From 310718cb00370138b8d6f0e8a8222e5ecdda843c Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Mon, 29 Feb 2016 19:39:56 -0500
Subject: Change to ACM style and inlining of the proofs

---
 approximation.tex | 477 +++++++++++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 471 insertions(+), 6 deletions(-)

(limited to 'approximation.tex')

diff --git a/approximation.tex b/approximation.tex
index 3a4376b..65c0e61 100644
--- a/approximation.tex
+++ b/approximation.tex
@@ -1,7 +1,7 @@
 Previous approaches towards designing truthful, budget feasible mechanisms for \textsc{Knapsack}~\cite{chen} and \textsc{Coverage}~\cite{singer-influence} build upon polynomial-time algorithms that compute an approximation of $OPT$, the optimal value in the full information case. Crucially, to be used in designing a truthful mechanism, such algorithms need also to be \emph{monotone}, in the sense that decreasing any cost $c_i$ leads to an increase  in the estimation of $OPT$; %In the cases of \textsc{Knapsack} and~\textsc{Coverage}, as well as in the case of \EDP{}, 
 the monotonicity property  precludes using traditional approximation algorithms.
 
-In the first part of this section, we address this issue by designing a convex relaxation of \EDP{}, and showing that its solution can be used to approximate $OPT$. The objective of this relaxation is concave and self-concordant \cite{boyd2004convex} and, as such, there exists an algorithm that solves this relaxed problem with arbitrary accuracy in polynomial time. Unfortunately, the output of this algorithm may not necessarily be monotone. Nevertheless, in the second part of this section, we show that a  solver of the relaxed problem can be used to construct a solver that is ``almost'' monotone. In Section~\ref{sec:main}, we show that this algorithm can be used to design a $\delta$-truthful mechanism for \EDP.
+In Section~\ref{sec:concave}, we address this issue by designing a convex relaxation of \EDP{}, and showing that its solution can be used to approximate $OPT$ (Proposition~\ref{prop:relaxation}). The objective of this relaxation is concave and self-concordant \cite{boyd2004convex} and, as such, there exists an algorithm that solves this relaxed problem with arbitrary accuracy in polynomial time. Unfortunately, the output of this algorithm may not necessarily be monotone. Nevertheless, in Section~\ref{sec:monotonicity}, we show that a  solver of the relaxed problem can be used to construct a solver that is ``almost'' monotone (Proposition~\ref{prop:monotonicity}). In Section~\ref{sec:main}, we show that this algorithm can be used to design a $\delta$-truthful mechanism for \EDP.
 
 
 %As noted above, \EDP{} is NP-hard. Designing a mechanism for this problem, as
@@ -29,12 +29,13 @@ In the first part of this section, we address this issue by designing a convex r
 %(Proposition~\ref{prop:relaxation}) and then showing how to approximately solve
 %this problem in a monotone way.
 
-\subsection{A Convex Relaxation of \EDP}\label{sec:concave}
+\subsection{A Convex Relaxation of \EDP}
+\label{sec:concave}
 
 A classical way of relaxing combinatorial optimization problems is 
 \emph{relaxing by expectation}, using the so-called \emph{multi-linear}
 extension of the objective function $V$ (see, \emph{e.g.}, \cite{calinescu2007maximizing,vondrak2008optimal,dughmi2011convex}).
-This is because this extension can yield approximation guarantees for a wide class of combinatorial problems through \emph{pipage rounding}, a technique by \citeN{pipage}.  In general, such relaxations preserve monotonicity which, as discussed, is required in mechanism design.
+This is because this extension can yield approximation guarantees for a wide class of combinatorial problems through \emph{pipage rounding}, a technique introduced by \citeN{pipage}.  In general, such relaxations preserve monotonicity which, as discussed, is required in mechanism design.
 
 Formally, let $P_\mathcal{N}^\lambda$ be a probability distribution over $\mathcal{N}$ parametrized by $\lambda\in [0,1]^n$, where a set $S\subseteq \mathcal{N}$ sampled from  $P_\mathcal{N}^\lambda$ is constructed as follows:  each $i\in \mathcal{N}$ is selected to be in $S$ independently with probability $\lambda_i$, \emph{i.e.},
 %\begin{displaymath}
@@ -73,7 +74,158 @@ For all $\lambda\in[0,1]^{n},$
         \,L(\lambda)\leq
         F(\lambda)\leq L(\lambda)$.
 \end{lemma}
+\begin{comment}
 The proof of this lemma can be found in \cite{arxiv}. In short,  exploiting the concavity of the $\log\det$ function over the set of positive semi-definite matrices, we first  bound the ratio of all partial derivatives of $F$ and $L$. We then show that the bound on the ratio of the derivatives also implies a bound on the ratio $F/L$.
+\end{comment}
+\begin{proof}
+    The bound $F(\lambda)\leq L(\lambda)$ follows by the concavity of the $\log\det$ function and Jensen's inequality.
+    To show the lower bound, 
+    we first prove that $\frac{1}{2}$ is a lower bound of the ratio $\partial_i
+    F(\lambda)/\partial_i L(\lambda)$, where we use
+    $\partial_i\, \cdot$ as a shorthand for  $\frac{\partial}{\partial \lambda_i}$, the partial derivative  with respect to the
+    $i$-th variable. 
+
+   Let us start by computing the partial derivatives of $F$ and
+    $L$ with respect to the $i$-th component. 
+    Observe that
+    \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{displaymath}
+        \partial_i F(\lambda) =
+        \sum_{\substack{S\subseteq\mathcal{N}\\ i\in S}}
+        P_{\mathcal{N}\setminus\{i\}}^\lambda(S\setminus\{i\})V(S)
+        - \sum_{\substack{S\subseteq\mathcal{N}\\ i\in \mathcal{N}\setminus S}}
+        P_{\mathcal{N}\setminus\{i\}}^\lambda(S)V(S).
+    \end{displaymath}
+    Now, using that every $S$ such that $i\in S$ can be uniquely written as
+    $S'\cup\{i\}$, we can write:
+    \begin{displaymath}
+        \partial_i F(\lambda) =
+        \sum_{\substack{S\subseteq\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
+        P_{\mathcal{N}\setminus\{i\}}^\lambda(S)\big(V(S\cup\{i\})
+        - V(S)\big).
+    \end{displaymath}
+Recall from \eqref{eq:marginal_contrib} that the marginal contribution of $i$ to $S$ is given by
+$$V(S\cup \{i\}) - V(S) =\frac{1}{2}\log(1 + \T{x_i}A(S)^{-1}x_i), $$
+where $A(S) = I_d+ \T{X_S}X_S$.
+%  $  V(S\cup\{i\}) - V(S) = \frac{1}{2}\log\left(1 +  \T{x_i} A(S)^{-1}x_i\right)$.
+Using this,
+    \begin{displaymath}
+        \partial_i F(\lambda) = \frac{1}{2}
+        \sum_{\substack{S\subseteq\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
+        P_{\mathcal{N}\setminus\{i\}}^\lambda(S)
+        \log\Big(1 + \T{x_i}A(S)^{-1}x_i\Big)
+    \end{displaymath}
+     The computation of the derivative of $L$ uses standard matrix
+    calculus: writing $\tilde{A}(\lambda) \defeq I_d+\sum_{i\in
+    \mathcal{N}}\lambda_ix_i\T{x_i}$,
+    \begin{displaymath}
+        \det \tilde{A}(\lambda + h\cdot e_i) = \det\big(\tilde{A}(\lambda)
+        + hx_i\T{x_i}\big)
+         =\det \tilde{A}(\lambda)\big(1+
+        h\T{x_i}\tilde{A}(\lambda)^{-1}x_i\big).
+    \end{displaymath}
+    Hence,
+    \begin{displaymath}
+       \log\det\tilde{A}(\lambda + h\cdot e_i)= \log\det\tilde{A}(\lambda)
+        + h\T{x_i}\tilde{A}(\lambda)^{-1}x_i + o(h),
+    \end{displaymath}
+    which implies
+    \begin{displaymath}
+        \partial_i L(\lambda)
+        =\frac{1}{2} \T{x_i}\tilde{A}(\lambda)^{-1}x_i.
+    \end{displaymath}
+Recall from \eqref{eq:inverse} that the monotonicity of the matrix inverse over positive definite matrices implies 
+\begin{gather*}
+        \forall S\subseteq\mathcal{N},\quad A(S)^{-1} \succeq A(S\cup\{i\})^{-1}
+\end{gather*}
+as $A(S)\preceq A(S\cup\{i\})$. Observe that since $1\leq \lambda_i\leq 1$, 
+$P_{\mathcal{N}\setminus\{i\}}^\lambda(S) \geq P_\mathcal{N}^\lambda(S)$ and
+$P_{\mathcal{N}\setminus\{i\}}^\lambda(S)\geq P_{\mathcal{N}}^\lambda(S\cup\{i\})$
+for all $S\subseteq\mathcal{N}\setminus\{i\}$. Hence,
+\begin{align*}
+    \partial_i F(\lambda) 
+    & \geq \frac{1}{4}
+    \sum_{\substack{S\subseteq\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
+    P_{\mathcal{N}}^\lambda(S)
+    \log\Big(1 + \T{x_i}A(S)^{-1}x_i\Big)\\
+    &+\frac{1}{4}
+    \sum_{\substack{S\subseteq\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
+    P_{\mathcal{N}}^\lambda(S\cup\{i\})
+    \log\Big(1 + \T{x_i}A(S\cup\{i\})^{-1}x_i\Big)\\
+    &\geq \frac{1}{4}
+    \sum_{S\subseteq\mathcal{N}}
+    P_\mathcal{N}^\lambda(S)
+    \log\Big(1 + \T{x_i}A(S)^{-1}x_i\Big).
+\end{align*}
+Using that $A(S)\succeq I_d$ we get that $\T{x_i}A(S)^{-1}x_i \leq
+\norm{x_i}_2^2 \leq 1$. Moreover, $\log(1+x)\geq x$ for all $x\leq 1$.
+Hence,
+\begin{displaymath}
+    \partial_i F(\lambda) \geq
+    \frac{1}{4}
+    \T{x_i}\bigg(\sum_{S\subseteq\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 (see Appendix~\ref{app:properties}):
+\begin{displaymath}
+    \partial_i F(\lambda) \geq
+    \frac{1}{4}
+    \T{x_i}\bigg(\sum_{S\subseteq\mathcal{N}}P_\mathcal{N}^\lambda(S)A(S)\bigg)^{-1}x_i
+    = \frac{1}{4}\T{x_i}\tilde{A}(\lambda)^{-1}x_i
+    = \frac{1}{2}
+    \partial_i L(\lambda).
+\end{displaymath}
+
+Having bound the ratio between the partial derivatives, we now bound the ratio
+$F(\lambda)/L(\lambda)$ from below. Consider the following three cases.
+
+First, if the minimum is attained as $\lambda$ converges to zero in,
+\emph{e.g.}, the $l_2$ norm, by the Taylor approximation, one can write:
+\begin{displaymath}
+    \frac{F(\lambda)}{L(\lambda)}
+    \sim_{\lambda\rightarrow 0}
+    \frac{\sum_{i\in \mathcal{N}}\lambda_i\partial_i F(0)}
+    {\sum_{i\in\mathcal{N}}\lambda_i\partial_i L(0)}
+    \geq \frac{1}{2},
+\end{displaymath}
+\emph{i.e.}, the ratio $\frac{F(\lambda)}{L(\lambda)}$ is necessarily bounded
+from below by 1/2 for small enough $\lambda$.
+
+Second, if the minimum of the ratio $F(\lambda)/L(\lambda)$ is attained at
+a vertex of the hypercube $[0,1]^n$ different from 0. $F$ and $L$ being
+relaxations of the value function $V$, they are equal to $V$ on the vertices
+which are exactly the binary points. Hence, the minimum is equal to 1 in this
+case; in particular, it is greater than $1/2$.
+
+Finally, if the minimum is attained at a point $\lambda^*$ with at least one
+coordinate belonging to $(0,1)$, let $i$ be one such coordinate and consider
+the function $G_i$:
+\begin{displaymath}
+    G_i: x \mapsto \frac{F}{L}(\lambda_1^*,\ldots,\lambda_{i-1}^*, x,
+    \lambda_{i+1}^*, \ldots, \lambda_n^*).
+\end{displaymath}
+Then this function attains a minimum at $\lambda^*_i\in(0,1)$ and its
+derivative is zero at this point. Hence:
+\begin{displaymath}
+    0 = G_i'(\lambda^*_i) = \partial_i\left(\frac{F}{L}\right)(\lambda^*).
+\end{displaymath}
+But $\partial_i(F/L)(\lambda^*)=0$ implies that
+\begin{displaymath}
+    \frac{F(\lambda^*)}{L(\lambda^*)} = \frac{\partial_i
+    F(\lambda^*)}{\partial_i L(\lambda^*)}\geq \frac{1}{2}
+\end{displaymath}
+using the lower bound on the ratio of the partial derivatives. This concludes
+the proof of the lemma.
+\end{proof}
 
 Armed with this result, we subsequently use  pipage rounding to show that any $\lambda$ that maximizes the multi-linear extension $F$ can be rounded to an ``almost'' integral solution. More specifically, given a set of costs $c\in \reals^n_+$, we say that a $\lambda\in [0,1]^n$ is feasible if it belongs to the set $\dom_c =\{\lambda \in [0,1]^n: \sum_{i\in \mathcal{N}} c_i\lambda_i\leq B\}$. Then, the following lemma holds:
 
@@ -82,15 +234,85 @@ Armed with this result, we subsequently use  pipage rounding to show that any $\
     $\bar{\lambda}\in \dom_c$ such that (a) $F(\lambda)\leq F(\bar{\lambda})$, and (b) at most one of the 
     coordinates  of $\bar{\lambda}$ is fractional. %, that is, lies in $(0,1)$ and:
 \end{lemma}
+\begin{comment}
 The proof, also in \cite{arxiv}, follows the main steps of the pipage rounding method of \citeN{pipage}. % this  rounding property is referred to in the literature as \emph{cross-convexity} (see, \emph{e.g.}, \cite{dughmi}), or $\varepsilon$-convexity by \citeN{pipage}. 
+\end{comment}
+\begin{proof}
+    We give a rounding procedure which, given a feasible $\lambda$ with at least
+    two fractional components, returns some feasible $\lambda'$ with one fewer fractional
+    component such that $F(\lambda) \leq F(\lambda')$.
+
+    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{equation}\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{equation}
+    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{displaymath}
+        \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{displaymath}
+    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 limits, at which either the $i$-th or $j$-th component of
+    $\lambda_\varepsilon$ becomes integral.
+\end{proof}
+
 Together, Lemma~\ref{lemma:relaxation-ratio} and Lemma~\ref{lemma:rounding} imply that $OPT$, the optimal value of \EDP, can be approximated by solving the following convex optimization problem:
 \begin{align}\tag{$P_c$}\label{eq:primal}
 \text{Maximize:} \quad L(\lambda)\quad \text{subject to:} \quad\lambda \in \dom_c
 \end{align}
-In particular, for $L_c^*\defeq \max_{\lambda\in \dom_c} L(\lambda)$, the following holds \cite{arxiv}:
+In particular, for $L_c^*\defeq \max_{\lambda\in \dom_c} L(\lambda)$, the following holds:
 \begin{proposition}\label{prop:relaxation}
 $OPT\leq L^*_c \leq 2 OPT + 2\max_{i\in\mathcal{N}}V(i)$.
 \end{proposition}
+\begin{proof}
+The lower bound on $L^*_c$ follows immediately from the fact that $L$ extends $V$ to $[0,1]^n$. For the upper bound, let us consider a feasible point $\lambda^*\in \dom_c$ such that
+$L(\lambda^*) = L^*_c$. 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(\lambda^*) \leq 2 F(\bar{\lambda}).
+\end{equation}
+    Let $\lambda_i$ denote the fractional component of $\bar{\lambda}$ and $S$
+    denote the set whose indicator vector is $\bar{\lambda} - \lambda_i e_i$.
+    By definition of the multi-linear extension $F$:
+    \begin{displaymath}
+        F(\bar{\lambda}) = (1-\lambda_i)V(S) +\lambda_i V(S\cup\{i\}).
+    \end{displaymath}
+    By submodularity of $V$, $V(S\cup\{i\})\leq V(S) + V(\{i\})$. Hence,
+    \begin{displaymath}
+        F(\bar{\lambda}) \leq V(S) + V(i).
+    \end{displaymath}
+    Note that since $\bar{\lambda}$ is feasible, $S$ is also feasible and
+    $V(S)\leq OPT$. Hence,
+    \begin{equation}\label{eq:e2}
+        F(\bar{\lambda}) \leq  OPT + \max_{i\in\mathcal{N}} V(i).
+    \end{equation}
+Together, \eqref{eq:e1} and \eqref{eq:e2} imply the proposition.
+\end{proof}
+
 As we discuss in the next section, $L^*_c$ can be computed by a poly-time algorithm at arbitrary accuracy. However, the outcome of this computation may not necessarily be monotone; we address this by converting this poly-time estimator of $L^*_c$ to one that is ``almost'' monotone.%The optimization program \eqref{eq:non-strategic} extends naturally to such
 %a relaxation. We define:
 %\begin{equation}\tag{$P_c$}\label{eq:primal}
@@ -136,13 +358,256 @@ Note that $\dom_c=\dom_{c,0}.$ Consider the following perturbed problem:
 \end{split}
 \end{align}
 
-Restricting the feasible set to $\dom_{c,\alpha}$ ensures that the gradient of the optimal solution with respect to $c$ is bounded from below. This implies that an approximate solution to $P_{c,\alpha}$ given by the barrier method is $\delta$-decreasing with respect to the costs. On the other hand, by taking $\alpha$ small enough, we ensure that the approximate solution to $P_{c,\alpha}$ is still an $\epsilon$-accurate approximation of $L_c^*$. This methodology is summarized in the following proposition, whose proof can be found in \cite{arxiv}.
+Restricting the feasible set to $\dom_{c,\alpha}$ ensures that the gradient of the optimal solution with respect to $c$ is bounded from below. This implies that an approximate solution to $P_{c,\alpha}$ given by the barrier method is $\delta$-decreasing with respect to the costs. On the other hand, by taking $\alpha$ small enough, we ensure that the approximate solution to $P_{c,\alpha}$ is still an $\epsilon$-accurate approximation of $L_c^*$. 
+
+We note that the execution of the barrier method on the restricted set $\dom_{c,\alpha}$ is necessary. The algorithm's output when executed over the entire domain may not necessarily be $\delta$-decreasing, even when the approximation accuracy is small. This is because costs become saturated when the optimal $\lambda\in \dom_c$ lies at the boundary: increasing them has no effect on the objective. Forcing the optimization to happen ``off'' the boundary ensures that this does not occur, while taking $\alpha$ to be small ensures that this perturbation does not cost much in terms of approximation accuracy.
 
 \begin{proposition}\label{prop:monotonicity}
-    For any $\delta\in(0,1]$ and any $\varepsilon\in(0,1]$, using the barrier
+    For any $\delta\in(0,1]$ and $\varepsilon\in(0,1]$, using the barrier
     method to solve \eqref{eq:perturbed-primal} for $\alpha\defeq\varepsilon
     (\delta/B+n^2)^{-1}$ with accuracy $\frac{1}{2^{n+1}B}\alpha\delta b$
     yields a $\delta$-decreasing, $\varepsilon$-accurate approximation of
     $L^*_c$. The running time of the algorithm is $O\big(poly(n, d,
     \log\log\frac{B}{b\varepsilon\delta})\big)$.
 \end{proposition}
+
+A formal description of the algorithm used in
+Proposition~\ref{prop:monotonicity} is given in Algorithm~\ref{alg:monotone}.
+We then proceed by showing that the optimal value of
+\eqref{eq:perturbed-primal} is close to the optimal value of \eqref{eq:primal}
+(Lemma~\ref{lemma:proximity}) while being well-behaved with respect to changes
+of the cost (Lemma~\ref{lemma:monotonicity}). These lemmas together imply
+Proposition~\ref{prop:monotonicity}.
+
+\begin{algorithm}[t]
+    \caption{}\label{alg:monotone}
+    \begin{algorithmic}[1]
+	\Require{ $B\in \reals_+$, $c\in[0,B]^n$, $\delta\in (0,1]$, $\epsilon\in (0,1]$ }
+        \State $\alpha \gets \varepsilon (\delta/B+n^2)^{-1}$ 
+        \State Use the barrier method to solve \eqref{eq:perturbed-primal} with
+        accuracy $\varepsilon'=\frac{1}{2^{n+1}B}\alpha\delta b$; denote the output by $\hat{L}^*_{c,\alpha}$
+	\State \textbf{return} $\hat{L}^*_{c,\alpha}$
+    \end{algorithmic}
+\end{algorithm}
+
+
+
+Note that the choice of $\alpha$ given in Algorithm~\ref{alg:monotone} implies
+that $\alpha<\frac{1}{n}$. This in turn implies that the feasible set
+$\mathcal{D}_{c, \alpha}$ of \eqref{eq:perturbed-primal} is non-empty: it
+contains the strictly feasible point $\lambda=(\frac{1}{n},\ldots,\frac{1}{n})$.
+
+\begin{lemma}\label{lemma:derivative-bounds}
+    Let $\partial_i L(\lambda)$ denote the $i$-th derivative of $L$, for $i\in\{1,\ldots, n\}$, then:
+    \begin{displaymath}
+        \forall\lambda\in[0, 1]^n,\;\frac{b}{2^n} \leq \partial_i L(\lambda) \leq 1
+    \end{displaymath}
+\end{lemma}
+
+\begin{proof}
+    Recall that we had defined:
+    \begin{displaymath}
+        \tilde{A}(\lambda)\defeq I_d + \sum_{i=1}^n \lambda_i x_i\T{x_i}
+        \quad\mathrm{and}\quad
+        A(S) \defeq I_d + \sum_{i\in S} x_i\T{x_i}
+    \end{displaymath}
+    Let us also define $A_k\defeq A(\{x_1,\ldots,x_k\})$.
+    We have $\partial_i L(\lambda) = \T{x_i}\tilde{A}(\lambda)^{-1}x_i$. Since
+    $\tilde{A}(\lambda)\succeq I_d$, $\partial_i L(\lambda)\leq \T{x_i}x_i \leq 1$, which
+    is the right-hand side  of the lemma.
+    For the left-hand side, note that $\tilde{A}(\lambda) \preceq A_n$. Hence
+    $\partial_iL(\lambda)\geq \T{x_i}A_n^{-1}x_i$.
+    Using the Sherman-Morrison formula \cite{sm}, for all $k\geq 1$:
+    \begin{displaymath}
+        \T{x_i}A_k^{-1} x_i = \T{x_i}A_{k-1}^{-1}x_i 
+        - \frac{(\T{x_i}A_{k-1}^{-1}x_k)^2}{1+\T{x_k}A_{k-1}^{-1}x_k}
+    \end{displaymath}
+    By the Cauchy-Schwarz inequality:
+    \begin{displaymath}
+        (\T{x_i}A_{k-1}^{-1}x_k)^2 \leq \T{x_i}A_{k-1}^{-1}x_i\;\T{x_k}A_{k-1}^{-1}x_k
+    \end{displaymath}
+    Hence:
+    \begin{displaymath}
+        \T{x_i}A_k^{-1} x_i \geq \T{x_i}A_{k-1}^{-1}x_i 
+        - \T{x_i}A_{k-1}^{-1}x_i\frac{\T{x_k}A_{k-1}^{-1}x_k}{1+\T{x_k}A_{k-1}^{-1}x_k}
+    \end{displaymath}
+    But $\T{x_k}A_{k-1}^{-1}x_k\leq 1$ and $\frac{a}{1+a}\leq \frac{1}{2}$ if
+    $0\leq a\leq 1$, so:
+    \begin{displaymath}
+        \T{x_i}A_{k}^{-1}x_i \geq \T{x_i}A_{k-1}^{-1}x_i
+        - \frac{1}{2}\T{x_i}A_{k-1}^{-1}x_i\geq \frac{\T{x_i}A_{k-1}^{-1}x_i}{2}
+    \end{displaymath}
+    By induction:
+    \begin{displaymath}
+        \T{x_i}A_n^{-1} x_i \geq \frac{\T{x_i}x_i}{2^n}
+    \end{displaymath}
+    Using that $\T{x_i}{x_i}\geq b$ concludes the proof of the left-hand side
+    of the lemma's inequality.
+\end{proof}
+Let us introduce the Lagrangian of problem \eqref{eq:perturbed-primal}:
+
+\begin{displaymath}
+    \mathcal{L}_{c, \alpha}(\lambda, \mu, \nu, \xi) \defeq L(\lambda) 
+    + \T{\mu}(\lambda-\alpha\mathbf{1}) + \T{\nu}(\mathbf{1}-\lambda) + \xi(B-\T{c}\lambda)
+\end{displaymath}
+so that:
+\begin{displaymath}
+    L^*_{c,\alpha} = \min_{\mu, \nu, \xi\geq 0}\max_\lambda \mathcal{L}_{c, \alpha}(\lambda, \mu, \nu, \xi)
+\end{displaymath}
+Similarly, we define $\mathcal{L}_{c}\defeq\mathcal{L}_{c, 0}$ the lagrangian of \eqref{eq:primal}.
+
+Let $\lambda^*$ be primal optimal for \eqref{eq:perturbed-primal}, and $(\mu^*,
+\nu^*, \xi^*)$ be dual optimal for the same problem. In addition to primal and
+dual feasibility, the Karush-Kuhn-Tucker (KKT) conditions \cite{boyd2004convex} give $\forall i\in\{1, \ldots, n\}$:
+\begin{gather*}
+    \partial_i L(\lambda^*) + \mu_i^* - \nu_i^* - \xi^* c_i = 0\\
+    \mu_i^*(\lambda_i^* - \alpha) = 0\\
+    \nu_i^*(1 - \lambda_i^*) = 0
+\end{gather*}
+
+\begin{lemma}\label{lemma:proximity}
+We have:
+\begin{displaymath}
+    L^*_c - \alpha n^2\leq L^*_{c,\alpha} \leq L^*_c
+\end{displaymath}
+In particular, $|L^*_c - L^*_{c,\alpha}| \leq \alpha n^2$.
+\end{lemma}
+
+\begin{proof}
+    $\alpha\mapsto L^*_{c,\alpha}$ is a decreasing function as it is the
+    maximum value of the $L$ function over a set-decreasing domain, which gives
+    the rightmost inequality.
+
+    Let $\mu^*, \nu^*, \xi^*$ be dual optimal for $(P_{c, \alpha})$, that is:
+    \begin{displaymath}
+        L^*_{c,\alpha} = \max_\lambda \mathcal{L}_{c, \alpha}(\lambda, \mu^*, \nu^*, \xi^*)
+    \end{displaymath}
+
+    Note that $\mathcal{L}_{c, \alpha}(\lambda, \mu^*, \nu^*, \xi^*)
+    = \mathcal{L}_{c}(\lambda, \mu^*, \nu^*, \xi^*)
+    - \alpha\T{\mathbf{1}}\mu^*$, and that for any $\lambda$ feasible for
+    problem \eqref{eq:primal}, $\mathcal{L}_{c}(\lambda, \mu^*, \nu^*, \xi^*)
+    \geq L(\lambda)$. Hence,
+    \begin{displaymath}
+        L^*_{c,\alpha} \geq L(\lambda) - \alpha\T{\mathbf{1}}\mu^*
+    \end{displaymath}
+    for any $\lambda$ feasible for \eqref{eq:primal}. In particular, for $\lambda$ primal optimal for $\eqref{eq:primal}$:
+    \begin{equation}\label{eq:local-1}
+        L^*_{c,\alpha} \geq L^*_c - \alpha\T{\mathbf{1}}\mu^*
+    \end{equation}
+
+    Let us denote by the $M$ the support of $\mu^*$, that is $M\defeq
+    \{i|\mu_i^* > 0\}$, and by $\lambda^*$ a primal optimal point for
+    $\eqref{eq:perturbed-primal}$.  From the KKT conditions we see that:
+    \begin{displaymath}
+        M \subseteq \{i|\lambda_i^* = \alpha\}
+    \end{displaymath}
+
+
+    Let us first assume that $|M| = 0$, then $\T{\mathbf{1}}\mu^*=0$ and the lemma follows.
+
+    We will now assume that $|M|\geq 1$. In this case $\T{c}\lambda^*
+    = B$, otherwise we could increase the coordinates of $\lambda^*$ in $M$,
+    which would increase the value of the objective function and contradict the
+    optimality of $\lambda^*$. Note also, that $|M|\leq n-1$, otherwise, since
+    $\alpha< \frac{1}{n}$, we would have $\T{c}\lambda^*\ < B$, which again
+    contradicts the optimality of $\lambda^*$. Let us write:
+    \begin{displaymath}
+        B = \T{c}\lambda^* = \alpha\sum_{i\in M}c_i + \sum_{i\in \bar{M}}\lambda_i^*c_i
+        \leq \alpha |M|B + (n-|M|)\max_{i\in \bar{M}} c_i
+    \end{displaymath}
+    That is:
+    \begin{equation}\label{local-2}
+        \max_{i\in\bar{M}} c_i \geq \frac{B - B|M|\alpha}{n-|M|}> \frac{B}{n}
+    \end{equation}
+    where the last inequality uses again that $\alpha<\frac{1}{n}$. From the
+    KKT conditions, we see that for $i\in M$, $\nu_i^* = 0$ and:
+    \begin{equation}\label{local-3}
+        \mu_i^* = \xi^*c_i - \partial_i L(\lambda^*)\leq \xi^*c_i\leq \xi^*B
+    \end{equation}
+    since $\partial_i L(\lambda^*)\geq 0$ and $c_i\leq 1$.
+
+    Furthermore, using the KKT conditions again, we have that:
+    \begin{equation}\label{local-4}
+        \xi^* \leq \inf_{i\in \bar{M}}\frac{\partial_i L(\lambda^*)}{c_i}\leq \inf_{i\in \bar{M}} \frac{1}{c_i}
+        = \frac{1}{\max_{i\in\bar{M}} c_i}
+    \end{equation}
+    where the last inequality uses Lemma~\ref{lemma:derivative-bounds}.
+    Combining \eqref{local-2}, \eqref{local-3} and \eqref{local-4}, we get that:
+    \begin{displaymath}
+        \sum_{i\in M}\mu_i^* \leq |M|\xi^*B \leq n\xi^*B\leq \frac{nB}{\max_{i\in\bar{M}} c_i} \leq n^2
+    \end{displaymath}
+    This implies that:
+    \begin{displaymath}
+        \T{\mathbf{1}}\mu^* = \sum_{i=1}^n \mu^*_i = \sum_{i\in M}\mu_i^*\leq n^2
+    \end{displaymath}
+    which along with \eqref{eq:local-1} proves the lemma.
+\end{proof} 
+
+\begin{lemma}\label{lemma:monotonicity}
+    If $c'$ = $(c_i', c_{-i})$, with $c_i'\leq c_i - \delta$, we have:
+    \begin{displaymath}
+        L^*_{c',\alpha} \geq L^*_{c,\alpha} + \frac{\alpha\delta b}{2^nB}
+    \end{displaymath}
+\end{lemma}
+
+\begin{proof}
+    Let $\mu^*, \nu^*, \xi^*$ be dual optimal for $(P_{c', \alpha})$. Noting that:
+    \begin{displaymath} 
+    \mathcal{L}_{c', \alpha}(\lambda, \mu^*, \nu^*, \xi^*) \geq
+    \mathcal{L}_{c, \alpha}(\lambda, \mu^*, \nu^*, \xi^*) + \lambda_i\xi^*\delta,
+    \end{displaymath}
+    we get similarly to Lemma~\ref{lemma:proximity}:
+    \begin{displaymath}
+        L^*_{c',\alpha} \geq L(\lambda) + \lambda_i\xi^*\delta
+    \end{displaymath}
+    for any $\lambda$ feasible for \eqref{eq:perturbed-primal}. In particular, for $\lambda^*$ primal optimal for \eqref{eq:perturbed-primal}:
+    \begin{displaymath}
+        L^*_{c',\alpha} \geq L^*_{c,\alpha} + \alpha\xi^*\delta
+    \end{displaymath}
+    since $\lambda_i^*\geq \alpha$.
+
+    Using the KKT conditions for $(P_{c', \alpha})$, we can write:
+    \begin{displaymath}
+        \xi^* = \inf_{i:\lambda^{'*}_i>\alpha} \frac{\T{x_i}S(\lambda^{'*})^{-1}x_i}{c_i'}
+    \end{displaymath}
+    with $\lambda^{'*}$ optimal for $(P_{c', \alpha})$. Since $c_i'\leq B$,
+    using Lemma~\ref{lemma:derivative-bounds}, we get that $\xi^*\geq
+    \frac{b}{2^nB}$, which concludes the proof.
+\end{proof}
+
+We are now ready to conclude the  proof of Proposition~\ref{prop:monotonicity}.
+
+\begin{proof}
+Let $\hat{L}^*_{c,\alpha}$ be the approximation computed by
+Algorithm~\ref{alg:monotone}.
+\begin{enumerate}
+    \item using Lemma~\ref{lemma:proximity}:
+\begin{displaymath}
+        |\hat{L}^*_{c,\alpha} - L^*_c| \leq |\hat{L}^*_{c,\alpha} - L^*_{c,\alpha}| + |L^*_{c,\alpha} - L^*_c|
+        \leq \frac{\alpha\delta}{B} + \alpha n^2 = \varepsilon
+\end{displaymath}
+which proves the $\varepsilon$-accuracy.
+
+\item for the $\delta$-decreasingness, let $c' = (c_i', c_{-i})$ with $c_i'\leq
+    c_i-\delta$, then:
+\begin{displaymath}
+    \hat{L}^*_{c',\alpha} \geq L^*_{c',\alpha} - \frac{\alpha\delta b}{2^{n+1}B} 
+                     \geq L^*_{c,\alpha} + \frac{\alpha\delta b}{2^{n+1}B}
+    \geq \hat{L}^*_{c,\alpha}
+\end{displaymath}
+where the first and last inequalities follow from the accuracy of the approximation, and
+the inner inequality follows from Lemma~\ref{lemma:monotonicity}.
+
+\item the accuracy of the approximation $\hat{L}^*_{c,\alpha}$ is:
+\begin{displaymath}
+    \varepsilon' =\frac{\varepsilon\delta b}{2^{n+1}(\delta + n^2B)}
+\end{displaymath}
+
+Note that:
+\begin{displaymath}
+    \log\log (\varepsilon')^{-1} = O\bigg(\log\log\frac{B}{\varepsilon\delta b} + \log n\bigg)
+\end{displaymath}
+Using Lemma~\ref{lemma:barrier} concludes the proof of the running time.\qedhere
+\end{enumerate}
+\end{proof}
-- 
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