    \item we make the following contributions: ...
    \item extension to a more general setup which includes a wider class of
    machine learning problems
\end{itemize}

\section{Problem formulation}

\subsection{Notations}

Throughout the paper, we will make use of the following notations: if $x$ is
a (column) vector in $\mathbf{R}^d$, $x^*$ denotes its transposed (line)
vector. Thus, the standard inner product between two vectors $x$ and $y$ is
simply $x^* y$. $\norm{x}_2 = x^*x$ will denote the $L_2$ norm of $x$.

We will also often use the following order over symmetric matrices: if $A$ and
$B$ are two $d\times d$ and $B$ are two $d\times d$ real symmetric matrices, we
write that $A\leq B$ iff:
\begin{displaymath}
    \forall x\in\mathbf{R}^d,\quad
    x^*Ax \leq x^*Bx
\end{displaymath}
That is, iff $B-A$ is symmetric semi-definite positive.

This order let us define the notion of a \emph{decreasing} or \emph{convex}
matrix function similarly to their real counterparts. In particular, let us
recall that the matrix inversion is decreasing and convex over symmetric
definite positive matrices.

\subsection{Data model}

There is a set of $n$ users, $\mathcal{N} = \{1,\ldots, n\}$. Each user
$i\in\mathcal{N}$ has a public vector of features $x_i\in\mathbf{R}^d$ and an
undisclosed piece of information $y_i\in\mathbf{R}$. We assume that the data
has already been normalized so that $\norm{x_i}_2\leq 1$ for all
$i\in\mathcal{N}$.

The experimenter is going to select a set of users and ask them to reveal their
private piece of information. We are interested in a \emph{survey setup}: the
experimenter has not seen the data yet, but he wants to know which users he
should be selecting. His goal is to learn the model underlying the data. Here,
we assume a linear model:
\begin{displaymath}
    \forall i\in\mathcal{N},\quad y_i = \beta^* x_i + \varepsilon_i
\end{displaymath}
where $\beta\in\mathbf{R}^d$ and $\varepsilon_i\in\mathbf{R}$ follows a normal
distribution of mean $0$ and variance $\sigma^2$. Furthermore, we assume the
error $\varepsilon$ to be independent of the user:
$(\varepsilon_i)_{i\in\mathcal{N}}$ are mutually independent.

After observing the data, the experimenter could simply do linear regression to
learn the model parameter $\beta$. However, in a more general setup, the
experimenter has a prior knowledge about $\beta$, a distribution over
$\mathbf{R}^d$. After observing the data, the experimenter performs
\emph{maximum a posteriori estimation}: computing the point which maximizes the
posterior distribution of $\beta$ given the observations.

Here, we will assume, as it is often done, that the prior distribution is
a multivariate normal distribution of mean zero and covariance matrix $\kappa
I_d$. Maximum a posteriori estimation leads to the following maximization
problem:
\begin{displaymath}
    \beta_{\text{max}} = \argmax_{\beta\in\mathbf{R}^d} \sum_i (y_i - \beta^*x_i)^2
    + \frac{1}{\mu}\sum_i \norm{\beta}_2^2
\end{displaymath}
which is the well-known \emph{ridge regression}. $\mu
= \frac{\kappa}{\sigma^2}$ is the regularization parameter. Ridge regression
can thus be seen as linear regression with a regularization term which
prevents $\beta$ from having a large $L_2$-norm.

\subsection{Value of data}

Because the user private variables $y_i$ have not been observed yet when the
experimenter has to decide which users to include in his experiment, we treat
$\beta$ as a random variable whose distribution is updated after observing the
data.

Let us recall that if $\beta$ is random variable over $\mathbf{R}^d$ whose
probability distribution has a density function $f$ with respect to the
Lebesgue measure, its entropy is given by:
\begin{displaymath}
    \mathbb{H}(\beta) \defeq - \int_{b\in\mathbf{R}^d} \log f(b) f(b)\text{d}b
\end{displaymath}

A usual way to measure the decrease of uncertainty induced by the observation
of data is to use the entropy. This leads to the following definition of the
value of data called the \emph{value of information}:
\begin{displaymath}
    \forall S\subset\mathcal{N},\quad V(S) = \mathbb{H}(\beta)
    - \mathbb{H}(\beta\,|\,
    Y_S)
\end{displaymath}
where $Y_S = \{y_i,\,i\in S\}$ is the set of observed data.

\begin{theorem}
    Under the ridge regression model explained in section TODO, the value of data
    is equal to:
    \begin{align*}
        \forall S\subset\mathcal{N},\; V(S)
        & = \frac{1}{2}\log\det\left(I_d
            + \mu\sum_{i\in S} x_ix_i^*\right)\\
        & \defeq \frac{1}{2}\log\det A(S)
    \end{align*}
\end{theorem}

\begin{proof}

Let us denote by $X_S$ the matrix whose rows are the vectors $(x_i^*)_{i\in
S}$. Observe that $A_S$ can simply be written as:
\begin{displaymath}
    A_S  = I_d + \mu X_S^* X_S
\end{displaymath}

Let us recall that the entropy of a multivariate normal variable $B$ over
$\mathbf{R}^d$ of covariance $\Sigma I_d$ is given by:
\begin{equation}\label{eq:multivariate-entropy}
    \mathbb{H}(B) = \frac{1}{2}\log\big((2\pi e)^d \det \Sigma I_d\big)
\end{equation}

Using the chain rule for conditional entropy, we get that:
\begin{displaymath}
    V(S) = \mathbb{H}(Y_S) - \mathbb{H}(Y_S\,|\,\beta)
\end{displaymath}

Conditioned on $\beta$, $(Y_S)$ follows a multivariate normal
distribution of mean $X\beta$ and of covariance matrix $\sigma^2 I_n$. Hence:
\begin{equation}\label{eq:h1}
    \mathbb{H}(Y_S\,|\,\beta) 
    = \frac{1}{2}\log\left((2\pi e)^n \det(\sigma^2I_n)\right)
\end{equation}

$(Y_S)$ also follows a multivariate normal distribution of mean zero. Let us
compute its covariance matrix, $\Sigma_Y$:
\begin{align*}
    \Sigma_Y & = \expt{YY^*} = \expt{(X_S\beta + E)(X_S\beta + E)^*}\\
             & = \kappa X_S X_S^* + \sigma^2I_n
\end{align*}
Thus, we get that:
\begin{equation}\label{eq:h2}
    \mathbb{H}(Y_S) 
    = \frac{1}{2}\log\left((2\pi e)^n \det(\kappa X_S X_S^* + \sigma^2 I_n)\right)
\end{equation}

Combining \eqref{eq:h1} and \eqref{eq:h2} we get:
\begin{displaymath}
    V(S) = \frac{1}{2}\log\det\left(I_n+\frac{\kappa}{\sigma^2}X_S
    X_S^*\right)
\end{displaymath}

Finally, we can use Sylvester's determinant theorem to get the result.
\end{proof}

It is also interesting to look at the marginal contribution of a user to a set: the
increase of value induced by adding a user to an already existing set of users.
We have the following lemma.

\begin{lemma}[Marginal contribution]
    \begin{displaymath}
        \Delta_i V(S)\defeq V(S\cup\{i\}) - V(S) 
        = \frac{1}{2}\log\left(1 + \mu x_i^*A(S)^{-1}x_i\right)
    \end{displaymath}
\end{lemma}

\begin{proof}
    We have:
    \begin{align*}
        V(S\cup\{i\}) & = \frac{1}{2}\log\det A(S\cup\{i\})\\
        & = \frac{1}{2}\log\det\left(A(S) + \mu x_i x_i^*\right)\\
        & = V(S) + \frac{1}{2}\log\det\left(I_d + \mu A(S)^{-1}x_i
    x_i^*\right)\\
        & = V(S) + \frac{1}{2}\log\left(1 + \mu x_i^* A(S)^{-1}x_i\right)
    \end{align*}
    where the last equality comes from Sylvester's determinant formula.
\end{proof}

Because $A(S)$ is symmetric definite positive, the marginal contribution is
positive, which proves that the value function is set increasing. Furthermore,
it is easy to see that if $S\subset S'$, then $A(S)\leq A(S')$. Using the fact
that matrix inversion is decreasing, we see that the marginal contribution of
a fixed user is a set decreasing function. This is the \emph{submodularity} of
the value function.

TODO? Explain what are the points which are the most valuable : points which
are aligned along directions where the spread over the already existing points
is small.

\subsection{Auction}

TODO Explain the optimization problem, why it has to be formulated as an auction
problem. Explain the goals:
\begin{itemize}
    \item truthful
    \item individually rational
    \item budget feasible
    \item has a good approximation ratio

TODO Explain what is already known: it is ok when the function is submodular. When
should we introduce the notion of submodularity?
\end{itemize}

\section{Main result}

All the budget feasible mechanisms studied recently (TODO ref Singer Chen)
rely on using a greedy heuristic extending the idea of the greedy heuristic for
the knapsack problem. In knapsack, objects are selected based on their
\emph{value-per-cost} ratio. The quantity which plays a similar role for
general submodular functions is the \emph{marginal-contribution-per-cost}
ratio: let us assume that you have already selected a set of points $S$, then
the \emph{marginal-contribution-per-cost} ratio per cost of a new point $i$ is
defined by:
\begin{displaymath}
    \frac{V(S\cup\{i\}) - V(S)}{c_i}
\end{displaymath}

The greedy heuristic then simply repeatedly selects the point whose
marginal-contribution-per-cost ratio is the highest until it reaches the budget
limit. Mechanism considerations aside, this is known to have an unbounded
approximation ratio. However, lemma TODO ref in Chen or Singer shows that the
maximum between the greedy heuristic and the point with maximum value (as
a singleton set) provides a $\frac{5e}{e-1}$ approximation ratio.

Unfortunately, TODO Singer 2011 points out that taking the maximum between the
greedy heuristic and the most valuable point is not incentive compatible.
Singer and Chen tackle this issue similarly: instead of comparing the most
valuable point to the greedy solution, they compare it to a solution which is
close enough to keep a constant approximation ratio:
\begin{itemize}
    \item Chen suggests using $OPT(V,\mathcal{N}\setminus\{i\}, B)$.
        Unfortunately, in the general case, this cannot be computed exactly in
        polynomial time.
    \item Singer uses using the optimal value of a relaxed objective function
        which can be proven to be close to the optimal of the original
        objective function. The function used is tailored to the specific
        problem of coverage.
\end{itemize}

Here, we use a relaxation of the objective function which is tailored to the
problem of ridge regression. We define:
\begin{displaymath}
    \forall\lambda\in[0,1]^{|\mathcal{N}|}\,\quad L_{\mathcal{N}}(\lambda) \defeq
    \log\det\left(I_d + \mu\sum_{i\in\mathcal{N}} \lambda_i x_i x_i^*\right)
\end{displaymath}

We can now present the mechanism we use, which has a same flavor to Chen's and
Singer's

\begin{algorithm}\label{mechanism}
    \caption{Mechanism for ridge regression}
    \begin{algorithmic}[1]
    \State $i^* \gets \argmax_{j\in\mathcal{N}}V(j)$
    \State $x^* \gets \argmax_{x\in[0,1]^n} \{L_{\mathcal{N}\setminus\{i^*\}}(x)
                                    \,|\, c(x)\leq B\}$
        \Statex
        \If{$L(x^*) < CV(i^*)$}
            \State \textbf{return} $\{i^*\}$
        \Else
            \State $i \gets \argmax_{1\leq j\leq n}\frac{V(j)}{c_j}$
            \State $S \gets \emptyset$
            \While{$c_i\leq \frac{B}{2}\frac{V(S\cup\{i\})-V(S)}{V(S\cup\{i\})}$}
                \State $S \gets S\cup\{i\}$
                \State $i \gets \argmax_{j\in\mathcal{N}\setminus S}
                \frac{V(S\cup\{j\})-V(S)}{c_j}$
            \EndWhile
            \State \textbf{return} $S$
        \EndIf
    \end{algorithmic}
\end{algorithm}

Notice, that the stopping condition in the while loop is more sophisticated
than just ensuring that the sum of the costs does not exceed the budget. This
is because the selected users will be payed more than their costs, and this
stopping condition ensures budget feasibility when the users are paid their
threshold payment.

We can now state the main result of this section:
\begin{theorem}
    The mechanism in \ref{mechanism} is truthful, individually rational, budget
    feasible. Furthermore, choosing:
    \begin{multline*}
        C = C^* =  \frac{5e-1 + C_\mu(2e+1)}{2C_\mu(e-1)}\\
        + \frac{\sqrt{C_\mu^2(1+2e)^2
        + 2C_\mu(14e^2+5e+1) + (1-5e)^2}}{2C_\mu(e-1)}
    \end{multline*}
    we get an approximation ratio of:
    \begin{multline*}
        1 + C^* = \frac{5e-1 + C_\mu(4e-1)}{2C_\mu(e-1)}\\
        + \frac{\sqrt{C_\mu^2(1+2e)^2
        + 2C_\mu(14e^2+5e+1) + (1-5e)^2}}{2C_\mu(e-1)}
    \end{multline*}
    where:
    \begin{displaymath}
        C_\mu = \frac{\log(1+\mu)}{2\mu}
    \end{displaymath}
\end{theorem}

The proof will consist of the claims of the theorem broken down into lemmas.

Because the user strategy is parametrized by a single parameter, truthfulness
is equivalent to monotonicity (TODO ref). The proof here is the same as in TODO
and is given for the sake of completeness.

\begin{lemma}
The mechanism is monotone.
\end{lemma}

\begin{proof}
    We assume by contradiction that there exists a user $i$ that has been
    selected by the mechanism and that would not be selected had he reported
    a cost $c_i'\leq c_i$ (all the other costs staying the same).

    If $i\neq i^*$ and $i$ has been selected, then we are in the case where
    $L(x^*) \geq C V(i^*)$ and $i$ was included in the result set by the greedy
    part of the mechanism. By reporting a cost $c_i'\leq c_i$, using the
    submodularity of $V$, we see that $i$ will satisfy the greedy selection
    rule:
    \begin{displaymath}
        i = \argmax_{j\in\mathcal{N}\setminus S} \frac{V(S\cup\{j\})
        - V(S)}{c_j}
    \end{displaymath}
    in an earlier iteration of the greedy heuristic. Let us denote by $S_i$
    (resp. $S_i'$) the set to which $i$ is added when reporting cost $c_i$
    (resp. $c_i'$). We have $S_i'\subset S_i$. Moreover:
    \begin{align*}
        c_i' & \leq c_i \leq
        \frac{B}{2}\frac{V(S_i\cup\{i\})-V(S_i)}{V(S_i\cup\{i\})}\\
        & \leq \frac{B}{2}\frac{V(S_i'\cup\{i\})-V(S_i')}{V(S_i'\cup\{i\})}
    \end{align*}
    Hence $i$ will still be included in the result set.

    If $i = i^*$, $i$ is included iff $L(x^*) \leq C V(i^*)$. Reporting $c_i'$
    instead of $c_i$ does not change the value $V(i^*)$ nor $L(x^*)$ (which is
    computed over $\mathcal{N}\setminus\{i^*\}$). Thus $i$ is still included by
    reporting a different cost.
\end{proof}

\begin{lemma}
The mechanism is budget feasible.
\end{lemma}

The proof is the same as in Chen and is given here for the sake of
completeness.
\begin{proof}

\end{proof}

Using the characterization of the threshold payments from Singer TODO, we can
prove individual rationality similarly to Chen TODO.
\begin{lemma}
The mechanism is individually rational
\end{lemma}

\begin{proof}

\end{proof}

The following lemma requires to have a careful look at the relaxation function
we chose in the mechanism. The next section will be dedicated to studying this
relaxation and will contain the proof of the following lemma:
\begin{lemma}
    We have:
    \begin{displaymath}
        OPT(L_\mathcal{N}, B) \leq \frac{1}{C_\mu}\big(2 OPT(V,\mathcal{N},B)
        + \max_{i\in\mathcal{N}}V(i)\big)
    \end{displaymath}
\end{lemma}

\begin{lemma}
    Let us denote by $S_M$ the set returned by the mechanism. Let us also
    write:
    \begin{displaymath}
        C_{\textrm{max}} = \max\left(1+C,\frac{e}{e-1}\left( 3 + \frac{12e}{C\cdot
        C_\mu(e-1) -5e  +1}\right)\right)
    \end{displaymath}

    Then:
    \begin{displaymath}
        OPT(V, \mathcal{N}, B) \leq
        C_\text{max}\cdot V(S_M) 
    \end{displaymath}
\end{lemma}

\begin{proof}

    If the condition on line 3 of the algorithm holds, then:
    \begin{displaymath}
        V(i^*) \geq \frac{1}{C}L(x^*) \geq
        \frac{1}{C}OPT(V,\mathcal{N}\setminus\{i\}, B)
    \end{displaymath}

    But:
    \begin{displaymath}
        OPT(V,\mathcal{N},B) \leq OPT(V,\mathcal{N}\setminus\{i\}, B) + V(i^*)
    \end{displaymath}

    Hence:
    \begin{displaymath}
        V(i^*) \geq \frac{1}{C+1} OPT(V,\mathcal{N}, B)
    \end{displaymath}

    If the condition of the algorithm does not hold:
    \begin{align*}
        V(i^*) & \leq \frac{1}{C}L(x^*) \leq \frac{1}{C\cdot C_\mu}
        \big(2 OPT(V,\mathcal{N}, B) + V(i^*)\big)\\
        & \leq \frac{1}{C\cdot C_\mu}\left(\frac{2e}{e-1}\big(3 V(S_M)
        + 2 V(i^*)\big)
        + V(i^*)\right)
    \end{align*}
    
    Thus:
    \begin{align*}
        V(i^*) \leq \frac{6e}{C\cdot C_\mu(e-1)- 5e + 1} V(S_M)
    \end{align*}

    Finally, using again that:
    \begin{displaymath}
        OPT(V,\mathcal{N},B) \leq \frac{e}{e-1}\big(3 V(S_M) + 2 V(i^*)\big)
    \end{displaymath}
    
    We get:
    \begin{displaymath}
        OPT(V, \mathcal{N}, B) \leq \frac{e}{e-1}\left( 3 + \frac{12e}{C\cdot
        C_\mu(e-1) -5e  +1}\right) V(S_M)
    \end{displaymath}
\end{proof}

The optimal value for $C$ is:
\begin{displaymath}
    C^* = \arg\min C_{\textrm{max}}
\end{displaymath}

This equation has two solutions. Only one of those is such that:
\begin{displaymath}
    C\cdot C_\mu(e-1) -5e  +1 \geq 0
\end{displaymath}
which is needed in the proof of the previous lemma. Computing this solution,
we can state the main result of this section.

\subsection{Relaxations of the value function}

To prove lemma TODO, we will use a general method called pipage rounding,
introduced in TODO. Two consecutive relaxations are used: the one that we are
interested in, whose optimization can be computed efficiently, and the
multilinear extension which presents a \emph{cross-convexity} like behavior
which allows for rounding of fractional solution without decreasing the value
of the objective function and thus ensures a constant approximation of the
value function. The difficulty resides in showing that the ratio of the two
relaxations is bounded.

We say that $R_\mathcal{N}:[0,1]^n\rightarrow\mathbf{R}$ is a relaxation of the
value function $V$ over $\mathcal{N}$ if it coincides with $V$ at binary
points. Formally, for any $S\subset\mathcal{N}$, let $\mathbf{1}_S$ denote the
indicator vector of $S$. $R_\mathcal{N}$ is a relaxation of $V$ over
$\mathcal{N}$ iff:
\begin{displaymath}
    \forall S\subset\mathcal{N},\; R_\mathcal{N}(\mathbf{1}_S) = V(S)
\end{displaymath}

We can extend the optimisation problem defined above to a relaxation by
extending the cost function:
\begin{displaymath}
    \forall \lambda\in[0,1]^n,\; c(\lambda)
    = \sum_{i\in\mathcal{N}}\lambda_ic_i
\end{displaymath}
The optimisation problem becomes:
\begin{displaymath}
    OPT(R_\mathcal{N}, B) =
    \max_{\lambda\in[0,1]^n}\left\{R_\mathcal{N}(\lambda)\,|\, c(\lambda)\leq B\right\}
\end{displaymath}
The relaxations we will consider here rely on defining a probability
distribution over subsets of $\mathcal{N}$.

Let $\lambda\in[0,1]^n$, let us define:
\begin{displaymath}
    P_\mathcal{N}^\lambda(S) = \prod_{i\in S}\lambda_i
    \prod_{i\in\mathcal{N}\setminus S}(1-\lambda_i)
\end{displaymath}
$P_\mathcal{N}^\lambda(S)$ is the probability of picking the set $S$ if we select
a subset of $\mathcal{N}$ at random by deciding independently for each point to
include it in the set with probability $\lambda_i$ (and to exclude it with
probability $1-\lambda_i$).

We will consider two relaxations of the value function $V$ over $\mathcal{N}$:
\begin{itemize}
    \item the \emph{multi-linear extension} of $V$:
        \begin{align*}
            F_\mathcal{N}(\lambda) 
            & = \mathbb{E}_{S\sim P_\mathcal{N}^\lambda}\big[\log\det A(S)\big]\\
            & = \sum_{S\subset\mathcal{N}} P_\mathcal{N}^\lambda(S) V(S)\\
            & = \sum_{S\subset\mathcal{N}} P_\mathcal{N}^\lambda(S) \log\det A(S)\\
        \end{align*}
    \item the \emph{concave relaxation} of $V$:
        \begin{align*}
            L_{\mathcal{N}}(\lambda) 
            & = \log\det \mathbb{E}_{S\sim P_\mathcal{N}^\lambda}\big[A(S)\big]\\
            & = \log\det\left(\sum_{S\subset N}
                P_\mathcal{N}^\lambda(S)A(S)\right)\\
            & = \log\det\left(I_d + \mu\sum_{i\in\mathcal{N}}
                \lambda_ix_ix_i^*\right)\\
            & \defeq \log\det \tilde{A}(\lambda)
        \end{align*}
\end{itemize}

\begin{lemma}
    The \emph{concave relaxation} $L_\mathcal{N}$ is concave\footnote{Hence
    this relaxation is well-named!}.
\end{lemma}

\begin{proof}
    This follows from the concavity of the $\log\det$ function over symmetric
    positive semi-definite matrices. More precisely, if $A$ and $B$ are two
    symmetric positive semi-definite matrices, then:
    \begin{multline*}
        \forall\alpha\in [0, 1],\; \log\det\big(\alpha A + (1-\alpha) B\big)\\
        \geq \alpha\log\det A + (1-\alpha)\log\det B
    \end{multline*}
\end{proof}

\begin{lemma}[Rounding]\label{lemma:rounding}
    For any feasible $\lambda\in[0,1]^n$, there exists a feasible
    $\bar{\lambda}\in[0,1]^n$ such that at most one of its component is
    fractional, that is, lies in $(0,1)$ and:
    \begin{displaymath}
        F_{\mathcal{N}}(\lambda)\leq F_{\mathcal{N}}(\bar{\lambda})
    \end{displaymath}
\end{lemma}

\begin{proof}
    We give a rounding procedure which given a feasible $\lambda$ with at least
    two fractional components, returns some $\lambda'$ with one less fractional
    component, feasible such that:
    \begin{displaymath}
        F_\mathcal{N}(\lambda) \leq F_\mathcal{N}(\lambda')
    \end{displaymath}
    Applying this procedure recursively yields the lemma's result.

    Let us consider such a feasible $\lambda$. Let $i$ and $j$ be two
    fractional components of $\lambda$ and let us define the following
    function:
    \begin{displaymath}
        F_\lambda(\varepsilon) = F(\lambda_\varepsilon)
        \quad\textrm{where} \quad
        \lambda_\varepsilon = \lambda + \varepsilon\left(e_i-\frac{c_i}{c_j}e_j\right)
    \end{displaymath}

    It is easy to see that if $\lambda$ is feasible, then:
    \begin{multline}\label{eq:convex-interval}
        \forall\varepsilon\in\Big[\max\Big(-\lambda_i,(\lambda_j-1)\frac{c_j}{c_i}\Big), \min\Big(1-\lambda_i, \lambda_j
        \frac{c_j}{c_i}\Big)\Big],\;\\
            \lambda_\varepsilon\;\;\textrm{is feasible}
    \end{multline}

    Furthermore, the function $F_\lambda$ is convex, indeed:
    \begin{align*}
        F_\lambda(\varepsilon)
        & = \mathbb{E}_{S'\sim P_{\mathcal{N}\setminus\{i,j\}}^\lambda(S')}\Big[
        (\lambda_i+\varepsilon)\Big(\lambda_j-\varepsilon\frac{c_i}{c_j}\Big)V(S'\cup\{i,j\})\\
        & + (\lambda_i+\varepsilon)\Big(1-\lambda_j+\varepsilon\frac{c_i}{c_j}\Big)V(S'\cup\{i\})\\
        & + (1-\lambda_i-\varepsilon)\Big(\lambda_j-\varepsilon\frac{c_i}{c_j}\Big)V(S'\cup\{j\})\\
        & + (1-\lambda_i-\varepsilon)\Big(1-\lambda_j+\varepsilon\frac{c_i}{c_j}\Big)V(S')\Big]\\
    \end{align*}
    Thus, $F_\lambda$ is a degree 2 polynomial whose dominant coefficient is:
    \begin{multline*}
        \frac{c_i}{c_j}\mathbb{E}_{S'\sim
        P_{\mathcal{N}\setminus\{i,j\}}^\lambda(S')}\Big[
            V(S'\cup\{i\})+V(S'\cup\{i\})\\
        -V(S'\cup\{i,j\})-V(S')\Big]
    \end{multline*}
    which is positive by submodularity of $V$. Hence, the maximum of
    $F_\lambda$ over the interval given in \eqref{eq:convex-interval} is
    attained at one of its limit, at which either the $i$-th or $j$-th component of
    $\lambda_\varepsilon$ becomes integral.
\end{proof}

\begin{lemma}\label{lemma:relaxation-ratio}
    The following inequality holds:
    \begin{displaymath}
        \forall\lambda\in[0,1]^n,\; 
        \frac{\log\big(1+\mu\big)}{2\mu}
        \,L_\mathcal{N}(\lambda)\leq
        F_\mathcal{N}(\lambda)\leq L_{\mathcal{N}}(\lambda)
    \end{displaymath}
\end{lemma}

\begin{proof}

    We will prove that:
    \begin{displaymath}
        \frac{\log\big(1+\mu\big)}{2\mu}
    \end{displaymath}
    is a lower bound of the ratio $\partial_i F_\mathcal{N}(\lambda)/\partial_i
    L_\mathcal{N}(\lambda)$.

    This will be enough to conclude, by observing that:
    \begin{displaymath}
        \frac{F_\mathcal{N}(\lambda)}{L_\mathcal{N}(\lambda)}
        \sim_{\lambda\rightarrow 0}
        \frac{\sum_{i\in \mathcal{N}}\lambda_i\partial_i F_\mathcal{N}(0)}
        {\sum_{i\in\mathcal{N}}\lambda_i\partial_i L_\mathcal{N}(0)}
    \end{displaymath}
    and that an interior critical point of the ratio
    $F_\mathcal{N}(\lambda)/L_\mathcal{N}(\lambda)$ is defined by:
    \begin{displaymath}
        \frac{F_\mathcal{N}(\lambda)}{L_\mathcal{N}(\lambda)}
        = \frac{\partial_i F_\mathcal{N}(\lambda)}{\partial_i
        L_\mathcal{N}(\lambda)}
    \end{displaymath}

    Let us start by computing the derivatives of $F_\mathcal{N}$ and
    $L_\mathcal{N}$ with respect to
    the $i$-th component.

    For $F$, it suffices to look at the derivative of
    $P_\mathcal{N}^\lambda(S)$:
    \begin{displaymath}
        \partial_i P_\mathcal{N}^\lambda(S) = \left\{
            \begin{aligned}
                & P_{\mathcal{N}\setminus\{i\}}^\lambda(S\setminus\{i\})\;\textrm{if}\; i\in S \\
                & - P_{\mathcal{N}\setminus\{i\}}^\lambda(S)\;\textrm{if}\;
                i\in \mathcal{N}\setminus S \\
            \end{aligned}\right.
    \end{displaymath}

    Hence:
    \begin{multline*}
        \partial_i F_\mathcal{N} =
        \sum_{\substack{S\subset\mathcal{N}\\ i\in S}}
        P_{\mathcal{N}\setminus\{i\}}^\lambda(S\setminus\{i\})V(S)\\
        - \sum_{\substack{S\subset\mathcal{N}\\ i\in \mathcal{N}\setminus S}}
        P_{\mathcal{N}\setminus\{i\}}^\lambda(S)V(S)\\
    \end{multline*}

    Now, using that every $S$ such that $i\in S$ can be uniquely written as
    $S'\cup\{i\}$, we can write:
    \begin{multline*}
        \partial_i F_\mathcal{N} =
        \sum_{\substack{S\subset\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
        P_{\mathcal{N}\setminus\{i\}}^\lambda(S)V(S\cup\{i\})\\
        - \sum_{\substack{S\subset\mathcal{N}\\ i\in \mathcal{N}\setminus S}}
        P_{\mathcal{N}\setminus\{i\}}^\lambda(S)V(S)\\
    \end{multline*}

    Finally, by using the expression for the marginal contribution of $i$ to
    $S$:
    \begin{displaymath}
        \partial_i F_\mathcal{N}(\lambda) = 
        \sum_{\substack{S\subset\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
        P_{\mathcal{N}\setminus\{i\}}^\lambda(S)
        \log\Big(1 + \mu x_i^*A(S)^{-1}x_i\Big)
    \end{displaymath}

    The computation of the derivative of $L_\mathcal{N}$ uses standard matrix
    calculus and gives:
    \begin{displaymath}
        \partial_i L_\mathcal{N}(\lambda)
        = \mu x_i^* \tilde{A}(\lambda)^{-1}x_i
    \end{displaymath}
    
    Using the following inequalities:
    \begin{gather*}
        \forall S\subset\mathcal{N}\setminus\{i\},\quad
        P_{\mathcal{N}\setminus\{i\}}^\lambda(S)\geq
        P_{\mathcal{N}\setminus\{i\}}^\lambda(S\cup\{i\})\\
        \forall S\subset\mathcal{N},\quad P_{\mathcal{N}\setminus\{i\}}^\lambda(S) 
        \geq P_\mathcal{N}^\lambda(S)\\
        \forall S\subset\mathcal{N},\quad A(S)^{-1} \geq A(S\cup\{i\})^{-1}\\
    \end{gather*}
    we get:
    \begin{align*}
        \partial_i F_\mathcal{N}(\lambda) 
        & \geq \sum_{\substack{S\subset\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
        P_\mathcal{N}^\lambda(S)
        \log\Big(1 + \mu x_i^*A(S)^{-1}x_i\Big)\\
        & \geq \frac{1}{2}
        \sum_{\substack{S\subset\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
        P_\mathcal{N}^\lambda(S)
        \log\Big(1 + \mu x_i^*A(S)^{-1}x_i\Big)\\
        &\hspace{-3.5em}+\frac{1}{2}
        \sum_{\substack{S\subset\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
        P_\mathcal{N}^\lambda(S\cup\{i\})
        \log\Big(1 + \mu x_i^*A(S\cup\{i\})^{-1}x_i\Big)\\
        &\geq \frac{1}{2}
        \sum_{S\subset\mathcal{N}}
        P_\mathcal{N}^\lambda(S)
        \log\Big(1 + \mu x_i^*A(S)^{-1}x_i\Big)\\
    \end{align*}

    Using that $A(S)\geq I_d$ we get that:
    \begin{displaymath}
        \mu x_i^*A(S)^{-1}x_i \leq \mu
    \end{displaymath}
    
    Moreover:
    \begin{displaymath}
        \forall x\leq\mu,\; \log(1+x)\geq
    \frac{\log\big(1+\mu\big)}{\mu} x
    \end{displaymath}

    Hence:
    \begin{displaymath}
        \partial_i F_\mathcal{N}(\lambda) \geq
        \frac{\log\big(1+\mu\big)}{2\mu}
        x_i^*\bigg(\sum_{S\subset\mathcal{N}}P_\mathcal{N}^\lambda(S)A(S)^{-1}\bigg)x_i
    \end{displaymath}
    
    Finally, using that the inverse is a matrix convex function over symmetric
    positive definite matrices:
    \begin{align*}
        \partial_i F_\mathcal{N}(\lambda) &\geq
        \frac{\log\big(1+\mu\big)}{2\mu}
        x_i^*\bigg(\sum_{S\subset\mathcal{N}}P_\mathcal{N}^\lambda(S)A(S)\bigg)^{-1}x_i\\
        & \geq \frac{\log\big(1+\mu\big)}{2\mu}
        \partial_i L_\mathcal{N}(\lambda)
    \end{align*}
\end{proof}

We can now prove lemma TODO from previous section.

\begin{proof}
    Let us consider a feasible point $\lambda^*\in[0,1]^n$ such that $L_\mathcal{N}(\lambda^*)
    = OPT(L_\mathcal{N}, B)$. By applying lemma~\ref{lemma:relaxation-ratio}
    and lemma~\ref{lemma:rounding} we get a feasible point $\bar{\lambda}$ with at most
    one fractional component such that:
    \begin{equation}\label{eq:e1}
        L_\mathcal{N}(\lambda^*) \leq \frac{1}{C_\mu}
        F_\mathcal{N}(\bar{\lambda})
    \end{equation}

    Let $\lambda_i$ denote the fractional component of $\bar{\lambda}$ and $S$
    denote the set whose indicator vector is $\bar{\lambda} - \lambda_i e_i$.
    Using the fact that $F_\mathcal{N}$ is linear with respect to the $i$-th
    component and is a relaxation of the value function, we get:
    \begin{displaymath}
        F_\mathcal{N}(\bar{\lambda}) = V(S) +\lambda_i V(S\cup\{i\})
    \end{displaymath}

    Using the submodularity of $V$:
    \begin{displaymath}
        F_\mathcal{N}(\bar{\lambda}) \leq 2 V(S) + V(i)
    \end{displaymath}

    Note that since $\bar{\lambda}$ is feasible, $S$ is also feasible and
    $V(S)\leq OPT(V,\mathcal{N}, B)$. Hence:
    \begin{equation}\label{eq:e2}
        F_\mathcal{N}(\bar{\lambda}) \leq 2 OPT(V,\mathcal{N}, B)
        + \max_{i\in\mathcal{N}} V(i)
    \end{equation}

    Putting \eqref{eq:e1} and \eqref{eq:e2} together gives the results.
\end{proof}

\section{General setup}

\section{Conclusion}

\end{document}