Robust Adaptive Submodular Maximization

07/23/2021 ∙ by Shaojie Tang, et al. ∙ 0

Most of existing studies on adaptive submodular optimization focus on the average-case, i.e., their objective is to find a policy that maximizes the expected utility over a known distribution of realizations. However, a policy that has a good average-case performance may have very poor performance under the worst-case realization. In this study, we propose to study two variants of adaptive submodular optimization problems, namely, worst-case adaptive submodular maximization and robust submodular maximization. The first problem aims to find a policy that maximizes the worst-case utility and the latter one aims to find a policy, if any, that achieves both near optimal average-case utility and worst-case utility simultaneously. We introduce a new class of stochastic functions, called worst-case submodular function. For the worst-case adaptive submodular maximization problem subject to a p-system constraint, we develop an adaptive worst-case greedy policy that achieves a 1/p+1 approximation ratio against the optimal worst-case utility if the utility function is worst-case submodular. For the robust adaptive submodular maximization problem subject to a cardinality constraint, if the utility function is both worst-case submodular and adaptive submodular, we develop a hybrid adaptive policy that achieves an approximation close to 1-e^-1/2 under both worst case setting and average case setting simultaneously. We also describe several applications of our theoretical results, including pool-base active learning, stochastic submodular set cover and adaptive viral marketing.

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1 Introduction

Submodular maximization subject to practical constraints has been extensively studied in the literature (Tang and Yuan 2020, Yuan and Tang 2017, Krause and Guestrin 2007, Leskovec et al. 2007, Badanidiyuru and Vondrák 2014, Mirzasoleiman et al. 2016, Ene and Nguyen 2018, Mirzasoleiman et al. 2015)

. Many machine learning and AI tasks such as viral marketing

Golovin and Krause (2011b), data summarizationBadanidiyuru et al. (2014) and active learning Golovin and Krause (2011b) can be formulated as a submodular maximization problem. While most of existing studies focus on non-adaptive submodular maximization by assuming a deterministic utility function, Golovin and Krause (2011b) extends this study to the adaptive setting where the utility function is stochastic. Concretely, the input of an adaptive optimization problem is a set of items, and each item is in a particular state drawn from some known prior distribution. There is an utility function which is defined over items and their states. One must select an item before revealing its realized state. Their objective is to adaptively select a group of items, each selection is based on the feedback from the past, to maximize the average-case utility over the distribution of realizations. Note that a policy that has a good average-case performance may perform poorly under the worst-case realization. This raises our first research question:

Is it possible to design a policy that maximizes the worst-case utility?

Moreover, even if we can find such a policy, this worst-case guarantee often comes at the expense of degraded average-case performance. This raises our second research question:

Is it possible to design a policy, if any, that achieves both good average-case and worst-case performance simultaneously?

In this paper, we provide affirmative answers to both questions by studying two variants of adaptive submodular optimization problems: worst-case adaptive submodular maximization and robust submodular maximization. In the first problem, our goal is to find a policy that maximizes the utility under the worst-case realization. The second problem aims to find a policy that achieves both near optimal average-case utility and worst-case utility simultaneously. To tackle these two problems, we introduce a new class of stochastic functions, called worst-case submodular function, and we show that this property can be found in a wide range of real-world applications, including pool-based active learning (Golovin and Krause 2011b) and adaptive viral marketing (Golovin and Krause 2011b).

Based on the above notation, we first study the worst-case adaptive submodular maximization problem subject to a -system constraint, and develop an adaptive worst-case greedy policy that achieves a approximation ratio against the optimal worst-case utility if the utility function is worst-case submodular. Note that the -system constraint is general enough to subsume many practical constraints, including cardinality, matroid, intersection of matroids, -matchoid and -extendible constraints, as special cases. We also show that both the approximation ratio and the running time can be improved for the case of a single cardinality constraint. Then we initiate the study of robust adaptive submodular maximization problem subject to a cardinality constraint. If the utility function is both worst-case submodular and adaptive submodular, we develop a hybrid adaptive policy that achieves a nearly approximation ratio under both worst and average cases simultaneously.

2 Related Works

Golovin and Krause (2011b) introduce the problem of adaptive submodular maximization, where they extend the notion of submodularity and monotonicity to the adaptive setting by introducing adaptive submodularity and adaptive monotonicity. They develop a simple adaptive greedy algorithm that achieves a approximation for maximizing a monotone adaptive submodular function subject to a cardinality constraint in average case. In Golovin and Krause (2011a), they further extend their results to a -system constraint. When the utility function is not adaptive monotone, Tang (2021) develops the first constant factor approximation algorithm in average case. While most of existing studies on adaptive submodular maximization focus on the average case setting, Guillory and Bilmes (2010), Golovin and Krause (2011b) studied the worst-case min-cost submodular set cover problem, their objective, which is different from ours, is to adaptively select a group of cheapest items until the resulting utility function achieves some threshold. Cuong et al. (2014) studied the pointwise submodular maximization problem subject to a cardinality constraint in worst case setting. They claimed that if the utility function is pointwise submodular, then a greedy policy achieves a constant approximation ratio in worst case setting. Unfortunately, we show in Section 4.2 that this result does not hold in general. In particular, we construct a counter example to show that the performance of the aforementioned greedy policy is arbitrarily bad even if the utility function is pointwise submodular. To tackle this problem, we introduce the notation of worst-case submodularity and develop a series of effective solutions for maximizing a worst-case submodular function subject to a -system constraint. Our results are not restricted to any particular applications. Moreover, perhaps surprisingly, we propose the first algorithm that achieves good approximation ratios in both average case and worst case settings simultaneously.

3 Preliminaries

In the rest of this paper, we use to denote the set , and we use to denote the cardinality of a set .

3.1 Items and States

We consider a set of items, and each item has a random state . Let a function , denote a realization, where for each , represents the realization of ’s state. The realization of

is unknown initially. However, there is a known prior probability distribution

over realizations . Given any , let denote a partial realization and is called the domain of . Given a partial realization and a realization , we say is consistent with , denoted , if they are equal everywhere in . We use to denote the conditional distribution over realizations conditional on a partial realization : .

3.2 Policy and Adaptive/Worst-case Submodularity

We represent a policy using a function that maps a set of partial realizations to : . Intuitively, is a mapping from the observations collected so far to the next item to select. Note that any randomized policy can be represented as a distribution of a group of deterministic policies, thus we will focus on deterministic policies.

[Policy Concatenation] Given two policies and , let denote a policy that runs first, and then runs , ignoring the observation obtained from running .

[Level--Truncation of a Policy] Given a policy , we define its level--truncation as a policy that runs until it selects items.

Let denote the subset of items selected by under realization . The expected utility of a policy can be written as

where the expectation is taken over possible realizations. The worst-case utility of a policy can be written as

Let

denote the conditional expected marginal utility of on top of a partial realization , where the expectation is taken over with respect to . We next introduce the notations of adaptive submodularity and adaptive monotonicity Golovin and Krause (2011b).

Golovin and Krause (2011b)[Adaptive Submodularity and Adaptive Monotonicity] Consider any two partial realizations and such that . A function is called adaptive submodular if for each , we have

(1)

A function is called adaptive monotone if for all and , we have .

By extending the definition of , let . We next introduce the worst-case marginal utility of on top of a partial realization .

(2)

where denotes the set of possible states of conditional on a partial realization .

We next introduce a new class of stochastic functions. [Worst-case Submodularity and Worst-case Monotonicity] Consider any two partial realizations and such that . A function is called worst-case submodular if for each , we have

(3)

A function is called worst-case monotone if for each partial realization and , .

Note that if for each partial realization and , , then because , it is also true that for each partial realization and , . Therefore, worst-case monotonicity implies adaptive monotonicity. We next introduce the property of minimal dependency Cuong et al. (2014) which states that the utility of any group of items does not depend on the states of any items outside that group. [Minimal Dependency] For any partial realization and any realization such that , we have .

3.3 Worst-case and Robust Adaptive Submodular Maximization

Now we are ready to introduce the two optimization problems studied in this paper. We first introduce the notation of independence system. [Independence System] Consider a ground set and a collection of sets , the pair is an independence system if it satisfies the following two conditions:

  1. The empty set is independent,, i.e., ;

  2. is downward-closed, that is, and implies that .

If and imply that , then is called a base. Moreover, a set is called a base of if and is a base of the independence system . Let denote the collection of all bases of . We next present the definition of -system.

[-system] A -system for an integer is an independence system such that for every set , .

In this paper, we study two adaptive submodular maximization problems: worst-case adaptive submodular maximization problem and robust adaptive submodular maximization problem.

3.3.1 Worst-case adaptive submodular maximization

The worst-case adaptive submodular maximization problem subject to a -system constraint can be formulated as follows:

3.3.2 Robust adaptive submodular maximization

We next introduce the robust adaptive submodular maximization problem subject to a cardinality constraint . The aim of this problem is to find a policy that performs well under both worst case setting and average setting. Let

denote the optimal worst-case adaptive policy and let

denote the optimal average-case adaptive policy subject to a cardinality constraint . Define as the robustness ratio of . Intuitively, a larger indicates that has better robustness. The robust adaptive submodular maximization problem is to find a policy that maximizes .

4 Worst-case Adaptive Submodular Maximization

We first study the case of worst-case adaptive submodular maximization and present an Adaptive Worst-case Greedy Policy for this problem. The detailed implementation of is listed in Algorithm 1. It starts with an empty set and at each round , selects an item that maximizes the worst-case marginal utility on top of the current observation , i.e.,

After observing the state of , update the current partial realization using . This process iterates until the current solution can not be further expanded.

Let denote the optimal policy, we next show that achieves a approximation ratio, i.e., , if the utility function is worst-case monotone and worst-case submodular, and it satisfies the property of minimal dependency.

1:  .
2:  while  do
3:     select ;
4:     ;
5:     ;
6:     ;
Algorithm 1 Adaptive Worst-case Greedy Policy for -System Constraint

If the utility function is worst-case monotone, worst-case submodular with respect to and it satisfies the property of minimal dependency, then . Proof: Assume is the worst-case realization of , i.e., , and it selects items conditional on , i.e., . For each , let denote the first items selected by conditional on , and let denote the partial realization of conditional on , i.e., . Thus, is a collection of all items selected by and is the partial realization of conditional on . Let denote the -th item selected by conditional on , i.e., . We first show that .

The second equality is due to the assumption of minimal dependency. To prove this theorem, it suffice to demonstrate that for any optimal policy , there exists a realization such that . The rest of the proof is devoted to constructing such a realization . First, we ensure that is consistent with by setting for each . Next, we complete the construction of by simulating the execution of conditional on . Let denote the first items selected by during the process of construction and let denote the partial realization of . Starting with and let , assume the optimal policy picks as the -th item after observing , we set the state of to , then update the observation using . This construction process continues until does not select new items. Intuitively, in each round of , we choose a state for to minimize the marginal utility of on top of the partial realization . Assume , i.e., selects items conditional on , thus, is a collection of all items selected by and is the partial realization of conditional on . Note that there may exist multiple realizations which meet the above description, we choose an arbitrary one as . Then we have

(4)

The second inequality is due to is worst-case monotone. The first and the last equalities are due to the assumption of minimal dependency. The forth equality is due to is a state of minimizing the marginal utility , i.e., .

We next show that for all and ,

(5)

If , the above inequality holds due to is worst-case submodular and . If , then and due to is worst-case monotone, thus (5) also holds.

Calinescu et al. (2007) show that there exists a sequence of sets whose nonempty members partition , such that for all and such that , we have , which implies due to , and . Now consider a fixed and any , because , where the equality is due to the selection rule of . This together with (5) implies that

(6)

Because , we have where the second inequality is due to (6). It follows that

(7)

The first equality is due to the nonempty members of partition and the inequality is due to (7). This together with (4) implies that .

4.1 Improved Results for Cardinality Constraint

1:  .
2:  while  do
3:     select ;
4:     ;
5:     ;
Algorithm 2 Adaptive Worst-case Greedy Policy for Cardinality Constraint

In this section, we provide enhanced results for the following worst-case adaptive submodular maximization problem subject to a cardinality constraint .

Note that a single cardinality constraint is a -system constraint. We show that the approximation ratio of (Algorithm 1) can be further improved to under a single cardinality constraint. For ease of presentation, we provide a simplified version of (Algorithm 1) in Algorithm 2. Note that follows the same greedy rule as described in to select items.

If the utility function is worst-case monotone, worst-case submodular with respect to and it satisfies the property of minimal dependency, then . Proof: Assume is the worst-case realization of , i.e., . For each , let denote the partial realization of the first items selected by conditional on . Given any optimal policy and a partial realization after running for rounds, we construct a realization as follows. First, we ensure that is consistent with by setting for each . Next, we complete the construction of by simulating the execution of . Starting with and let , assume the optimal policy picks as the th item after observing , we set the state of to . After each round , we update the observation using . This construction process continues until does not select new items. Intuitively, in each round of , we choose a state for to minimize the marginal utility of on top of the partial realization . Note that there may exist multiple realizations which meet the above description, we choose an arbitrary one as .

To prove this theorem, it suffice to show that for all ,

(8)

This is because by induction on , we have that for any ,

(9)

Then this theorem follows because .

Thus, we focus on proving (8) in the rest of the proof.

The second equality is due to the selection rule of , i.e., and the forth inequality is due to is worst-case monotone.

A faster algorithm that maximizes the expected worst-case utility.

Inspired by the sampling technique developed in Mirzasoleiman et al. (2015), Tang (2021), we next present a faster randomized algorithm that achieves a approximation ratio when maximizing the expected worst-case utility subject to a cardinality constraint. We first extend the definition of so that it can represent any randomized policies. In particular, we re-define as a mapping function that maps a set of partial realizations to a distribution of : . The expected worst-case utility of a randomized policy is defined as

where the expectation is taken over the internal randomness of the policy .

1:  .
2:  while  do
3:      a random set of size sampled uniformly at random from ;
4:     select ;
5:     ;
6:     ;
Algorithm 3 Adaptive Stochastic Worst-case Greedy Policy

Now we are ready to present our Adaptive Stochastic Worst-case Greedy Policy . Starting with an empty set and at each round , first samples a random set of size uniformly at random from , where is some positive constant. Then it selects the -th item from that maximizes the worst-case marginal utility on top of the current observation , i.e., . This process continues until selects items.

We next show that the expected worst-case utility of is at least . Moreover, the running time of is bounded by which is linear in the problem size and independent of the cardinality constraint . We move the proof of the following theorem to appendix. If the utility function is worst-case monotone, worst-case submodular with respect to and it satisfies the property of minimal dependency, then . The running time of is bounded by .

4.2 Pointwise submodularity does not imply worst-case submodularity

We first introduce the notations of pointwise submodularity and pointwise monotonicity. Then we construct an example to show that the performance of could be arbitrarily bad even if is worst-case monotone, pointwise submodular and it satisfies the property of minimal dependency. As a result, the main result claimed in Cuong et al. (2014) does not hold in general. This observation together with Theorem 3 also indicates that pointwise submodularity does not imply worst-case submodularity. In Section 4.3, we show that adding one more condition, i.e., for all and all possible partial realizations such that , , on top of the above three conditions makes it a sufficient condition to ensure the worst-case submodularity of . (Golovin and Krause 2011b)[Pointwise Submodularity and Pointwise Monotonicity] A function is pointwise submodular if is submodular for any given realization such that . Formally, consider any realization such that , any two sets and such that , and any item , we have