Streaming Adaptive Submodular Maximization

We consider a set $E$ of $n$ items. Each items $e\in E$ belongs to a random state $\Phi(e)\in O$ where $O$ represents the set of all possible states. Denote by $\phi$ a realization of $\Phi$, i.e., for each $e\in E$, $\phi(e)$ is a realization of $\Phi(e)$. In the application of experimental design, an item $e$ represents a test, such as the blood pressure, and $\Phi(e)$ is the result of the test, such as, high. We assume that there is a known prior probability distribution $p(\phi)=\Pr(\Phi=\phi)$ over realizations $\phi$. The distribution $p$ completely factorizes if realizations are independent. However, we consider a general setting where the realizations are dependent. For any subset of items $S\subseteq E$, we use $\psi:S\rightarrow O$ to represent a partial realization and $\mathrm{dom}(\psi)=S$ is called the domain of $\psi$. For any pair of a partial realization $\psi$ and a realization $\phi$, we say $\phi$ is consistent with $\psi$, denoted $\phi\sim\psi$, if they are equal everywhere in $\mathrm{dom}(\psi)$. For any two partial realizations $\psi$ and $\psi^{\prime}$, we say that $\psi$ is a subrealization of $\psi^{\prime}$, and denoted by $\psi\subseteq\psi^{\prime}$, if $\mathrm{dom}(\psi)\subseteq\mathrm{dom}(\psi^{\prime})$ and they are consistent in $\mathrm{dom}(\psi)$. In addition, each item $e\in E$ has a cost $c(e)$. For any $S\subseteq E$, let $c(S)=\sum_{e\in S}c(e)$ denote the total cost of $S$.

In the stream-based setting, we assume that items arrive one by one in an adversarial order $\sigma$. A policy has to make an irrevocable decision on whether to select an item or not when an item arrives. If an item is selected, then we are able to observe its realized state; otherwise, we can not reveal its realized state. Formally, a stream-based policy is a partial mapping that maps a pair of partial realizations $\psi$ and an item $e$ to some distribution of $\{0,1\}$: $\pi:2^{E}\times O^{E}\times E\rightarrow\mathcal{P}(\{0,1\})$, specifying whether to select the arriving item $e$ based on the current observation $\psi$. For example, assume that the current observation is $\psi$ and the newly arrived item is $e$, then $\pi(\psi,e)=1$ (resp. $\pi(\psi,e)=0$) indicates that $\pi$ selects (res. does not select) $e$.

Assume that there is a utility function $f:2^{E\times O}\rightarrow\mathbb{R}_{\geq 0}$ which is defined over items and states. Letting $E(\pi,\phi,\sigma)$ denote the subset of items selected by a stream-based policy $\pi$ conditioned on a realization $\phi$ and a sequence of arrivals $\sigma$, the expected utility $f_{avg}(\pi)$ of a stream-based policy $\pi$ conditioned on a sequence of arrivals $\sigma$ can be written as

where the expectation is taken over all possible realizations $\Phi$ and the internal randomness of the policy $\pi$.

We next introduce the concept of policy concatenation which will be used in our proofs.

Our objective is to find an stream-based policy that maximizes the worst-case expected utility subject to a budget constraint $B$, i.e.,

where $\Omega^{s}=\{\pi\mid\forall\phi,\sigma^{\prime}:c(E(\pi,\Phi,\sigma^{\prime}))\leq B\}$ represents a set of all feasible stream-based policies subject to a knapsack constraint $(c,B)$. That is, a feasible policy must satisfy the budget constraint under all possible realizations and sequences of arrivals.

We next introduce some additional notations and important assumptions in order to facilitate our study.

We next introduce a new class of stochastic functions.

In the rest of this paper, we always assume that our utility function $f:2^{E\times O}\rightarrow\mathbb{R}_{\geq 0}$ is adaptive monotone, adaptive submodular and semi-policywise submodular with respect to a prior $p(\phi)$ and a knapsack constraint $(c,B)$. In appendix, we show that this type of function can be found in a variety of important real world applications. All missing materials are moved to appendix.

Recall that $\pi^{*}\in\operatorname*{arg\,max}_{\pi\in\Omega^{p}}\ f_{avg}(\pi)$ represents an optimal pool-based policy subject to a budget constraint $B$, suppose we can estimate $f_{avg}(\pi^{*})$ approximately, i.e., we know a value $v$ such that $\beta\cdot f_{avg}(\pi^{*})\geq v\geq\alpha\cdot f_{avg}(\pi^{*})$ for some $\alpha\in[0,1]$ and $\beta\in[1,2]$. Our policy, called Online Adaptive Policy $\pi^{c}$, starts with an empty set $S=\emptyset$. In each subsequent iteration $i$, after observing an arriving item $\sigma(i)$, $\pi^{c}$ adds $\sigma(i)$ to $S$ if the marginal value of $\sigma(i)$ on top of the current partial realization $\psi_{t}$ is at least $\frac{v}{2B}$; otherwise, it skips $\sigma(i)$. This process iterates until there are no more arriving items or it reaches the cardinality constraint. A detailed description of $\pi^{c}$ is listed in Algorithm 1.

We present the main result of this section in the following theorem.

Recall that the design of $\pi^{c}$ requires that we know a good approximation of $f_{avg}(\pi^{*})$. We next explain how to obtain such an estimation. It is well known that a simple greedy pool-based policy $\pi^{g}$ (which is outlined in Algorithm 2) provides a $(1-1/e)$ approximation for the pool-based adaptive submodular maximization problem subject to a cardinality constraint [6], i.e., $f_{avg}(\pi^{g})\geq(1-1/e)f_{avg}(\pi^{*})$. Hence, $f_{avg}(\pi^{g})$ is a good approximation of $f_{avg}(\pi^{*})$. In particular, if we set $v=f_{avg}(\pi^{g})$, then we have $f_{avg}(\pi^{*})\geq v\geq(1-1/e)f_{avg}(\pi^{*})$. This, together with Theorem 5.1, implies that $\pi^{c}$ achieves a $\frac{1-1/e}{4}$ approximation ratio against $\pi^{*}$. One can estimate the value of $f_{avg}(\pi^{g})$ by simulating $\pi^{g}$ on every possible realization $\phi$ to obtain $E(\pi^{g},\phi)$ and letting $f_{avg}(\pi^{g})=\sum_{\phi}p(\phi)f(E(\pi^{g},\phi),\phi)$. When the number of possible realizations is large, one can sample a set of realizations according to $p(\phi)$ then run the simulation. Although obtaining a good estimation of $f_{avg}(\pi^{g})$ may be time consuming, this only needs to be done once in an offline manner. Thus, it does not contribute to the running time of the online implementation of $\pi^{c}$.

Suppose we can estimate $f_{avg}(\pi^{*})$ approximately, i.e., we know a value $v$ such that $\beta\cdot f_{avg}(\pi^{*})\geq v\geq\alpha\cdot f_{avg}(\pi^{*})$ for some $\alpha\in[0,1]$ and $\beta\in[1,2]$. For each $e\in E$, let $f(e)$ denote $\mathbb{E}_{\Phi}[f(\{e\},\Phi)]$ for short. Our policy randomly selects a solution from $\{e^{*}\}$ and $\pi^{k}$ with equal probability, where $e^{*}=\operatorname*{arg\,max}_{e\in E}f(e)$ is the best singleton and $\pi^{k}$, which is called Online Adaptive Policy with Nonuniform Cost, is a density-greedy policy. Hence, the expected utility of our policy is $(f(e^{*})+\mathbb{E}[f_{avg}(\pi^{k})\mid\sigma])/2$ for any given sequence of arrivals $\sigma$. We next explain the design of $\pi^{k}$. $\pi^{k}$ starts with an empty set $S=\emptyset$. In each subsequent iteration $i$, after observing an arriving item $\sigma(i)$, it adds $\sigma(i)$ to $S$ if the marginal value per unit budget of $\sigma(i)$ on top of the current realization $\psi_{t}$ is at least $\frac{v}{2B}$, i.e., $\frac{\Delta(\sigma(i)\mid\psi_{t})}{c(\sigma(i))}\geq\frac{v}{2B}$, and adding $\sigma(i)$ to $S$ does not violate the budget constraint; otherwise, if $\frac{\Delta(\sigma(i)\mid\psi_{t})}{c(\sigma(i))}<\frac{v}{2B}$, $\pi^{k}$ skips $\sigma(i)$. This process iterates until there are no more arriving items or it reaches the first item (excluded) that violates the budget constraint. A detailed description of $\pi^{k}$ is listed in Algorithm 3.

Before presenting the main theorem, we first introduce a technical lemma.

Proof: We first introduce an auxiliary policy $\pi^{k+}$ that follows the same procedure of $\pi^{k}$ except that $\pi^{k+}$ is allowed to add the first item that violates the budget constraint. Although $\pi^{k+}$ is not necessarily feasible, we next show that the expected utility $\mathbb{E}[f_{avg}(\pi^{k+})\mid\sigma]$ of $\pi^{k+}$ is upper bounded by $\max\{f(e^{*}),\mathbb{E}[f_{avg}(\pi^{k})\mid\sigma]\}$ for any sequence of arrivals $\sigma$, i.e., $\mathbb{E}[f_{avg}(\pi^{k+})\mid\sigma]\leq\max\{f(e^{*}),\mathbb{E}[f_{avg}(\pi^{k})\mid\sigma]\}$.

Proposition 1, whose proof is deferred to appendix, implies that to prove this lemma, it suffices to show that $\mathbb{E}[f_{avg}(\pi^{k+})\mid\sigma]\geq\min\{\frac{\alpha}{4},\frac{2-\beta}{4}\}f_{avg}(\pi^{*})$. The rest of the analysis is devoted to proving this inequality for any fixed sequence of arrivals $\sigma$. We use $\lambda=\{\psi_{1}^{\lambda},\psi_{2}^{\lambda},\psi_{3}^{\lambda},\cdots,\psi_{z^{\lambda}}^{\lambda}\}$ to denote a fixed run of $\pi^{k+}$, where $\psi_{t}^{\lambda}$ is the partial realization of the first $t$ selected items and $z^{\lambda}$ is the total number of selected items under $\lambda$. Let $U=\{\lambda\mid\Pr[\lambda]>0\}$ represent all possible stories of running $\pi^{k+}$, $U^{+}$ represent those stories where $\pi^{k+}$ meets or violates the budget, i.e., $U^{+}=\{\lambda\in U\mid c(\mathrm{dom}(\psi_{z^{\lambda}}^{\lambda}))\geq B\}$, and $U^{-}$ represent those stories where $\pi^{k+}$ does not use up the budget, i.e., $U^{-}=\{\lambda\in U\mid c(\mathrm{dom}(\psi_{z^{\lambda}}^{\lambda}))<B\}$. Therefore, $U=U^{+}\cup U^{-}$. For each $\lambda$ and $t\in[z^{\lambda}]$, let $e^{\lambda}_{t}$ denote the $t$-th selected item under $\lambda$. Define $\psi_{0}^{\lambda}=\emptyset$ for any $\lambda$. Using the above notations, we can represent $\mathbb{E}[f_{avg}(\pi^{k+})\mid\sigma]$ as follows:

Then we consider two cases. We first consider the case when $\sum_{\lambda\in U^{+}}\Pr[\lambda]\geq 1/2$ and show that the value of $I$ is lower bounded by $\frac{\alpha}{4}f_{avg}(\pi^{*})$. According to the definition of $U^{+}$, we have $\sum_{t\in[z^{\lambda}]}c(e^{\lambda}_{t})\geq B$ for any $\lambda\in U^{+}$. Moreover, recall that for all $t\in[z^{\lambda}]$, $\frac{\Delta(e^{\lambda}_{t}\mid\psi_{t-1}^{\lambda})}{c(e^{\lambda}_{t})}\geq\frac{v}{2B}$ due to the design of our algorithm. Therefore, for any $\lambda\in U^{+}$,

Because we assume that $\sum_{\lambda\in U^{+}}\Pr[\lambda]\geq 1/2$, we have

The first inequality is due to (7) and the third inequality is due to the assumption that $v\geq\alpha\cdot f_{avg}(\pi^{*})$. We conclude that the value of $I$ (and thus $\mathbb{E}[f_{avg}(\pi^{k+})\mid\sigma]$) is no less than $\frac{\alpha}{4}f_{avg}(\pi^{*})$, i.e.,

We next consider the case when $\sum_{\lambda\in U^{+}}\Pr[\lambda]<1/2$. We show that under this case,

Because $f:2^{E\times O}\rightarrow\mathbb{R}_{\geq 0}$ is adaptive monotone, we have $\mathbb{E}[f_{avg}(\pi^{k+}@\pi^{*})\mid\sigma]\geq f_{avg}(\pi^{*})$. To prove (10), it suffices to show that

Observe that we can represent the gap between $f_{avg}(\pi^{k+}@\pi^{*})$ and $f_{avg}(\pi^{*})$ conditioned on $\sigma$ as follows:

Because $f:2^{E\times O}\rightarrow\mathbb{R}_{\geq 0}$ is semi-policywise submodular with respect to $p(\phi)$ and $(c,B)$, we have $\max_{\pi\in\Omega^{p}}\Delta(\pi\mid\psi_{z^{\lambda}}^{\lambda})\leq f_{avg}(\pi^{*})$. Moreover, because $\Delta(\pi^{*}\mid\psi_{z^{\lambda}}^{\lambda})\leq\max_{\pi\in\Omega^{p}}\Delta(\pi\mid\psi_{z^{\lambda}}^{\lambda})$, we have

It follows that

Next, we show that $III$ is upper bounded by $(\sum_{\lambda\in U^{-}}\Pr[\lambda])\frac{\beta}{2}f_{avg}(\pi^{*})$. For any $\psi_{z^{\lambda}}^{\lambda}$, we number all items $e\in E$ by decreasing ratio $\frac{\Delta(e\mid\psi_{z^{\lambda}}^{\lambda})}{c(e)}$, i.e., $e(1)\in\arg\max_{e\in E}\frac{\Delta(e\mid\psi_{z^{\lambda}}^{\lambda})}{c(e)}$. Let $l=\min\{i\in\mathbb{N}\mid\sum_{j=1}^{i}c(e(i))\geq B\}$. Define $D(\psi_{z^{\lambda}}^{\lambda})=\{e(i)\in E\mid i\in[l]\}$ as the set containing the first $l$ items. Intuitively, $D(\psi_{z^{\lambda}}^{\lambda})$ represents a set of best-looking items conditional on $\psi_{z^{\lambda}}^{\lambda}$. Consider any $e\in D(\psi_{z^{\lambda}}^{\lambda})$, assuming $e$ is the $i$-th item in $D(\psi_{z^{\lambda}}^{\lambda})$, let

where $\cup_{j\in[i-1]}\{e(j)\}$ represents the first $i-1$ items in $D(\psi_{z^{\lambda}}^{\lambda})$.

In analogy to Lemma 1 of [9],

Note that for every $\lambda\in U^{-}$, we have $\sum_{t\in[z^{\lambda}]}c(e^{\lambda}_{t})<B$, that is, $\pi^{k+}$ does not use up the budget under $\lambda$. This, together with the design of $\pi^{k+}$, indicates that for any $e\in E$, its benefit-to-cost ratio on top of $\psi_{z^{\lambda}}^{\lambda}$ is less than $\frac{v}{2B}$, i.e., $\frac{\Delta(e\mid\psi_{z^{\lambda}}^{\lambda})}{c(e)}<\frac{v}{2B}$. Therefore,

(15) and (16) imply that

We next provide an upper bound of $III$,

where the first inequality is due to (17) and the second inequality is due to $v\leq\beta\cdot f_{avg}(\pi^{*})$.

Now we are in position to bound the value of $\mathbb{E}[f_{avg}(\pi^{k+}@\pi^{*})-f_{avg}(\pi^{k+})\mid\sigma]$,

The first inequality is due to (14) and (19). The second inequality is due to $\sum_{\lambda\in U^{+}}\Pr[\lambda]+\sum_{\lambda\in U^{-}}\Pr[\lambda]=1$ and the assumptions that $\sum_{\lambda\in U^{+}}\Pr[\lambda]<1/2$ and $\beta\in[1,2]$. Because $\mathbb{E}[f_{avg}(\pi^{k+}@\pi^{*})\mid\sigma]\geq\mathbb{E}[f_{avg}(\pi^{*})\mid\sigma]$, which is due to $f:2^{E\times O}\rightarrow\mathbb{R}_{\geq 0}$ is adaptive monotone, we have

where the second inequality is due to (23). This, together with the fact that $\mathbb{E}[f_{avg}(\pi^{*})\mid\sigma]=f_{avg}(\pi^{*})$, i.e., the optimal pool-based policy is not dependent on the sequence of arrivals, implies (10).

Combining the above two cases ((9) and (10)), we have

This, together with Proposition 1, immedinately conclues this lemma. $\Box$

Recall that our final policy randomly picks a solution from $\{e^{*}\}$ and $\pi^{k}$ with equal probability, thus, its expected utility is $\frac{f(e^{*})+\mathbb{E}[f_{avg}(\pi^{k})\mid\sigma]}{2}$ which is lower bounded by $\frac{\max\{f(e^{*}),\mathbb{E}[f_{avg}(\pi^{k})\mid\sigma]\}}{2}$. This, together with Lemma 1, implies the following main theorem.