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diff --git a/appendix.tex b/appendix.tex index 5cb60f8..73e879d 100755 --- a/appendix.tex +++ b/appendix.tex @@ -264,7 +264,72 @@ Together, \eqref{eq:e1} and \eqref{eq:e2} imply the proposition.\qed %For any $\varepsilon>0$, the barrier method computes an $\varepsilon$-accurate %approximation of $L^*_c$ in time $O(poly(n,d,\log\log\varepsilon^{-1})$. %\end{lemma} -We proceed by showing that the optimal value of \eqref{eq:perturbed-primal} is close to the +Our next technical result establishes that, using the barrier method, it is possible to construct an algorithm that computes $L^*_c$ at arbitrary accuracy in polynomial time \emph{and} is $\delta$-decreasing. We achieve this by restricting the optimization over a subset of $\dom_c$ at which the concave relaxation $L$ is ``sufficiently'' concave. Formally, for $\alpha\geq 0$ let $$\textstyle\dom_{c,\alpha} \defeq \{\lambda \in [\alpha,1]^n: \sum_{i\in \mathcal{N}}c_i\lambda_i \leq B\}\subseteq \dom_c . $$ +Note that $\dom_c=\dom_{c,0}.$ Consider the following perturbation of the concave relaxation \eqref{eq:primal}: +\begin{align}\tag{$P_{c,\alpha}$}\label{eq:perturbed-primal} +\begin{split} \text{Maximize:} &\qquad L(\lambda)\\ +\text{subject to:} & \qquad\lambda \in \dom_{c,\alpha} +\end{split} +\end{align} + +%Note, that the feasible set in Problem~\eqref{eq:primal} increases (for the +%inclusion) when the cost decreases. +%non-increasing. + +%Furthermore, \eqref{eq:primal} being a convex optimization problem, using +%standard convex optimization algorithms (Lemma~\ref{lemma:barrier} gives +%a formal statement for our specific problem) we can compute +%a $\varepsilon$-accurate approximation of its optimal value as defined below. + +%\begin{definition} +%$a$ is an $\varepsilon$-accurate approximation of $b$ iff $|a-b|\leq \varepsilon$. +%\end{definition} + +%Note however that an $\varepsilon$-accurate approximation of a non-increasing +%function is not in general non-increasing itself. The goal of this section is +%to approximate $L^*_c$ while preserving monotonicity. The estimator we +%construct has a weaker form of monotonicity that we call +%\emph{$\delta$-monotonicity}. + +%\begin{definition} +%Let $f$ be a function from $\reals^n$ to $\reals$, we say that $f$ is +%\emph{$\delta$-increasing along the $i$-th coordinate} iff: +%\begin{equation}\label{eq:dd} +% \forall x\in\reals^n,\; +% \forall \mu\geq\delta,\; +% f(x+\mu e_i)\geq f(x) +%\end{equation} +%where $e_i$ is the $i$-th canonical basis vector of $\reals^n$. By extension, +%$f$ is $\delta$-increasing iff it is $\delta$-increasing along all coordinates. + +%We define \emph{$\delta$-decreasing} functions by reversing the inequality in +%\eqref{eq:dd}. +%\end{definition} + +%We consider a perturbation of \eqref{eq:primal} by introducing: +%\begin{equation}\tag{$P_{c, \alpha}$}\label{eq:perturbed-primal} +% L^*_{c,\alpha} \defeq \max_{\lambda\in[\alpha, 1]^{n}} +% \left\{L(\lambda) \Big| \sum_{i=1}^{n} \lambda_i c_i +% \leq B\right\} +%\end{equation} +%Note that we have $L^*_c = L^*_c(0)$. We will also assume that +%$\alpha<\frac{1}{nB}$ so that \eqref{eq:perturbed-primal} has at least one +%feasible point: $(\frac{1}{nB},\ldots,\frac{1}{nB})$. By computing +%an approximation of $L^*_{c,\alpha}$ as in Algorithm~\ref{alg:monotone}, we +%obtain a $\delta$-decreasing approximation of $L^*_c$. + +\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} + +Our construction of a $\delta$-decreasing, $\varepsilon$-accurate approximator of $L_c^*$ proceeds as follows: first, it computes an appropriately selected lower bound $\alpha$; using this bound, it solves the perturbed problem \eqref{eq:perturbed-primal} using the barrier method, also at an appropriately selected accuracy $\varepsilon'$, obtaining thus a $\varepsilon'$-accurate approximation of $L^*_{c,\alpha}\defeq \max_{\lambda\in \dom_{c,\alpha}} L(\lambda)$ . The corresponding output is returned as an approximation of $L^*_c$. A summary of the algorithm and the specific choices of $\alpha$ and $\varepsilon'$ are given in Algorithm~\ref{alg:monotone}. We 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 |