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diff --git a/paper/sections/introduction.tex b/paper/sections/introduction.tex new file mode 100644 index 0000000..6eda59d --- /dev/null +++ b/paper/sections/introduction.tex @@ -0,0 +1,54 @@ +The natural “diminishing return” property of submodular functions makes them +suitable to model many real-life scenarios. However, even when they are +suitable as a model, it is not realistic to assume that they are entirely +known. For example, most submodular functions depend on unknown parameters that +have to be learned from observations. + +A recent line of work has focused on learning submodular functions. The +PMAC-learnable framework formalizes what it means to learn a submodular +function in an “overall” sense. For specific examples of submodular functions, +results are known on how well the parameters of the function can be learned +from observations. + +When the experimenter wishes to optimize the unknown submodular function, the +well-known approximation algorithms cannot be readily applied because they rely +on a value query oracle, which is much stronger than what the present scenario +suggests (at best, random independent samples). The results on learning +submodular functions also seem to be insufficient in that they are myopic to +the ultimate use which will be made of the submodular function (optimizing). +The experimenter experimenter only cares about learning the function to the +extent that what he knows is sufficient to optimize it. As such, the main +question underlying the present work is the following: + +\begin{center} + \emph{What is a good model of learning for submodular functions when the + ultimate goal is to be able to optimize them?} +\end{center} + +We make the following contributions: +\begin{itemize} + \item We show how the PMAC learning framework does not provide a satisfying + framework for optimizing under uncertainty. In particular, we show that + there are functions which are PMAC-learnable but not optimizable and + vice-versa. + \item For specific class of submodular functions, we show how to optimize + them under uncertainty. +\end{itemize} + +\subsection{Related Work} + +\paragraph{Maximizing submodular functions} Vast literature on maximizing +submodular functions under many kind of constraints. All the algorithms rely on +some kind of oracle (e.g. value query or demand query) which is stronger than +what we have: access to polynomially many observations coming from +a distribution. In particular, an algorithm cannot make arbitrary (and +certainly not adaptive) queries to the oracle. + +\paragraph{PMAC-learning} Interesting line of work, but oblivious to the +ultimate goal. + +\paragraph{Bandits} Interesting in that it ties together learning and +optimizing. However, it assumes that arbitrary queries can be made to an oracle +allowing the experimenter to explore more efficiently (even though the standard +regret benchmark means that he will incur some cost for exploring vs +exploiting. diff --git a/paper/sections/negative.tex b/paper/sections/negative.tex new file mode 100644 index 0000000..555aa1d --- /dev/null +++ b/paper/sections/negative.tex @@ -0,0 +1,143 @@ +\subsection{Model} + +Let $\Omega$ be the universe of elements and $f$ a function defined on subsets +of $\Omega$: $f : S \in 2^{[\Omega]} \mapsto f(S) \in \mathbb{R}$. Let $K$ be a +collection of sets of $2^{[\Omega]}$, which we call \emph{constraints}. Let +$S^*_K$ be any solution to $\max_{S \in K} f(S)$, which we will also denote by +$S^*$ when there is no ambiguity. Let $L$ be the problem size, which is often +(but not always) equal to $|\Omega|$. + +In general, we say we can efficiently optimize a function $f$ under constraint +$K$ when we have a polynomial-time algorithm making adaptive value queries to +$f$,which returns a set $S$ such that $S \in K$ and $f(S) \geq \alpha f(S^*)$ +with high probability and $\alpha$ an absolute constant. + +Here, we consider the scenario where we cannot make adaptive value queries, and +in fact, where we cannot make queries at all! Instead, we suppose that we +observe a polynomial number of set-value pairs $(S, f(S))$ where $S$ is taken +from a known distribution $D$. We say we can efficiently \emph{passively +optimize} $f$ under distribution $D$ or $D-$optimize $f$ under constraints $K$ +when, after observing ${\cal O}(L^c)$ set-value pairs from $D$ where $c > 0$ is +an absolute constant, we can return a set $S$ such that $S \in K$ and $f(S) +\geq \alpha f(S^*)$ with high probability and $\alpha$ an absolute constant. + +In the case of \emph{passive} observations of set-value pairs under a +distribution $D$ for a function $f$, recent research has focused on whether we +can efficiently and approximately \emph{learn} $f$. This was formalized in the +PMAC model from \cite{balcan2011learning}. When thinking about passive +optimization, it is necessary to understand the link between being able to + $D-PMAC$ learn $f$ and being able to $D-$optimize $f$. + +\subsection{Learnable $\nRightarrow$ Optimizable} + +Consider the following function: +\begin{displaymath} + g(S) = \max\left\{\sqrt{n}|X\cap S|, |S|\right\} +\end{displaymath} +where $|X|=\sqrt{n}$. Assume that you are given polynomially many samples where +each element is included with probability $1/2$. Then with high probability all +the samples will have size roughly $n/2$, so you will observe $g(S)=|S|$ with +high probability. + +\paragraph{Claim 1:} $g$ is PAC-learnable because if you output $|S|$ then you +will be correct with high probability is $S$ is drawn from the same +distribution as above. + +\paragraph{Claim 2:} $g$ is not optimizable under budget $\sqrt{n}$ because you +never learn anything about $X$. + +\paragraph{Open Question:} Cook a similar example where $g$ is submodular and +where you are observing sets of the same size as your budget. + +\paragraph{Positive results:} Try to obtain guarantees about learning +parameters of parametric submodular functions and show whether or not these +guarantees are sufficient for optimization. First look at learning weights in +a cover function. Maybe facility location? Sums of concave over modular are +probably too hard because of the connection to neural networks. + +\subsection{Optimizable $\nRightarrow$ Learnable} + +Recall the matroid construction from~\cite{balcan2011learning}: +\begin{theorem} + For any $k$ with $k = 2^{o(n^{1/3})}$, there exists a family of sets ${\cal A} + \subseteq2^{[n]}$ and a family of matroids $\cal{M} = \{M_{\cal{B}} : + \cal{B} \subseteq\cal{A} \}$ such that: + \begin{itemize} + \item $|{\cal A}| = k$ and $|A| = n^{1/3}$ for every $A \in \cal{A}$ + \item For every $\cal{B} \subseteq\cal{A}$ and every $A \in \cal{A}$, we + have: + \begin{align*} + \text{rank}_{M_{\cal{B}}}(A) & = 8 \log k \ if A\in{\cal B} \\ + & = |A| \ if A\in {\cal A}\backslash{\cal B} + \end{align*} + \end{itemize} +\end{theorem} + +Consider the following subset of the above family of matroids: ${\cal +M}^{\epsilon} = \{M_{\cal B} : {\cal B} \subseteq{\cal A} \wedge |{\cal +A}\backslash{\cal B}| \geq \epsilon|{\cal A}|\}$ for $k = 2^{n^{1/6}}$. +Consider an \emph{unknown} function $f$, corresponding to the rank function of +one of the matroids $M_{\cal B}$ from ${\cal M}^{\epsilon}$. Note that as long +as we observe \emph{one} set from ${\cal A} \backslash {\cal B}$, we can +optimize $f$ exactly! Indeed, the largest possible value for $f$ under +cardinality constraint $n^{1/3}$ is $\max_{A\in 2^{[n]}} |A| = n^{1/3}$. + +One example of a distribution under which this occurs with probability at least +a constant is $D_u$, the uniform distribution over all sets of ${\cal A}$. For +$c>0$, after $n^c$ observations $O \equiv (S_i, f(S_i))_i$ for $S_i \sim D_u$, +we will observe at least one element from $\cal{A}\backslash\cal{B}$ with +constant probability: + +\begin{equation} \nonumber \mathbb{P}(\bigcup_{S_i} S_i \in {\cal +A}\backslash{\cal B}) \geq 1 - (1 - \epsilon)^{n^c} \approx \epsilon n^c +\end{equation} + +However, it follows from the analysis of~\cite{balcan2011learning} that we +cannot learn $f$ under any distribution, even with active value queries! +Indeed, for any polynomially-sized set of observations, there exists a +super-polynomially number of functions in ${\cal M^1}$ which coincide on this +set of observations, but which take very different values outside of this set +of observations: $8n^{1/6}$ for $A \in {\cal B}$ and $n^{1/3}$ for $A \in {\cal +A}\backslash {\cal B}$. + +{\bf TODO:} A cleaner simpler example would be nice. + +\subsection{Which learning guarantees imply optimization?} + +Here, we consider the following question: which learning guarantees imply +optimization? For example, \cite{TODO} provides the following guarantee for +cover functions: + +\begin{theorem} +There exists an algorithm such that for any $\epsilon>0$ given random and +uniform examples of a coverage function $c$ outputs a hypothesis, which is also +a coverage function $h$, such that with probability $2/3$: $\mathbb{E}_{\cal +U}[|h(x) - c(x)|] \leq \epsilon$. The algorithm runs in time $\tilde {\cal O}(n) +\cdot \text{poly}(s/\epsilon)$ and uses $\log(n)\cdot \text{poly}(s/epsilon)$ +and examples, where $s = \min\{size(c), (1/\epsilon)^{\log(1/\epsilon)}\}$. +\end{theorem} + +We would like to understand to what extent this $\ell1$-bound allows us to +optimize the coverage function $c$ under cardinality constraints using the +hypothesis $h$. Let us consider the simpler case of an additive function, and +suppose we have a similar guarantee: $\mathbb{E}_{x \sim {\cal U}}[|h(x) - +c(x)|]$ where $\forall x, c(x) \equiv \sum_i w_i x_i$ and $h(x) \equiv \sum_i +\hat w_i x_i$. Can we find a bound on $\|w - \hat w\|_{\infty}$? + +As it turns out, yes. Let $\forall i, w_i - \hat w_i \equiv \alpha_i$ and let +$V(x) \equiv |h(x) - c(x)| = |\sum_{i} \alpha_i x_i |$. Consider the collection +of good sets ${\cal G} \equiv \{S : v(S) < 4\epsilon\}$. We claim that $|{\cal G}| +\geq \frac{3}{4}\cdot2^n$. Suppose the contrary, there is at least a quarter of +the sets which have value $v(S) > 4\epsilon$ such that $\mathbb{E}_{x \sim {\cal +U}}[|v(x)|] \geq \frac{1}{2^n}\sum_{S \in {\cal G}^c} |v(S)| > +\frac{1}{4}\cdot4\epsilon = \epsilon$ which is a contradiction. Consider element +$i$ such that $|\alpha_i| \equiv \|\alpha\|_{\infty}$. Consider the collection +of sets which contain $i$: ${\cal S}_i \equiv \{S : i \in S\}$. Notice that +$|{\cal S}_i| = |{\cal S}_i^c| = 2^{n-1}$. Therefore, $|{\cal G} \cap {\cal +S}_i^c| \geq \frac{1}{4}\cdot 2^n$. For all sets $S$ in ${\cal G} \cap {\cal +S}_i^c$, $v(S\cup\{j\}) \geq \alpha_i - 4\epsilon$. It follows that +$\mathbb{E}_{x \sim {\cal U}}[|v(x)|] \geq \frac{1}{4}(\alpha_j - 4\epsilon)$ +and therefore we have $\|w - \hat w\|_{\infty} \leq 8 \epsilon$. + + + diff --git a/paper/sections/positive.tex b/paper/sections/positive.tex new file mode 100644 index 0000000..c4e8e0f --- /dev/null +++ b/paper/sections/positive.tex @@ -0,0 +1,365 @@ +\subsection{Applications} + +\paragraph{TODO:} Add citations! + +\subsubsection{Building blocks} + +These are generic examples and serve as building blocks for the applications +below: +\begin{itemize} + \item $f(S) = g\left(\sum_{i\in S} w_i\right)$ for a concave + $g:\reals\to\reals$ and weights $(w_i)_{i\in N}\in \reals_+^{|N|}$. Note + that $f$ is monotone iff $g$ is monotone. In this case, $g$ does not + matter for the purpose of optimization: the sets are in the same order + with or without $g$, the only things which matter are the weights which + serve as natural parameters for this class of functions. This class of + functions contains: + \begin{itemize} + \item additive functions (when $g$ is the identity function). + \item $f(S) = |S\cap X|$ for some set $X\subseteq N$. This is the + case where the weights are $0/1$ and $g$ is the identity + function. + \item symmetric submodular functions: when the weights are all one. + \item budget-additive functions, when $g:x\mapsto \min(B, x)$ for + some $B$. + \end{itemize} + \item $f(S) = \max_{i\in S} w_i$ for weights $(w_i)_{i\in N}\in + \reals_+^{|N|}$. This class of functions is also naturally parametrized + by the weights. + \item (weighted) matroid rank functions. Given a matroid $M$ over a ground + set $N$, we define its rank function to be: + \begin{displaymath} + \forall S\subseteq N,\; + r(S) = \max_{\substack{I\subseteq S\\I\in M}} |I| + \end{displaymath} + more generally, given a weight function $w:N\to\reals_+$, we define the + weighted matroid rank function: + \begin{displaymath} + \forall S\subseteq N,\; + r(S) = \max_{\substack{I\subseteq S\\I\in M}} \sum_{i\in I} w_i + \end{displaymath} +\end{itemize} + +\paragraph{Remark} The function $f(S)= \max_{i\in S}w_i$ is a weighted matroid +rank function for the $1$-uniform matroid (the matroid where the independent +sets are the sets of size at most one). + +\subsubsection{Examples} + +\begin{itemize} + \item \emph{Coverage functions:} they can be written as a positive linear + combination of matroid rank functions: + \begin{displaymath} + f(S) = \sum_{u\in\mathcal{U}} w_u c_u(S) + \end{displaymath} + where $c_u$ is the rank function of the matroid $M = \big\{ \emptyset, + \{u\}\big\}$. + \item \emph{Facility location:} (cite Bilmes) there is a universe + $\mathcal{U}$ of locations and a proximity score $s_{i,j}$ for each + pair of locations. We pick a subset of locations $S$ and each point in + the universe is allocated to its closest location (the one with highest + proximity): + \begin{displaymath} + f(S) = \sum_{u\in\mathcal{U}} \max_{v\in S} s_{u,v} + \end{displaymath} + This can be seen as a sum of weighted matroid rank functions: one for + each location in the universe associated with a $1$-uniform matroid + (other applications: job scheduling). + + \item \emph{Image segmentation:} (cite Jegelka) can be (in some cases) + written as a graph cut function. For image segmentation the goal is to + minimize the cut. + \begin{displaymath} + f(S) = \sum_{e\in E} w_e c_e(S) + \end{displaymath} + where $c_e(S)$ is one iff $e\in E(S,\bar{S})$. \textbf{TODO:} I think + this can be written as a matroid rank function. + \item \emph{Learning} (cite Krause) there is + a hypothesis $A$ (a random variable) which is “refined” when more + observations are made. Imagine that there is a finite set $X_1,\dots, + X_n$ of possible observations (random variables). Then, assuming that + the observations are independent conditioned on $A$, the information + gain: + \begin{displaymath} + f(S) = H(A) - H\big(A\,|\,(X_i)_{i\in S}\big) + \end{displaymath} + is submodular. The $\log\det$ is the specific case of a linear + hypothesis observed with additional independent Gaussian noise. + \item \emph{Entropy:} Closely related to the previous one. If $(X_1,\dots, + X_n)$ are random variables, then: + $ f(S) = H(X_S) $ is submodular. In particular, if $(X_1,\dots, + X_n)$ are jointly multivariate gaussian, then: + \begin{displaymath} + f(S) = c|S| + \frac{1}{2}\log\det X_SX_S^T + \end{displaymath} + for $c= 2\pi e...$ and we fall back to the usual $\log\det$ function. + \item \emph{data subset selection/summarization:} in statistical machine + translation, Bilmes used sum of concave over modular: + \begin{displaymath} + f(S) = \sum_{f} \lambda_f \phi\left(\sum_{e\in S}w_f(e)\right) + \end{displaymath} + where each $f$ represents a feature, $w_f(e)$ represents how much of + $f$ element $e$ has, and $\phi$ captures decreasing marginal gain when + we have a lot of a given feature. + Facility location functions are also commonly used for subset selection. + \item \emph{concave spectral functions} One would be tempted to say that + any multivariate concave function of a modular function is submodular. + This would be the natural generalization of concave over modular. + However \textbf{this is not true in general}. However, a possible nice + generalization is the following. Let $M$ be a symmetric $n\times + n$ matrix, and $g$ is a ``matrix concave'' function. Then: + \begin{displaymath} + f(S) = \mathrm{Tr}\big(g(M_S)\big) + \end{displaymath} + is submodular. This contains the $\log\det$ (when $g$ is the matrix + $\log$) but tons of other functions (like quantum entropy). +\end{itemize} +In summary, the two most general classes of submodular functions (which capture +all the examples known to man) are: sums of matrix concave functions and sums +of matroid rank functions. Sums of concave over modular are also nice if we +want to start with a simpler example. + +\subsection{Additive Functions} + +We consider here the simple case of additive functions. A function $f$ is +additive if there exists a weight vector $(w_s)_{s \in \Omega}$ such that +$\forall S \subseteq \Omega, \ f(S) = \sum_{s \in S} w_s$. We will sometimes +adopt the notation $w \equiv f(\Omega)$. + +Suppose we have observed $n^c$ set-value pairs ${(S_j, f(S_j))}_{j \in N}$ with +$S_j \sim D$ where $n \equiv |\Omega|$. Consider the $n^c \times n$ matrix $A$ +where for all $i$, the row $A_i = \chi_{S_i}$, the indicator vector of set +$S_i$. Let $B$ be the $n^c \times 1$ vector such that $\forall i, b_j \equiv +f(S_j)$. It is easy to see that if $w$ is the $n \times 1$ corresponding weight +vector for $f$ then: +\begin{displaymath} + A w = B +\end{displaymath} + +Note that if $A$ has full column rank, then we can solve for $w$ exactly and +also optimize $f$ under any cardinality constraint. We can therefore cast the +question of $D-$learning and $D-$optimizing $f$ as a random matrix theory +problem: what is the probability that after $n^c$ for $c > 0$ independent +samples from $D$, the matrix $A$ will have full rank? See +section~\ref{sec:matrix_theory} + +\paragraph{Extension} +Note that the previous reasoning can easily be extended to any \emph{almost} +additive function. Consider a function $g$ such that there exists $\alpha > 0$ +and $\beta > 0$ and an additive function $f$ such that $\forall S, \alpha f(S) +\leq g(S) \leq \beta f(S)$, then by solving for $\max_{S\in K} f(S)$ we have a +$\alpha/\beta$-approximation to the optimum of $g$ since: + +\begin{displaymath} +g(S^*) \geq \alpha f(S^*) \geq \alpha f(OPT) \geq +\frac{\alpha}{\beta} g(OPT) +\end{displaymath} + +where $OPT \in \arg \max_{S \in K} g(S)$. This can be taken one step further by +considering a function $g$ such that there exists $\alpha, \beta >0$, an +additive function $f$ and a bijective univariate function $\phi$, such that +$\forall S, \alpha \phi(f(S)) \leq g(S) \leq \beta \phi(f(S))$. In this case, we +solve the system $A w = \phi^{-1}(B)$ and obtain once again an +$\alpha/\beta$-approximation to the optimum of $g$. + + +\subsubsection{Average weight method} + +We consider here another method to $D-$optimizing $f$, which only requires +$D-$learning $f$ approximately. Note that for every element $i \in \Omega$, and +for a \emph{product} distribution $D$: +\begin{displaymath} + \mathbb{E}_{S \sim D}(f(S)|i \in S) - \mathbb{E}_{S \sim D}(f(S) | i \notin + S) = \sum_{j \neq i \in \Omega} w_s \left(\mathbb{P}(j\in S|i \in S) - + \mathbb{P}(j \in S | i \notin S)\right) + w_i(1 - 0) = w_i +\end{displaymath} + +Let $O$ be the collection of all sets $S$ for which we have observed $f(S)$ and +let $O_i \equiv \{S \in O : i \in S\}$ and $O^c_i \equiv \{ S\in O: i \notin S +\}$. If $O_i$ and $O^c_i$ are non-empty, define the following weight estimator: +\begin{displaymath} + \hat W_i \equiv \frac{1}{|O_i|} \sum_{S \in O_i} f(S) - \frac{1}{|O_i^c|} + \sum_{S \in O_i^c} f(S) +\end{displaymath} + +If $D$ is a product distribution such that $ \exists c > 0, \forall i, +\mathbb{P}(i) \geq c$, it is +easy to show that $\forall i \in +\Omega,\ \hat W_i \rightarrow_{|O| \rightarrow +\infty} w_i$ +By using standard concentration bounds, we can hope to obtain within $poly(n, +\frac{1}{\epsilon})$ observations: + +\subsubsection{Generic Random Matrix Theory} +\label{sec:matrix_theory} + +We cite the following result from~\cite{vershynin2010introduction} (Remark 5.40): + +\begin{proposition}\label{prop:non_isotropic_isometry} +Assume that $A$ is an $N \times n$ matrix whose rows $A_i$ are independent +sub-gaussian random vectors in $\mathbb{R}^n$ with second moment matrix +$\Sigma$. Then for every $t \geq 0$, the following inequality holds with +probability at least: $1- 2 \exp(-ct^2)$: \begin{equation} \|\frac{1}{N} A^T A +- \Sigma \| \leq \max(\delta, \delta^2) \ \text{where}\ \delta = +C\sqrt{\frac{n}{N}} + \frac{t}{\sqrt{N}} \end{equation} where $C$, $c$ depend +only on the subgaussian norm of the rows $K \equiv \max_i \| A_i\|_{\psi_2}$. +\end{proposition} + +The following result is a simple corollary of +Proposition~\ref{prop:non_isotropic_isometry}: + +\begin{corollary}\label{cor:number_of_rows_needed} Let $n \in \mathbb{N}$ and + $k \in ]0,n[$. Assume that $A$ is an $N \times n$ matrix whose rows $A_i$ are + independent Bernoulli variable vectors in $\mathbb{R}^n$ such that + $\mathbb{P}(A_{i,j} = 1) = \frac{k}{n} = 1 - \mathbb{P}(A_{i,j} = 0)$ and let + $\sigma \equiv \frac{k}{n}(1- \frac{k}{n}) \neq 0$, then if $N > {(C+1)}^2 + n/\sigma^2$, the matrix A has full row rank with probability at least $1 - + e^{-cn}$, where $C, c$ are constant depending only on the subgaussian norm + of the rows $K \equiv \sup_{p \geq 1} + {\frac{1}{\sqrt{p}}(\mathbb{E}|A_1|^p)}^\frac{1}{p} = k$\footnote{Is this true? + And how do the constants behave?} \end{corollary} + +\begin{proof} It is easy to compare the kernels of $A$ and $A^TA$ and notice + that $rank(A) = rank(A^T A)$. Since $A^TA$ is an $n \times n$ matrix, it + follows that if $A^TA$ is invertible, then $A$ has full row rank. In other + words, if $A$ has smallest singular value $\sigma_{\min}(A) > 0$, then $A$ + has full row rank. Consider Prop.~\ref{prop:non_isotropic_isometry} with $t + \equiv \sqrt{n}$ and with $N > {(C+1)}^{2} n$, then with probability $1 - + 2\exp(-cn)$: \begin{equation} \|\frac{1}{N} A^T A - \sigma I \| \leq + (C+1)\sqrt{\frac{n}{N}} \end{equation} + +It follows that for any vector $x \in \mathbb{R}^n$, $|\frac{1}{N}\|A x\|^2_2 +-\sigma | \leq (C+1)\sqrt{\frac{n}{N}} \implies \|A x\|^2_2 \geq N\left(\sigma +- (C+1)\sqrt{\frac{n}{N}}\right)$. If $N > {(C+1)}^2n/\sigma^2$, then +$\sigma_{\min}(A) > 0$ with probability at least $1 - e^{-cn}$ \end{proof} + +\subsubsection{Algebraic Approach} + +Here we work on $\mathbb{F}_2$. An $n\times n$ binary matrix can be seen as +a matrix over $\mathbb{F}_2$. Let us assume that each row of $A$ is chosen +uniformly at random among all binary rows of length $n$. A standard counting +arguments shows that the number of non-singular matrices over $\mathbb{F}_2$ +is: +\begin{displaymath} + N_n = (2^n-1)(2^n - 2)\dots (2^n- 2^{n-1}) + = (2^n)^n\prod_{i=1}^n\left(1-\frac{1}{2^i}\right) +\end{displaymath} + +Hence the probability that our random matrix $A$ is invertible is: +\begin{displaymath} + P_n = \prod_{i=1}^n\left(1-\frac{1}{2^i}\right) +\end{displaymath} +It is easy to see that $P_n$ is a decreasing sequence. Its limit is +$\phi(\frac{1}{2})$ where $\phi$ is the Euler function. We have +$\phi(\frac{1}{2})\approx +0.289$ and $P_n$ converges +exponentially fast to this constant (one way to see this is to use the +Pentagonal number theorem). + +Hence, if we observe $\ell\cdot n$ uniformly random binary rows, the +probability that they will have full column rank is at least: +\begin{displaymath} + P_{\ell\cdot n}\geq 1 - \left(1-\phi\left(\frac{1}{2}\right)\right)^{\ell} +\end{displaymath} + +Note that this is the probability of having full column rank over +$\mathbb{F}_2$. A standard linear algebra argument shows that this implies full +column rank over $\mathbb{R}$. + +\paragraph{TODO:} Study the case where we only observe sets of size exactly +$k$, or at most $k$. This amounts to replacing $2^n$ in the computation above +by: +\begin{displaymath} +{n\choose k}\quad\text{or}\quad\sum_{j=0}^k{n\choose j} +\end{displaymath} + +Thinking about it, why do we assume that the sample sets are of size exactly +$k$. I think it would make more sense from the learning perspective to consider +uniformly random sets. In this case, the above approach allows us to conclude +directly. + +More generally, I think the “right” way to think about this is to look at $A$ +as the adjacency matrix of a random $k$-regular graph. There are tons of +results about the probability of the adjacency matrix being non singular. + +\subsection{Influence Functions} + +Let $G = (V, E)$ be a weighted directed graph. Without loss of generality, we +can assign a weight $p_{u,v} \in [0,1]$ to every possible edge $(u,v) \in V^2$. +Let $m$ be the number of non-zero edges of $G$. Let $\sigma_G(S, p)$ be the +influence of the set $S \subseteq V$ in $G$ under the IC model with parameters +$p$. + +Recall from \cite{Kempe:03} that: +\begin{equation} + \sigma_G(S, p) = \sum_{v\in V} \P_{G_p}\big[r_{G_p}(S\leadsto v)\big] +\end{equation} +where $G_p$ is a random graph where each edge $(u,v)\in E$ appears with +probability $p_{u,v}$ and $r_{G_p}(S\leadsto v)$ is the indicator variable of +\emph{there exists a path from $S$ to $v$ in $G_p$}. + +\begin{claim} +\label{cla:oracle} +If for all $(u,v) \in V^2$, $p_{u,v} \geq p'_{u,v} \geq p_{u,v} +- \frac{1}{\alpha m}$, then: +\begin{displaymath} + \forall S \subseteq V,\, \sigma_{G}(S, p) \geq \sigma_{G}(S,p') \geq (1 + - 1/\alpha) \sigma_{G}(S,p) +\end{displaymath} +\end{claim} + +\begin{proof} + We define two coupled random graphs $G_p$ and $G_p'$ as follows: for each + edge $(u,v)\in E$, draw a uniform random variable $\mathcal{U}_{u,v}\in + [0,1]$. Include $(u,v)$ in $G_p$ (resp. $G_{p'}$) iff + $\mathcal{U}_{u,v}\leq p_{u,v}$ (resp. $\mathcal{U}_{u,v}\leq p'_{u,v}$). + It is clear from the construction that each edge $(u,v)$ will be present in + $G_p$ (resp. $G_p'$) with probability $p_{u,v}$ (resp. $p'_{u,v}$). Hence: + \begin{displaymath} + \sigma_G(S, p) = \sum_{v\in V} \P_{G_p}\big[r_{G_p}(S\leadsto v)\big] + \text{ and } + \sigma_G(S, p') = \sum_{v\in V} \P_{G_p'}\big[r_{G_p'}(S\leadsto v)\big] + \end{displaymath} + + By construction $G_{p'}$ is always a subgraph of $G_p$, i.e + $r_{G_p'}(S\leadsto v)\leq r_{G_p}(S\leadsto v)$. This proves the left-hand + side of the Claim's inequality. + + The probability that an edge is present in $G_p$ but not in $G_p'$ is at + most $\frac{1}{\alpha m}$, so with probability $\left(1-\frac{1}{\alpha + m}\right)^m$, $G_p$ and $G_p'$ have the same set of edges. This implies that: + \begin{displaymath} \P_{G_{p'}}\big[r_{G_p'}(S\leadsto v)\big]\geq + \left(1-\frac{1}{\alpha m}\right)^m\P_{G_{p}}\big[r_{G_p}(S\leadsto v)\big] + \end{displaymath} + which proves the right-hand side of the claim after observing that + $\left(1-\frac{1}{\alpha m}\right)^m\geq 1-\frac{1}{\alpha}$ with equality + iff $m=1$. +\end{proof} + + We can use Claim~\ref{cla:oracle} to find a constant factor approximation + algorithm to maximising influence on $G$ by maximising influence on $G'$. For + $k \in \mathbb{N}^*$, let $O_k \in \arg\max_{|S| \leq k} \sigma_G(S)$ and + $\sigma_{G}^* = \sigma_G(O_k)$ where we omit the $k$ when it is unambiguous. + + +\begin{proposition} +\label{prop:approx_optim} +Suppose we have an unknown graph $G$ and a known graph $G'$ such that $V = V'$ +and for all $(u,v) \in V^2, |p_{u,v} - p_{u,v}'| \leq \frac{1}{\alpha m}$. +Then for all $k \in \mathbb{N}^*$, we can find a set $\hat O_k$ such that +$\sigma_{G}(\hat O_k) \geq (1 - e^{\frac{2}{\alpha} - 1}) \sigma^*_{G}$ +\end{proposition} + +\begin{proof} For every edge $(u,v) \in V^2$, let $\hat p = p'_{u,v} - + \frac{1}{\alpha m}$. We are now in the conditions of Claim~\ref{cla:oracle} + with $\alpha \leftarrow \alpha/2$. We return the set $\hat O_k$ obtained by + greedy maximisation on $\hat G$. It is a classic exercise to show then that + $\sigma_G(\hat O_k) \geq 1 - e^{\frac{2}{\alpha} - 1}$ (see Pset 1, CS284r). +\end{proof} + +A small note on the approximation factor: it is only $>0$ for $\alpha > 2$. +Note that $\alpha \geq 7 \implies 1 - e^{\frac{2}{\alpha} - 1} \geq +\frac{1}{2}$ and that it converges to $(1 - 1/e)$ which is the best possible +polynomial-time approximation ratio to influence maximisation unless $P = NP$. +Also note that in the limit of large $m$, $(1 -\frac{1}{\alpha m})^m +\rightarrow \exp(-1/\alpha)$ and the approximation ratio goes to $1 - +\exp(-\exp(-2/\alpha))$. |