Adaptive Submodularity: Theory and Applications in Active Learning and Stochastic Optimization

In many problems arising in artificial intelligence one needs to adaptively make a sequence of decisions, taking into account observations about the outcomes of past decisions. Often these outcomes are uncertain, and one may only know a probability distribution over them. Finding optimal policies for decision making in such partially observable stochastic optimization problems is notoriously intractable (see, e.g., Littman et al. (1998)). A fundamental challenge is to identify classes of planning problems for which simple solutions obtain (near-) optimal performance. In this paper, we introduce the concept of adaptive submodularity, and prove that if a partially observable stochastic optimization problem satisfies this property, a simple adaptive greedy algorithm is guaranteed to obtain near-optimal solutions. In fact, under reasonable complexity-theoretic assumptions, no polynomial time algorithm is able to obtain better solutions in general. Adaptive submodularity generalizes the classical notion of submodularity 1 , which has been successfully used to develop approximation algorithms for a variety of non-adaptive optimization problems. Submodularity, informally, is an intuitive notion of diminishing returns, which states that adding an element to a small set helps more than adding that same element to a larger (super-) set. A celebrated result of the work of Nemhauser et al. (1978) guarantees that for such submodular functions, a simple greedy algorithm, which adds the element that maximally increases the objective value, selects a near-optimal set of k elements. Similarly, it is guaranteed to find a set of near-minimal cost that achieves a desired quota of utility (Wolsey, 1982), using near-minimum average time to do so (Streeter and Golovin, 2008). Besides guaranteeing theoretical performance bounds, submodularity allows us to speed up algorithms without loss of solution quality by using lazy evaluations (Minoux, 1978), often leading to performance improvements of several orders of magnitude (Leskovec et al., 2007). Submodularity has been shown to be very useful in a variety of problems in artificial intelligence (Krause and Guestrin, 2009b). The challenge in generalizing submodularity to adaptive planning -where the action taken in each step depends on information obtained in the previous steps -is that feasible solutions are now policies (decision trees or conditional plans) instead of subsets. We propose a natural generalization of the diminishing returns property for adaptive problems, which reduces to the classical characterization of submodular set functions for deterministic distributions. We show how these results of Nemhauser et al. (1978), Wolsey (1982), Streeter and Golovin (2008), and Minoux (1978) generalize to the adaptive setting. Hence, we demonstrate how adaptive submodular optimization problems enjoy theoretical and practical benefits similar to those of classical, nonadaptive submodular problems. We further demonstrate the usefulness and generality of the concept by showing how it captures known results in stochastic optimization and active learning as special cases, admits tighter performance bounds, leads to natural generalizations and allows us to solve new problems for which no performance guarantees were known. As a first example, consider the problem of deploying (or controlling) a collection of sensors to monitor some spatial phenomenon. Each sensor can cover a region depending on its sensing range. Suppose we would like to find the best subset of k locations to place the sensors. In this application, intuitively, adding a sensor helps more if we have placed few sensors so far and helps less if we have already placed many sensors. We can formalize this diminishing returns property using the notion of submodularity -the total area covered by the sensors is a submodular function defined over all sets of locations. Krause and Guestrin (2007) show that many more realistic utility functions in sensor placement (such as the improvement in prediction accuracy w.r.t. some probabilistic model) are submodular as well. Now consider the following stochastic variant: Instead of deploying a fixed set of sensors, we deploy one sensor at a time. With a certain probability, deployed sensors can fail, and our goal is to maximize the area covered by the functioning sensors. Thus, when deploying the next sensor, we need to take into account which of the sensors we deployed in the past failed. This problem has been studied by Asadpour et al. (2008) for the case where each sensor fails independently at random. In this paper, we show that the coverage objective is adaptive submodular, and use this concept to handle much more general settings (where, e.g., rather than all-or-nothing failures there are different types of sensor failures of varying severity). We also consider a related problem where the goal is to place the minimum number of sensors to achieve the maximum possible sensor coverage (i.e., the coverage obtained by deploying sensors everywhere), or more generally the goal may be to achieve any fixed percentage of the maximum possible sensor coverage. Under the first goal, the problem is equivalent to one studied by Goemans and Vondrák (2006), and generalizes a problem studied by Liu et al. (2008). As another example, consider a viral marketing problem, where we are given a social network, and we want to influence as many people as possible in the network to buy some product. We do that by giving the product for free to a subset of the people, and hope that they convince their friends to buy the product as well. Formally, we have a graph, and each edge e is labeled by a number 0 ≤ p e ≤ 1. We "influence" a subset of nodes in the graph, and for each influenced node, their neighbors get randomly influenced according to the probability annotated on the edge connecting the nodes. This process repeats until no further node gets influenced. Kempe et al. (2003) show that the set function which quantifies the expected number of nodes influenced is submodular. A natural stochastic variant of the problem is where we pick a node, get to see which nodes it influenced, then adaptively pick the next node based on these observations and so on. We show that a large class of such adaptive influence maximization problems satisfies adaptive submodularity. Our third application is in active learning, where we are given an unlabeled data set, and we would like to adaptively pick a small set of examples whose labels imply all other labels. The same problem arises in automated diagnosis, where we have hypotheses about the state of the system (e.g., what illness a patient has), and would like to perform tests to identify the correct hypothesis. In both domains we want to pick examples / tests to shrink the remaining version space (the set of consistent hypotheses) as quickly as possible. Here, we show that the reduction in version space probability mass is adaptive submodular, and use that observation to prove that the adaptive greedy algorithm is a near-optimal querying policy. Our results for active learning namely Stochastic Submodular Maximization ( §6, page 16), Stochastic Submodular Coverage ( §7, page 18), Adaptive Viral Marketing ( §8, page 20), and Active Learning ( §9, page 24). In §10 (page 29) we report empirical results on two sensor selection problems. In §11 (page 31) we discuss the adaptivity gap of the problems we consider, and in §12 (page 33) we prove hardness results indicating that problems which are not adaptive submodular can be extremely inapproximable under reasonable complexity assumptions. We review related work in §13 (page 34) and provide concluding remarks in §14 (page 36). The Appendix (page 41) gives details of how to incorporate item costs and includes all of the proofs omitted from the main text. We start by introducing notation and defining the general class of adaptive optimization problems that we address in this paper. For sake of clarity, we will illustrate our notation using the sensor placement application mentioned in §1. We give examples of other applications in §6, §7, §8, and §9. Let E be a finite set of items (e.g., sensor locations). Each item e ∈ E is in a particular (initially unknown) state from a set O of possible states (describing whether a sensor placed at location e would malfunction or not). We represent the item states using a function φ : E → O, called a realization (of the states of all items in the ground set). Thus, we say that φ(e) is the state of e under realization φ. We use Φ to denote a random realization. We take a Bayesian approach and assume that there is a known prior probability distribution p (φ) := P [Φ = φ] over realizations (e.g., modeling that sensors fail independently with failure probability), so that we can compute posterior distributions2 . We will consider problems where we sequentially pick an item e ∈ E, get to see its state Φ(e), pick the next item, get to see its state, and so on (e.g., place a sensor, see whether it fails, and so on). After each pick, our observations so far can be represented as a partial realization ψ, a function from some subset of E (i.e., the set of items that we already picked) to their states (e.g., ψ encodes where we placed sensors and which of them failed). For notational convenience, we sometimes represent ψ as a relation, so that ψ ⊆ E × O equals {(e, o) : ψ(e) = o}. We use the notation dom(ψ) = {e : ∃o.(e, o) ∈ ψ} to refer to the domain of ψ (i.e., the set of items observed in ψ). A partial realization ψ is consistent with a realization φ if they are equal everywhere in the domain of ψ. In this case we write φ ∼ ψ. If ψ and ψ are both consistent with some φ, and dom(ψ) ⊆ dom(ψ ), we say ψ is a subrealization of ψ . Equivalently, ψ is a subrealization of ψ if and only if, when viewed as relations, ψ ⊆ ψ . Partial realizations are similar to the notion of "belief states" in Partially Observable Markov Decision Problems (POMDPs), as they encode the effect of all actions taken (items selected) and observations made, and determine our posterior belief about the state of the world (i.e., the state of all items e not yet selected, p (φ | ψ) := P [Φ = φ | Φ ∼ ψ]). We encode our adaptive strategy for picking items as a policy π, which is a function from a set of partial realizations to E, specifying which item to pick next under a particular set of observations (e.g., π chooses the next sensor location given where we have placed sensors so far, and whether they failed or not). We also allow randomized policies that are functions from a set of partial realizations to distributions on E, though our emphasis will primarily be on deterministic policies. If ψ is not in the domain of π, the policy terminates (stops picking items) upon observation of ψ. We use dom(π) to denote the domain of π. Technically, we require that dom(π) be closed under subrealizations. That is, if ψ ∈ dom(π) and ψ is a subrealization of ψ then ψ ∈ dom(π). We use the notation E(π, φ) to refer to the set of items selected by π under realization φ. Each deterministic policy π can be associated with a decision tree T π in a natural way (see Fig. 1 for an Figure 1: Illustration of a policy π, its corresponding decision tree representation, and the decision tree representation of π [2] , the level 2 truncation of π (as defined in §5.1). illustration). Here, we adopt a policy-centric view that admits concise notation, though we find the decision tree view to be valuable conceptually. Since partial realizations are similar to POMDP belief states, our definition of policies is similar to the notion of policies in POMDPs, which are usually defined as functions from belief states to actions. We will further discuss the relationship between the stochastic optimization problems considered in this paper and POMDPs in Section 13. We wish to maximize, subject to some constraints, a utility function f : 2 E × O E → R ≥0 that depends on which items we pick and which state each item is in (e.g., modeling the total area covered by the working sensors). Based on this notation, the expected utility of a policy π is f avg (π) := E [f (E(π, Φ), Φ)] where the expectation is taken with respect to p (φ). The goal of the Adaptive Stochastic Maximization problem is to find a policy π * such that π * ∈ arg max where k is a budget on how many items can be picked (e.g., we would like to adaptively choose k sensor locations such that the working sensors provide as much information as possible in expectation). Alternatively, we can specify a quota Q of utility that we would like to obtain, and try to find the cheapest policy achieving that quota (e.g., we would like to achieve a certain amount of information, as cheaply as possible in expectation). Formally, we define the average cost c avg (π) of a policy as the expected number of items it picks, so that c avg (π) := E [|E(π, Φ)|]. Our goal is then to find i.e., the policy π * that minimizes the expected number of items picked such that under all possible realizations, at least utility Q is achieved. We call Problem 2 the Adaptive Stochastic Minimum Cost Cover problem. We will also consider the problem where we want to minimize the worst-case cost c wc (π) := max φ |E(π, φ)|. This worst-case cost c wc (π) is the cost incurred under adversarially chosen realizations, or equivalently the depth of the deepest leaf in T π , the decision tree associated with π. Yet another important variant is to minimize the average time required by a policy to obtain its utility. Formally, let u(π, t) be the expected utility obtained by π after t steps3 , let Q = E [f (E, Φ)] be the maximum possible expected utility, and define the min-sum cost c Σ (π) of a policy as c Σ (π) := ∞ t=0 (Q -u(π, t)). We then define the Adaptive Stochastic Min-Sum Cover problem as the search for (3) Unfortunately, as we will show in §12, even for linear functions f , i.e., those where f (A, φ) = e∈A w e,φ is simply the sum of weights (depending on the realization φ), Problems (1), (2), and (3) are hard to approximate under reasonable complexity theoretic assumptions. Despite the hardness of the general problems, in the following sections we will identify conditions that are sufficient to allow us to approximately solve them. Instead of quantifying the cost of a set E(π, φ) by the number of elements |E(π, φ)|, we can also consider the case where each item e ∈ E has a cost c(e), and the cost of a set S ⊆ E is c(S) = e∈S c(e). We can then consider variants of Problems (1), (2), and (3) with the |E(π, φ)| replaced by c(E(π, φ)). For clarity of presentation, we will focus on the unit cost case, i.e., c(e) = 1 for all e, and explain how our results generalize to the non-uniform case in the Appendix. We first review the classical notion of submodular set functions, and then introduce the novel notion of adaptive submodularity. Let us first consider the very special case where p (φ) is deterministic or, equivalently, |O| = 1 (e.g., in our sensor placement applications, sensors never fail). In this case, the realization φ is known to the decision maker in advance, and thus there is no benefit in adaptive selection. Given the realization φ, Problem (1) is equivalent to finding a set A * ⊆ E such that For most interesting classes of utility functions f , this is an NP-hard optimization problem. However, in many practical problems, such as those mentioned in §1, i.e., adding e to the smaller set A increases f by at least as much as adding e to the superset B. Furthermore, f is called monotone, if, whenever A ⊆ B it holds that f (A) ≤ f (B) (e.g., adding a sensor can never reduce the amount of information obtained). A celebrated result by Nemhauser et al. (1978) states that for monotone submodular functions with f (∅) = 0, a simple greedy algorithm that starts with the empty set, A 0 = ∅ and chooses . Thus, the greedy set A k obtains at least a (1 -1/e) fraction of the optimal value achievable using k elements. Furthermore, Feige (1998) shows that this result is tight if P = NP; under this assumption no polynomial time algorithm can do strictly better than the greedy algorithm, i.e., achieve a (1 -1/e + )-approximation for any constant > 0, even for the special case of Maximum k-Cover where f (A) is the cardinality of the union of sets indexed by A. Similarly, Wolsey (1982) shows that the same greedy algorithm also near-optimally solves the deterministic case of Problem (2), called the Minimum Submodular Cover problem: Pick the first set A constructed by the greedy algorithm such that f (A ) ≥ Q. Then, for integer-valued submodular functions, is at most |A * |(1 + log max e f (e)), i.e., the greedy set is at most a logarithmic factor larger than the smallest set achieving quota Q. For the special case of Set Cover, where f (A) is the cardinality of a union of sets indexed by A, this result matches a lower bound by Feige (1998): Unless NP ⊆ DTIME(n O(log log n) ), Set Cover is hard to approximate by a factor better than (1 -ε) ln Q, where Q is the number of elements to be covered. Now let us relax the assumption that p (φ) is deterministic. In this case, we may still want to find a nonadaptive solution (i.e., a constant policy π A that always picks set A independently of Φ) maximizing f avg (π A ). If f is pointwise submodular, i.e., f (A, φ) is submodular in A for any fixed φ, the function f (A) = f avg (π A ) is submodular, since nonnegative linear combinations of submodular functions remain submodular. Thus, the greedy algorithm allows us to find a near-optimal non-adaptive policy. That is, in our sensor placement example, if we are willing to commit to all locations before finding out whether the sensors fail or not, the greedy algorithm can provide a good solution to this non-adaptive problem. However, in practice, we may be more interested in obtaining a non-constant policy π, that adaptively chooses items based on previous observations (e.g., takes into account which sensors are working before placing the next sensor). In many settings, selecting items adaptively offers huge advantages, analogous to the advantage of binary search over sequential (linear) search4 . Thus, the question is whether there is a natural extension of submodularity to policies. In the following, we will develop such a notion -adaptive submodularity. The key challenge is to find appropriate generalizations of monotonicity and of the diminishing returns condition (5). We begin by again considering the very special case where p (φ) is deterministic, so that the policies are non-adaptive. In this case a policy π simply specifies a sequence of items (e 1 , e 2 , . . . , e r ) which it selects in order. Monotonicity in this context can be characterized as the property that "the marginal benefit of selecting an item is always nonnegative," meaning that for all such sequences (e 1 , e 2 , . . . , e r ), items e and 1 ≤ i ≤ r it holds that f ({e j : j ≤ i} ∪ {e}) -f ({e j : j ≤ i}) ≥ 0. Similarly, submodularity can be viewed as the property that "selecting an item later never increases its marginal benefit," meaning that for all sequences (e 1 , e 2 , . . . , e r ), items e, and all i ≤ r, f ({e j : j ≤ i} ∪ {e}) -f ({e j : j ≤ i}) ≥ f ({e j : j ≤ r} ∪ {e}) -f ({e j : j ≤ r}). We take these views of monotonicity and submodularity when defining their adaptive analogues, by using an appropriate generalization of the marginal benefit. When moving to the general adaptive setting, the challenge is that the items' states are now random and only revealed upon selection. A natural approach is thus to condition on observations (i.e., partial realizations of selected items), and take the expectation with respect to the items that we consider selecting. Hence, we define our adaptive monotonicity and submodularity properties in terms of the conditional expected marginal benefit of an item. Definition 1 (Conditional Expected Marginal Benefit) Given a partial realization ψ and an item e, the conditional expected marginal benefit of e conditioned on having observed ψ, denoted ∆(e | ψ), is where the expectation is computed with respect to p (φ Similarly, the conditional expected marginal benefit of a policy π is which is the benefit term subtracted out in Eq. ( 8) and Eq. ( 9). Similarly, the expected total benefit obtained after observing ψ and then selecting e is The corresponding benefit for running π after observing ψ is slightly more complex. Under realization φ ∼ ψ, the final cumulative benefit will be f (dom(ψ) ∪ E(π, φ), φ). Taking the expectation with respect to p (φ | ψ) and subtracting out the benefit already obtained by dom(ψ) then yields the conditional expected marginal benefit of π. We are now ready to introduce our generalizations of monotonicity and submodularity to the adaptive setting: is adaptive monotone with respect to distribution p (φ) if the conditional expected marginal benefit of any item is nonnegative, i.e., for all ψ with P [Φ ∼ ψ] > 0 and all e ∈ E we have is adaptive submodular with respect to distribution p (φ) if the conditional expected marginal benefit of any fixed item does not increase as more items are selected and their states are observed. Formally, f is adaptive submodular w.r.t. p (φ) if for all ψ and ψ such that ψ is a subrealization of ψ (i.e., ψ ⊆ ψ ), and for all e ∈ E \ dom(ψ ), we have From the decision tree perspective, the condition ∆(e | ψ) ≥ ∆(e | ψ ) amounts to saying that for any decision tree T , if we are at a node v in T which selects an item e, and compare the expected marginal benefit of e selected at v with the expected marginal benefit e would have obtained if it were selected at an ancestor of v in T , then the latter must be no smaller than the former. Note that when comparing the two expected marginal benefits, there is a difference in both the set of items previously selected (i.e., dom(ψ) vs. dom(ψ )) and in the distribution over realizations (i.e., p (φ | ψ) vs. p (φ | ψ )). It is also worth emphasizing that adaptive submodularity is defined relative to the distribution p (φ) over realizations; it is possible that f is adaptive submodular with respect to one distribution, but not with respect to another. We will give concrete examples of adaptive monotone and adaptive submodular functions that arise in the applications introduced in §1 in §6, §7, §8, and §9. In the Appendix, we will explain how the notion of adaptive submodularity can be extended to handle non-uniform costs (since, e.g., the cost of placing a sensor at an easily accessible location may be smaller than at a location that is hard to get to). It can be seen that adaptive monotonicity and adaptive submodularity enjoy similar closure properties as monotone submodular functions. In particular, if w 1 , . . . , w m ≥ 0 and f 1 , . . . , f m are adaptive monotone submodular w.r. Similarly, for a fixed constant c ≥ 0 and adaptive monotone submodular function f , the function g(E, φ) = min(f (E, φ), c) is adaptive monotone submodular. Thus, adaptive monotone submodularity is preserved by nonnegative linear combinations and by truncation. Adaptive monotone submodularity is also preserved by restriction, so that if f : 2 E × O E → R ≥0 is adaptive monotone submodular w.r.t. p (φ), then for any e ∈ E, the function g : 2 E\{e} × O E → R ≥0 defined by g(A, φ) := f (A, φ) for all A ⊆ E \ {e} and all φ is also adaptive submodular w.r.t. p (φ). Finally, if f : 2 E × O E → R ≥0 is adaptive monotone submodular w.r.t. p (φ) then for each partial realization ψ the conditional function g(A, φ) Adaptive submodularity is a diminishing returns property for policies. Speaking informally, it can be applied in situations where there is an objective function to be optimized does not feature synergies in the benefits of items conditioned on observations. In some cases, the primary objective might not have this property, but a suitably chosen proxy of it does, as is the case with active learning with persistent noise (Golovin et al., 2010;Bellala and Scott, 2010). We give example applications in §6 through §9. It is also worth mentioning where adaptive submodularity is not directly applicable. An extreme example of synergistic effects between items conditioned on observations is the class of "treasure hunting" instances used to prove Theorem 26 on page 33, where the (binary) state of certain groups of items encode the treasure's location in a complex manner. Another problem feature which adaptive submodularity does not directly address is the possibility that items selection can alter the underlying realization φ, as is the case for the problem of optimizing policies for general POMDPs. The classical non-adaptive greedy algorithm (6) has a natural generalization to the adaptive setting. The greedy policy π greedy tries, at each iteration, to myopically increase the expected objective value, given its current observations. That is, suppose f : 2 E × O E → R ≥0 is the objective, and ψ is the partial realization indicating the states of items selected so far. Then the greedy policy will select the item e maximizing the expected increase in value, conditioned on the observed states of items it has already selected (i.e., conditioned on Φ ∼ ψ). That is, it will select e to maximize the conditional expected marginal benefit ∆(e | ψ) as defined in Eq. ( 8). Pseudocode of the adaptive greedy algorithm is given in Algorithm 1. The only difference to the classic, non-adaptive greedy algorithm studied by Nemhauser et al. (1978), is Line 1, where an observation Φ(e * ) of the selected item e * is obtained. Note that the algorithms in this section are presented for Adaptive Stochastic Maximization. For the coverage objectives, we simply keep selecting items as prescribed by π greedy until achieving the quota on objective value (for the min-cost objective) or until we have selected every item (for the min-sum objective). The adaptive greedy algorithm can be naturally modified to handle non-uniform item costs by replacing its selection rule by c(e) . In the following, we will focus on the uniform cost case (c ≡ 1), and defer the analysis with costs to the Appendix. In some applications, finding an item maximizing ∆(e | ψ) may be computationally intractable, and the best we can do is find an α-approximation to the best greedy selection. This means we find an e such that We call a policy which always selects such an item an α-approximate greedy policy. As we will show, α-approximate greedy policies have performance guarantees on several problems. The fact that these performance guarantees of greedy policies are robust to approximate greedy selection suggests a particular robustness guarantee against incorrect priors p (φ). Specifically, if our incorrect prior p is such that when we evaluate ∆(e | ψ) we err by a multiplicative factor of at most α, then when we compute the greedy policy with respect to p we are actually implementing an α-approximate greedy policy (with respect to the true prior), and hence obtain the corresponding guarantees. For example, a sufficient condition for erring by at most a multiplicative factor of α is that there exists c ≤ 1 and d ≥ 1 with α = d/c such that c p (φ) ≤ p (φ) ≤ d p (φ) for all φ, where p is the true prior. The definition of adaptive submodularity allows us to implement an "accelerated" version of the adaptive greedy algorithm using lazy evaluations of marginal benefits as originally suggested for the non-adaptive case by Minoux (1978). The idea is as follows. Suppose we run π greedy under some fixed realization φ, and select items e 1 , e 2 , . . . , e k . Let ψ i := {(e j , φ(e j ) : j ≤ i)} be the partial realizations observed during the run of π greedy . The adaptive greedy algorithm computes ∆(e | ψ i ) for all e ∈ E and 0 ≤ i < k, unless e ∈ dom(ψ i ). Naively, the algorithm thus needs to compute Θ(|E|k) marginal benefits (which can be expensive to compute). The key insight is that i → ∆(e | ψ i ) is nonincreasing for all e ∈ E, because of the adaptive submodularity of the objective. Hence, if when deciding which item to select as e i we know ∆(e | ψ j ) ≤ ∆(e | ψ i ) for some items e and e and j < i, then we may conclude ∆(e | ψ i ) ≤ ∆(e | ψ i ) and hence eliminate the need to compute ∆(e | ψ i ). The accelerated version of the adaptive greedy algorithm exploits this observation in a principled manner, by computing ∆(e | ψ) for items e in decreasing order of the upper bounds known on them, until it finds an item whose value is at least as great as the upper bounds of all other items. Pseudocode of this version of the adaptive greedy algorithm is given in Algorithm 2. In the non-adaptive setting, the use of lazy evaluations has been shown to significantly reduce running times in practice (Leskovec et al., 2007). We evaluated the naive and accelerated implementations of the adaptive greedy algorithm on two sensor selection problems, and obtained speedup factors that range from roughly 4 to 40 for those problems. See §10 on page 29 for details. In this section we show that if the objective function is adaptive submodular with respect to the probabilistic model of the environment in which we operate, then the greedy policy inherits precisely the performance guarantees of the greedy algorithm for classic (non-adaptive) submodular maximization and submodular coverage problems, such as Maximum k-Cover and Minimum Set Cover, as well as min-sum submodular coverage problems, such as Min-Sum Set Cover. In fact, we will show that this holds true more generally: α-approximate greedy policies inherit precisely the performance guarantees of α-approximate greedy algorithms for these classic problems. These guarantees suggest that adaptive submodularity is an appropriate generalization of submodularity to policies. In this section we focus on the unit cost case (i.e., every item has the same cost). In the Appendix we provide the proofs omitted in this section, and show how our results extend to non-uniform item costs if we greedily maximize the expected benefit/cost ratio. In this section we consider the maximum coverage objective, where the goal is to select k items adaptively to maximize their expected value. The task of maximizing expected value subject to more complex constraints, 1 foreach e ∈ E do Q. insert(e, +∞); such as matroid constraints and intersections of matroid constraints, is considered in the work of Golovin and Krause (2011b). Before stating our result, we require the following definition. Definition 4 (Policy Truncation) For a policy π, define the level-k-truncation π [k] of π to be the policy obtained by running π until it terminates or until it selects k items, and then terminating. Formally, dom(π [k] ) = {ψ ∈ dom(π) : |ψ| < k}, and π [k] (ψ) = π(ψ) for all ψ ∈ dom(π [k] ). We have the following result, which generalizes the classic result of the work of Nemhauser et al. (1978) that the greedy algorithm achieves a (1 -1/e)-approximation to the problem of maximizing monotone submodular functions under a cardinality constraint. By setting = k and α = 1 in Theorem 5, we see that the greedy policy which selects k items adaptively obtains at least (1 -1/e) of the value of the optimal policy that selects k items adaptively, measured with respect to f avg . For a proof see Theorem 38 in Appendix 15.3, which generalizes Theorem 5 to nonuniform item costs. Theorem 5 Fix any α ≥ 1. If f is adaptive monotone and adaptive submodular with respect to the distribution p (φ), and π is an α-approximate greedy policy, then for all policies π * and positive integers and k, In particular, with = k this implies any α-approximate greedy policy achieves a 1 -e -1/α approximation to the expected reward of the best policy, if both are terminated after running for an equal number of steps. If the greedy rule can be implemented only with small absolute error rather than small relative error, i.e., ∆(e | ψ) ≥ max e ∆(e | ψ) -ε, an argument similar to that used to prove Theorem 5 shows that This is important, since small absolute error can always be achieved (with high probability) whenever f can be evaluated efficiently, and sampling p (φ | ψ) is efficient. In this case, we can approximate where φ i are sampled i.i.d. from p (φ | ψ). For the maximum coverage objective, adaptive submodular functions have another attractive feature: they allow us to obtain data dependent bounds on the optimum, in a manner similar to the bounds for the non-adaptive case (Minoux, 1978). Consider the non-adaptive problem of maximizing a monotone submodular function f : 2 A → R ≥0 subject to the constraint |A| ≤ k. Let A * be an optimal solution, and fix any A ⊆ E. Then Note that unlike the original objective, we can easily compute max B:|B|≤k e∈B (f (A ∪ {e}) -f (A)) by computing δ(e) := f (A ∪ {e}) -f (A) for each e, and summing the k largest values. Hence we can quickly compute an upper bound on our distance from the optimal value, f (A * ) -f (A). In practice, such data-dependent bounds can be much tighter than the problem-independent performance guarantees of Nemhauser et al. (1978) for the greedy algorithm (Leskovec et al., 2007). Further note that these bounds hold for any set A, not just sets selected by the greedy algorithm. These data dependent bounds have the following analogue for adaptive monotone submodular functions. See Appendix 15.2 for a proof. Lemma 6 (The Adaptive Data Dependent Bound) Suppose we have made observations ψ after selecting dom(ψ). Let π * be any policy such that |E(π * , φ)| ≤ k for all φ. Then for adaptive monotone submodular f Thus, after running any policy π, we can efficiently compute a bound on the additional benefit that the optimal solution π * could obtain beyond the reward of π. We do that by computing the conditional expected marginal benefits for all elements e, and summing the k largest of them. Note that these bounds can be computed on the fly when running the greedy algorithm, in a similar manner as discussed by Leskovec et al. (2007) for the non-adaptive setting. Another natural objective is to minimize the number of items selected while ensuring that a sufficient level of value is obtained. This leads to the Adaptive Stochastic Minimum Cost Coverage problem described in §2, namely π * ∈ arg min π c avg (π) such that f (E(π, φ), φ) ≥ Q for all φ. Recall that c avg (π) is the expected cost of π, which in the unit cost case equals the expected number of items selected by π, i.e., c avg (π If the objective is adaptive monotone submodular, this is an adaptive version of the Minimum Submodular Cover problem (described on line (7) in §3.1). Recall that the greedy algorithm is known to give a (ln(Q) + 1)-approximation for Minimum Submodular Cover assuming the coverage function is integer-valued in addition to being monotone submodular (Wolsey, 1982). Adaptive Stochastic Minimum Cost Coverage is also related to the (Noisy) Interactive Submodular Set Cover problem studied by Guillory andBilmes (2010, 2011), which considers the worst-case setting (i.e., there is no distribution over states; instead states are realized in an adversarial manner). Similar results for active learning have been proved by Kosaraju et al. (1999) and Dasgupta (2004), as we discuss in more detail in §9. We assume throughout this section that there exists a quality threshold Q such that f (E, φ) = Q for all φ, and for all S ⊆ E and all φ, f (S, φ) ≤ Q. Note that, as discussed in Section 3, if we replace f (S, φ) by a new function g(S, φ) = min(f (S, φ), Q ) for some constant Q , g will be adaptive submodular if f is. Thus, if f (E, φ) varies across realizations, we can instead use the greedy algorithm on the function truncated at some threshold Q ≤ min φ f (E, φ) achievable by all realizations. In contrast to Adaptive Stochastic Maximization, for the coverage problem additional subtleties arise. In particular, it is not enough that a policy π achieves value Q for the true realization; in order for π to terminate, it also requires a proof of this fact. Formally, we require that π covers f : Definition 7 (Coverage) Let ψ = ψ(π, φ) be the partial realization encoding all states observed during the execution of π under realization φ. Given f : 2 E × O E → R, we say a policy π covers φ with respect to f if f (dom(ψ), φ ) = f (E, φ ) for all φ ∼ ψ. We say that π covers f if it covers every realization with respect to f . Coverage is defined in such a way that upon terminating, π might not know which realization is the true one, but has guaranteed that it has achieved the maximum reward in every possible case (i.e., for every realization consistent with its observations). We obtain results for both the average and worst-case cost objectives. Before presenting our approximation guarantee for the Adaptive Stochastic Minimum Cost Coverage, we introduce a special class of instances, called self-certifying instances. We make this distinction because the greedy policy has stronger performance guarantees for self-certifying instances, and such instances arise naturally in applications. For example, the Stochastic Submodular Cover and Stochastic Set Cover instances in §7, the Adaptive Viral Marketing instances in §8, and the Pool-Based Active Learning instances in §9 are all self-certifying. Definition 8 (Self-Certifying Instances) An instance of Adaptive Stochastic Minimum Cost Coverage is self-certifying if whenever a policy achieves the maximum possible value for the true realization it immediately has a proof of this fact. Formally, an instance (f, p (φ)) is self-certifying if for all φ, φ , and ψ such that φ ∼ ψ and φ ∼ ψ, we have One class of self-certifying instances which commonly arise are those in which f (A, φ) depends only on the state of items in A, and in which there is a uniform maximum amount of reward that can be obtained across realizations. Formally, we have the following observation. Proposition 9 Fix an instance (f, p (φ)). If there exists Q such that f (E, φ) = Q for all φ and there exists some g : 2 E×O → R ≥0 such that f (A, φ) = g ({(e, φ(e)) : e ∈ A}) for all A and φ, then (f, p (φ)) is self-certifying. Proof Fix φ, φ , and ψ such that φ ∼ ψ and φ ∼ ψ. Assuming the existence of g and treating ψ as a relation, For our results on minimum cost coverage, we also need a stronger monotonicity condition and a stronger submodularity condition: R is strongly adaptive monotone with respect to p (φ) if, informally "selecting more items never hurts" with respect to the expected reward. Formally, for all ψ, all e / ∈ dom(ψ), and all possible outcomes o ∈ O such that Strong adaptive monotonicity implies adaptive monotonicity, as the latter means that "selecting more items never hurts in expectation," i.e., To define strong adaptive submodularity, we first need the following extension of ∆(e | ψ): is strongly adaptive submodular with respect to distribution p (φ) if it is adaptive submodular and moreover the expected marginal benefit of any fixed item does not increase as more items are selected and their states are observed, conditioned on the (item, observation) pairs. Formally, f is adaptive submodular w.r.t. p (φ) if for all ψ and ψ such that ψ is a subrealization of ψ (i.e., ψ ⊆ ψ ), and for all e ∈ E \ dom(ψ ), we have In other words, conditioning on ψ , adding items dom(ψ ) \ dom(ψ) cannot increase the expected marginal benefit of e. A sufficient condition for strong adaptive submodularity with respect to p (φ) is that the function be adaptive submodular and pointwise submodular (i.e., f (A, φ) is submodular in A for any fixed φ), as we prove in Appendix 15.4. It is worth noting that pointwise submodularity is not sufficient to establish adaptive submodularity. We now state our main result for the average case cost c avg (π): is strongly adaptive submodular and strongly adaptive monotone with respect to p (φ) and there exists Q such that f (E, φ) = Q for all φ. Let η be any value such that f (S, φ) > Q -η implies f (S, φ) = Q for all S and φ. Let δ = min φ p (φ) be the minimum probability of any realization. Let π * avg be an optimal policy minimizing the expected number of items selected to guarantee every realization is covered. Let π be an α-approximate greedy policy with respect to the item costs. Then in general Note that if range(f ) ⊂ Z, then η = 1 is a valid choice, so for general and self-certifying instances we have Historical Note: An earlier version of Theorem 13 claimed logarithmic approximation factors rather than the squared-logarithmic factors present here. Unfortunately, the proof was flawed as pointed out by Nan and Saligrama (2017). Determining whether the logarithmic bounds hold remains an interesting open problem. In particular, it remains open whether c avg (π) ≤ α c avg (π * avg ) ln Q δη + 1 for general instances and c avg (π) ≤ α c avg (π * avg ) ln Q η + 1 for self-certifying instances under the conditions specified by Theorem 13. It also remains open whether the strong adaptive submodularity condition is required. For the worst-case cost c wc (π) := max φ |E(π, φ)|, strong adaptive monotonicity and strong submodularity are not required; adaptive monotonicity and adaptive submodularity suffice. We obtain the following result. Theorem 14 Suppose f : 2 E × O E → R ≥0 is adaptive monotone and adaptive submodular with respect to p (φ), and let η be any value such that f (S, φ) > f (E, φ) -η implies f (S, φ) = f (E, φ) for all S and φ. Let δ = min φ p (φ) be the minimum probability of any realization. Let π * wc be the optimal policy minimizing the worst-case number of queries to guarantee every realization is covered. Let π be an α-approximate greedy policy. Finally, let Q := E [f (E, φ)] be the maximum possible expected reward. Then The proofs of Theorems 13 and 14 are given in Appendix 15.4. Thus, even though adaptive submodularity is defined w.r.t. a particular distribution, perhaps surprisingly, the adaptive greedy algorithm is competitive even in the case of adversarially chosen realizations, against a policy optimized to minimize the worst-case cost. Theorem 14 therefore suggests that if we do not have a strong prior, we can obtain the strongest guarantees if we choose a distribution that is "as uniform as possible" (i.e., maximizes δ) while still guaranteeing adaptive submodularity. Note that the approximation factor for self-certifying instances in Theorem 14 reduces to the (ln(Q) + 1)approximation guarantee for the greedy algorithm for Set Cover instances with Q elements, in the case of a deterministic distribution p (φ). Moreover, with a deterministic distribution p (φ) there is no distinction between average-case and worst-case cost. Hence, an immediate corollary of the result of Feige (1998) mentioned in §3 is that for every constant > 0 there is no polynomial time (1 -) ln (Q/η) approximation algorithm for self-certifying instances of Adaptive Stochastic Min Cost Cover, under either the c avg (•) or the c wc (•) objective, unless NP ⊆ DTIME(n O(log log n) ). It remains open to determine whether or not Adaptive Stochastic Min Cost Cover with the worst-case cost objective admits a ln (Q/η) + 1 approximation for self-certifying instances via a polynomial time algorithm, and in particular whether the greedy policy has such an approximation guarantee. However, in Lemma 50 we show that Feige's result also implies there is no (1 -) ln (Q/δη) polynomial time approximation algorithm for general (non self-certifying) instances of Adaptive Stochastic Min Cost Cover under either objective, unless NP ⊆ DTIME(n O(log log n) ). In that sense, Theorem 14 is best-possible and Theorem 13 cannot be improved by more than a logarithmic factor and under reasonable complexity-theoretic assumptions. Yet another natural objective is the min-sum objective, in which an unrealized reward of x incurs a cost of x in each time step, and the goal is to minimize the total cost incurred. In the non-adaptive setting, perhaps the simplest form of a coverage problem with this objective is the Min-Sum Set Cover problem (Feige et al., 2004) in which the input is a set system (U, S), the output is a permutation of the sets S 1 , S 2 , . . . , S m , and the goal is to minimize the sum of element coverage times, where the coverage time of u is the index of the first set that contains it (e.g., it is j if u ∈ S j and u / ∈ S i for all i < j). In this problem and its generalizations the min-sum objective is useful in modeling processing costs in certain applications, for example in ordering diagnostic tests to identify a disease cheaply (Kaplan et al., 2005), in ordering multiple filters to be applied to database records while processing a query (Munagala et al., 2005), or in ordering multiple heuristics to run on boolean satisfiability instances as a means to solve them faster in practice (Streeter and Golovin, 2008). A particularly expressive generalization of min-sum set cover has been studied under the names Min-Sum Submodular Cover (Streeter and Golovin, 2008) and L 1 -Submodular Set Cover (Golovin et al., 2008). The former paper extends the greedy algorithm to a natural online variant of the problem, while the latter studies a parameterized family of L p -Submodular Set Cover problems in which the objective is analogous to minimizing the L p norm of the coverage times for Min-Sum Set Cover instances. In the Min-Sum Submodular Cover problem, there is a monotone submodular function f : 2 E → R ≥0 defining the reward obtained from a collection of elements 5 . There is an integral cost c(e) for each element, and the output is a sequence of all of the elements σ = e 1 , e 2 , . . . , e n . For each t ∈ R ≥0 , we define the set of 5. To encode Min-Sum Set Cover instance (U, S), let E := S and f (A) := | ∪ e∈A e|, where each e ∈ E is a subset of elements in U . elements in the sequence σ within a budget of t: The cost we wish to minimize is then ) . (17) Feige et al. (2004) proved that for Min-Sum Set cover, the greedy algorithm achieves a 4-approximation to the minimum cost, and also that this is optimal in the sense that no polynomial time algorithm can achieve a (4 -)-approximation, for any > 0, unless P = NP. Interestingly, the greedy algorithm also achieves a 4-approximation for the more general Min-Sum Submodular Cover problem as well (Streeter and Golovin, 2008;Golovin et al., 2008). In this article, we extend the result of Streeter and Golovin (2008) and Golovin et al. (2008) to an adaptive version of Min-Sum Submodular Cover. For clarity's sake we will consider the unit-cost case here (i.e., c(e) = 1 for all e); we show how to extend adaptive submodularity to handle general costs in the Appendix. In the adaptive version of the problem, π [t] plays the role of σ [t] , and f avg plays the role of f . The goal is to find a policy π minimizing We call this problem the Adaptive Stochastic Min-Sum Cover problem. The key difference between this objective and the minimum cost cover objective is that here, the cost at each step is only the fractional extent that we have not covered the true realization, whereas in the minimum cost cover objective we are charged in full in each step until we have completely covered the true realization (according to Definition 7). We prove the following result for the Adaptive Stochastic Min-Sum Cover problem with arbitrary item costs in Appendix 15.5. Theorem 15 Fix any α ≥ 1. If f is adaptive monotone and adaptive submodular with respect to the distribution p (φ), π is an α-approximate greedy policy with respect to the item costs, and π * is any policy, then c Σ (π) ≤ 4α c Σ (π * ). As our first application, consider the sensor placement problem introduced in §1. Suppose we would like to monitor a spatial phenomenon such as temperature in a building. We discretize the environment into a set E of locations. We would like to pick a subset A ⊆ E of k locations that is most "informative", where we use a set function f (A) to quantify the informativeness of placement A. Krause and Guestrin (2007) show that many natural objective functions (such as reduction in predictive uncertainty measured in terms of Shannon entropy with conditionally independent observations) are monotone submodular. Now consider the problem, where the informativeness of a sensor is unknown before deployment (e.g., when deploying cameras for surveillance, the location of objects and their associated occlusions may not be known in advance, or varying amounts of noise may reduce the sensing range). We can model this extension by assigning a state φ(e) ∈ O to each possible location, indicating the extent to which a sensor placed at location e is working. To quantify the value of a set of sensor deployments under a realization φ indicating to what extent the various sensors are working, we first define (e, o) for each e ∈ E and o ∈ O, which represents the placement of a sensor at location e which is in state o. We then suppose there is a function f : 2 E×O → R ≥0 which quantifies the informativeness of a set of sensor deployments in arbitrary states. (Note f is a set function taking a set of (sensor deployment, state) pairs as input.) The utility f (A, φ) of placing sensors at the locations in A under realization φ is then f (A, φ) := f ({(e, φ(e)) : e ∈ A}). We aim to adaptively place k sensors to maximize our expected utility. We assume that sensor failures at each location are independent of each other, i.e., P where P [φ(e) = o] is the probability that a sensor placed at location e will be in state o. Asadpour et al. (2008) studied a special case of our problem, in which sensors either fail completely (in which case they contribute no value at all) or work perfectly, under the name Stochastic Submodular Maximization. They proved that the adaptive greedy algorithm obtains a (1-1/e) approximation to the optimal adaptive policy, provided f is monotone submodular. We extend their result to multiple types of failures by showing that f (A, φ) is adaptive submodular with respect to distribution p (φ) and then invoking Theorem 5. Fig. 2 illustrates an instance of Stochastic Submodular Maximization where f (A, φ) is the cardinality of union of sets index by A and parameterized by φ. Theorem 16 Fix a prior such that P [Φ = φ] = e∈E P [Φ(e) = φ(e)] and an integer k, and let the objective function f : 2 E×O → R ≥0 be monotone submodular. Let π be any α-approximate greedy policy attempting to maximize f , and let π * be any policy. Then for all positive integers , In particular, if π is the greedy policy (i.e., α = 1) and = k, then Proof We prove Theorem 16 by first proving f is adaptive monotone and adaptive submodular in this model, and then applying Theorem 5. Adaptive monotonicity is readily proved after observing that f (•, φ) is monotone for each φ. Moving on to adaptive submodularity, fix any ψ, ψ such that ψ ⊆ ψ and any e / ∈ dom(ψ ). We aim to show ∆(e | ψ ) ≤ ∆(e | ψ). Intuitively, this is clear, as ∆(e | ψ ) is the expected marginal benefit of adding e to a larger base set than is the case with ∆(e | ψ), namely dom(ψ ) as compared to dom(ψ), and the realizations are independent. To prove it rigorously, we define a coupled distribution µ over pairs of realizations φ ∼ ψ and φ ∼ ψ such that φ(e ) = φ (e ) for all e / ∈ dom(ψ ). Formally, µ(φ, φ ) = e∈E\dom(ψ) P [Φ(e) = φ(e)] if φ ∼ ψ, φ ∼ ψ , and φ(e ) = φ (e ) for all e / ∈ dom(ψ ); otherwise µ(φ, φ ) = 0. (Note that µ(φ, φ ) > 0 implies φ(e ) = φ (e ) for all e ∈ dom(ψ) as well, since φ ∼ ψ, φ ∼ ψ , and ψ ⊆ ψ .) Also note that p (φ | ψ) = φ µ(φ, φ ) and p (φ | ψ ) = φ µ(φ, φ ). Calculating ∆(e | ψ ) and ∆(e | ψ) using µ, we see that for any (φ, φ ) in the support of µ, from the submodularity of f . Hence which completes the proof. Suppose that instead of wishing to adaptively place k unreliable sensors to maximize the utility of the information obtained, as discussed in §6, we have a quota on utility and wish to adaptively place the minimum number of unreliable sensors to achieve this quota. This amounts to a minimum-cost coverage version of the Stochastic Submodular Maximization problem introduced in §6, which we call Stochastic Submodular Coverage. As in §6, in the Stochastic Submodular Coverage problem we suppose there is a function f : 2 E×O → R ≥0 which quantifies the utility of a set of sensors in arbitrary states. Also, the states of each sensor are independent, so that The goal is to obtain a quota Q of utility at minimum cost. Thus, we define our objective as f (A, φ) := min Q, f ({(e, φ(e)) : e ∈ A}) , and want to find a policy π covering every realization and minimizing c avg (π) := E [|E(π, Φ)|]. We additionally assume that this quota can always be obtained using sufficiently many sensor placements; formally, this amounts to f (E, φ) = Q for all φ. We obtain the following result, whose proof we defer until the end of this section. Theorem 17 Fix a prior with independent sensor states s.t. P [Φ = φ] = e∈E P [Φ(e) = φ(e)], and let f : Let η be any value such that f (S, φ) > Q -η implies f (S, φ) = Q for all S and φ. Finally, let π be an α-approximate greedy policy for maximizing f , and let π * be any policy. Then The Stochastic Submodular Coverage problem is a generalization of the Stochastic Set Coverage problem (Goemans and Vondrák, 2006). In Stochastic Set Coverage the underlying submodular objective f is the number of elements covered in some input set system. In other words, there is a ground set U of n elements to be covered, and items E such that each item e is associated with a distribution over subsets of U . When an item is selected, a set is sampled from its distribution, as illustrated in Fig. 2. The problem is to adaptively select items until all elements of U are covered by sampled sets, while minimizing the expected number of items selected. Like us, Goemans and Vondrák also assume that the subsets are sampled independently for each item, and every element of U can be covered in every realization, so that f (E, φ) = |U | for all φ. Goemans and Vondrák primarily investigated the adaptivity gap (quantifying how much adaptive policies can outperform non-adaptive policies) of Stochastic Set Coverage, for variants in which items can be repeatedly selected or not, and prove adaptivity gaps of Θ(log n) in the former case, and between Ω(n) and O(n 2 ) in the latter. They also provide an n-approximation algorithm. More recently, Liu et al. (2008) considered a special case of Stochastic Set Coverage in which each item may be in one of two states. They were motivated by a streaming database problem, in which a collection of queries sharing common filters must all be evaluated on a stream element. They transform the problem to a Stochastic Set Coverage instance in which (filter, query) pairs are to be covered by filter evaluations; which pairs are covered by a filter depends on the (binary) outcome of evaluating it on the stream element. The resulting instances satisfy the assumption that every element of U can be covered in every realization. They study, among other algorithms, the adaptive greedy algorithm specialized to this setting, and show that if the subsets are sampled independently for each item, so that Liu et al. report that it empirically outperforms a number of other algorithms in their experiments. The adaptive submodularity framework allows us to prove approximate results for richer item distributions over subsets of U than considered by Liu et al. (2008) We now provide the proof of Theorem 17. Proof of Theorem 17: We will ultimately prove Theorem 17 by applying the bound from Theorem 13 for Stochastic Submodular Cover instances. The proof mostly consists of justifying this application. Without loss of generality we may assume f is truncated at Q, otherwise we may use ĝ(S) = min Q, f (S) in lieu of f . This removes the need to truncate f . Since we established the adaptive submodularity of f in the proof of Theorem 16, and by assumption f (E, φ) = Q for all φ, to apply Theorem 13 we need only show that f is strongly adaptive monotone and strongly adaptive submodular and that the instances under consideration are self-certifying. We begin by showing the strong adaptive monotonicity of f . Fix a partial realization ψ, an item e / ∈ dom(ψ) and a state o. Let ψ = ψ ∪ {(e, o)}. Then treating ψ and ψ as subsets of E × O, and using the monotonicity of f , we obtain which is equivalent to the strong adaptive monotonicity condition. Next we show the strong adaptive adaptive submodularity of f by showing it is pointwise submodular (having already proven adaptive submodularity for it). This is clearly true, since for all φ, S → f ({(e, φ(e)) : e ∈ S}) is monotone submodular by assumption. Finally we prove that these instances are self-certifying. Consider any ψ and φ, φ consistent with ψ. Then We have shown that f and p (φ) satisfy the assumptions of Theorem 13 on this self-certifying instance. Hence we may apply it to obtain the claimed approximation guarantee. the people influenced and the observations obtained after one person is selected. For our next application, consider the following scenario. Suppose we would like to generate demand for a genuinely novel product. Potential customers do not realize how valuable the new product will be to them, and conventional advertisements are failing to convince them to try it. In this case, we may try to spur demand by offering a special promotional deal to a select few people, and hope that demand builds virally, propagating through the social network as people recommend the product to their friends and associates. Supposing we know something about the structure of the social networks people inhabit, and how ideas, innovation, and new product adoption diffuse through them, this begs the question: to which initial set of people should we offer the promotional deal, in order to spur maximum demand for our product? This, broadly, is the viral marketing problem. The same problem arises in the context of spreading technological, cultural, and intellectual innovations, broadly construed. In the interest of unified terminology we follow Kempe et al. (2003) and talk of spreading influence through the social network, where we say people are active if they have adopted the idea or innovation in question, and inactive otherwise, and that a influences b if a convinces b to adopt the idea or innovation in question. There are many ways to model the diffusion dynamics governing the spread of influence in a social network. We consider a basic and well-studied model, the independent cascade model, described in detail below. For this model Kempe et al. (2003) obtain a very interesting result; they show that the eventual spread of the influence f (i.e., the ultimate number of customers that demand the product) is a monotone submodular function of the seed set S of people initially selected. This, in conjunction with the results of Nemhauser et al. (1978) implies that the greedy algorithm obtains at least 1 -1 e of the value of the best feasible seed set of size at most k, i.e., arg max S:|S|≤k f (S), where we interpret k as the budget for the promotional campaign. Though Kempe et al. consider only the maximum coverage version of the viral marketing problem, their result in conjunction with that of Wolsey (1982) also implies that the greedy algorithm will obtain a quota Q of value at a cost of at most ln(Q) + 1 times the cost of the optimal set arg min S {c(S) : f (S) ≥ Q} if f takes on only integral values. The viral marketing problem has a very natural adaptive analog. Instead of selecting a fixed set of people in advance, we may select a person to offer the promotion to, make some observations about the resulting spread of demand for our product, and repeat. See Fig. 3 for an illustration. In §8.2, we use the idea of adaptive submodularity to obtain results analogous to those of Kempe et al. (2003) in the adaptive setting. Specifically, we show that the greedy policy obtains at least 1 -1 e of the value of the best policy. Moreover, we extend this result by achieving that guarantee not only for the case where our reward is simply the number of influenced people, but also for any (nonnegative) monotone submodular function of the set of people influenced. In §8.3 we consider the minimum cost cover objective, and show that the greedy policy obtains a squared logarithmic approximation for it. To our knowledge, no approximation results for this adaptive variant of the viral marketing problem have been known. In this model, the social network is a directed graph G = (V, A) where each vertex in V is a person, and each edge (u, v) ∈ A has an associated binary random variable X uv indicating if u will influence v. That is, X uv = 1 if u will influence v once it has been influenced, and X uv = 0 otherwise. The random variables X uv are independent, and have known means p uv := E [X uv ]. We will call an edge (u, v) with X uv = 1 a live edge and an edge with X uv = 0 a dead edge. When a node u is activated, the edges X uv to each neighbor v of u are sampled, and v is activated if (u, v) is live. Influence can then spread from u's neighbors to their neighbors, and so on, according to the same process. Once active, nodes remain active throughout the process, however Kempe et al. (2003) show that this assumption is without loss of generality, and can be removed. In the Adaptive Viral Marketing problem under the independent cascades model, the items correspond to people we can activate by offering them the promotional deal. How we define the states φ(u) depends on what information we obtain as a result of activating u. Given the nature of the diffusion process, activating u can have wide-ranging effects, so the state φ(u) has more to do with the state of the social network on the whole than with u in particular. Specifically, we model φ(u) as a function φ u : A → {0, 1, ?}, where φ u ((u, v)) = 0 means that activating u has revealed that (u, v) is dead, φ u ((u, v)) = 1 means that activating u has revealed that (u, v) is live, and φ u ((u, v)) = ? means that activating u has not revealed the status of (u, v) (i.e., the value of X uv ). We require each realization to be consistent and complete. Consistency means that no edge should be declared both live and dead by any two states. That is, for all u, v ∈ V and a ∈ A, (φ u (a), φ v (a)) / ∈ {(0, 1), (1, 0)}. Completeness means that the status of each edge is revealed by some activation. That is, for all a ∈ A there exists u ∈ V such that φ u (a) ∈ {0, 1}. A consistent and complete realization thus encodes X uv for each edge (u, v). Let A(φ) denote the live edges as encoded by φ. There are several candidates for which edge sets we are allowed to observe when activating a node u. Here we consider what we call the Full-Adoption Feedback Model: After activating u we get to see the status (live or dead) of all edges exiting v, for all nodes v reachable from u via live edges (i.e., reachable from u in (V, A(φ)), where φ is the true realization. We illustrate the full-adoption feedback model in Fig. 3. In the simplest case, the reward for influencing a set U ⊆ V of nodes is f (U ) := |U |. Kempe et al. (2003) obtain an 1 -1 e -approximation for the slightly more general case in which each node u has a weight w u indicating its importance, and the reward is f (U ) := u∈U w u . We generalize this result further, to include arbitrary nonnegative monotone submodular reward functions f . This allows us, for example, to encode a value associated with the diversity of the set of nodes influenced, such as the notion that it is better to achieve 20% market penetration in five different (equally important) demographic segments than 100% market penetration in one and 0% in the others. We are now ready to formally state our result for the maximum coverage objective. Theorem 19 The greedy policy π greedy obtains at least 1 -1 e of the value of the best policy for the Adaptive Viral Marketing problem with arbitrary monotone submodular reward functions, in the independent cascade and full-adoption feedback models discussed above. That is, if σ(S, φ) is the set of all activated nodes when S is the seed set of activated nodes and φ is the realization, f : 2 V → R ≥0 is an arbitrary monotone submodular function indicating the reward for influencing a set, and the objective function is f (S, φ) := f (σ(S, φ)), then for all policies π and all k ∈ N we have More generally, if π is an α-approximate greedy policy then Proof Adaptive monotonicity follows immediately from the fact that f (•, φ) is monotonic for each φ. It thus suffices to prove that f is adaptive submodular with respect to the probability distribution on realizations p (φ), because then we can invoke Theorem 5 to complete the proof. We will say we have observed an edge (u, v) if we know its status, i.e., if it is live or dead. Fix any ψ, ψ such that ψ ⊆ ψ and any v / ∈ dom(ψ ). We must show To prove this rigorously, we define a coupled distribution µ over pairs of realizations φ ∼ ψ and φ ∼ ψ . Note that given the feedback model, the realization φ is a function of the random variables {X uw : (u, w) ∈ A} indicating the status of each edge. For conciseness we use the notation X = {X uw : (u, w) ∈ A}. We define µ implicitly in terms of a joint distribution μ on X × X , where φ = φ(X) and φ = φ (X ) are the realizations induced by the two distinct sets of random edge statuses, respectively. Hence µ(φ(X), φ(X )) = μ(X, X ). Next, let us say a partial realization ψ observes an edge e if some w ∈ dom(ψ) has revealed its status as being live or dead. For edges (u, w) observed by ψ, the random variable X uw is deterministically set to the status observed by ψ. Similarly, for edges (u, w) observed by ψ , the random variable X uw is deterministically set to the status observed by ψ . Note that since ψ ⊆ ψ , the state of all edges which are observed by ψ are the same in φ and φ . All (X, X ) ∈ support(μ) have these properties. Additionally, we will construct μ so that the status of all edges which are unobserved by both ψ and ψ are the same in X and X , meaning X uw = X uw for all such edges (u, w), or else μ(X, X ) = 0. The above constraints leave us with the following degrees of freedom: we may select X uw for all (u, w) ∈ A which are unobserved by ψ. We select them independently, such that E [X uw ] = p uw as with the prior p (φ). Hence for all (X, X ) satisfying the above constraints, μ(X, X ) = (u,w) unobserved by ψ p Xuw uw (1 -p uw ) 1-Xuw , and otherwise μ(X, X ) = 0. Note that p (φ | ψ) = φ µ(φ, φ ) and p (φ | ψ ) = φ µ(φ, φ ). We next claim that for all (φ, φ ) ∈ support(µ) Recall f (S, φ) := f (σ(S, φ)), where σ(S, φ) is the set of all activated nodes when S is the seed set of activated nodes and φ is the realization. Let B = σ(dom(ψ), φ) and C = σ(dom(ψ) ∪ {v} , φ) denote the active nodes before and after selecting v after dom(ψ) under realizations φ, and similarly define B and C with respect to ψ and φ . Let D := C \B, D := C \B . Then Eq. ( 19) is equivalent to By the submodularity of f , it suffices to show that B ⊆ B and D ⊆ D to prove the above inequality, which we will now do. We start by proving B ⊆ B . Fix w ∈ B. Then there exists a path from some u ∈ dom(ψ) to w in (V, A(φ)). Moreover, every edge in this path is not only live but also observed to be live, by definition of the feedback model. Since (φ, φ ) ∈ support(µ), this implies that every edge in this path is also live under φ , as edges observed by ψ must have the same status under both φ and φ . It follows that there is a path from u to w in (V, A(φ )). Since u is clearly also in dom(ψ ), we conclude w ∈ B , hence B ⊆ B . Next we show D ⊆ D. Fix some w ∈ D and suppose by way of contradiction that w / ∈ D. Hence there exists a path P from v to w in (V, A(φ )) but no such path exists in (V, A(φ)). The edges of P are all live under φ , and at least one must be dead under φ. Let (u, u ) be such an edge in P . Because the status of this edge differs in φ and φ , and (φ, φ ) ∈ support(µ), it must be that (u, u ) is observed by ψ but not observed by ψ. Because it is observed by ψ , in our feedback model it must be that u is active after dom(ψ ) is selected, i.e., u ∈ B . However, this implies that all nodes reachable from u via edges in P are also active after dom(ψ ) is selected, since all the edges in P are live. Hence all such nodes, including w, are in B . Since D and B are disjoint, this implies w / ∈ D , a contradiction. Having proved Eq. ( 19), we now proceed to use it to show which completes the proof. It is worth contrasting the Adaptive Viral Marketing problem with the Stochastic Submodular Maximization problem of §6. In the latter problem, we can think of the items as being random independently distributed sets. In Adaptive Viral Marketing by contrast, the random sets (of nodes influenced when a fixed node is selected) depend on the random status of the edges, and hence may be correlated through them. Nevertheless, we can obtain the same 1 -1 e approximation factor for both problems. A Comment on the Myopic Feedback Model. In the conference version of this article (Golovin and Krause, 2010), we considered an alternate feedback model called the myopic feedback model, in which after activating v we see the status of all edges exiting v in the social network, i.e., ∂ + (u We claimed that the objective f as defined previously is adaptive submodular in the independent cascade model with myopic feedback, and hence the greedy policy obtains a (1 -1 e ) approximation for it. We hereby retract this claim, and furthermore give a counterexample demonstrating that f is not adaptive submodular under myopic feedback. Consider a graph G = (V, E) with vertices V := {u, v, w}, and edges E := {(u, v), (v, w)}. The edge parameters are p uv = 1 and p vw = 1 -. Let f (U ) = |U | and construct f from f accordingly. We let ψ = {(u, φ u )}, where φ u ((u, v)) = 1 and φ u ((v, w)) = ?. Let ψ = {(u, φ u ), (v, φ v )} where φ v ((v, w)) = 0. Clearly, ψ ⊂ ψ . Note ∆(w | ψ) = , since the marginal benefit of w over dom(ψ) is one if (v, w) is dead, and zero if it is live, and the former occurs with probability . In contrast, ∆(w | ψ ) = 1, since ψ contains the observation that (v, w) is dead. Hence ∆(w | ψ) < ∆(w | ψ ), which violates adaptive submodularity. However, we conjecture that the greedy policy still obtains a constant factor approximation even in the myopic feedback model. We may also wish to adaptively run our campaign until a certain level of market penetration has been achieved, e.g., a certain number of people have adopted the product. We can formalize this goal using the minimum cost cover objective. For this objective, we have an instance of Adaptive Stochastic Minimum Cost Cover, in which we are given a quota Q ≤ f (V ) (quantifying the desired level of market penetration) and we must adaptively select nodes to activate until the set of all active nodes S satisfies f (S) ≥ Q. We obtain the following result. Theorem 20 Fix a monotone submodular function f : 2 V → R ≥0 indicating the reward for influencing a set, and a quota Q ≤ f (V ). Suppose the objective is f (S, φ) := min Q, f (σ(S, φ)) , where σ(S, φ) is the set of all activated nodes when S is the seed set of activated nodes and φ is the realization. Let η be any value such that f (S) > Q -η implies f (S) ≥ Q for all S. Then any α-approximate greedy policy π on average costs at most α ln Q η + 1 2 times the average cost of the best policy obtaining Q reward for the Adaptive Viral Marketing problem in the independent cascade model with full-adoption feedback as described above. That is, c avg (π) ≤ α ln Q η + 1 2 c avg (π * ) for any π * that covers every realization. Proof We prove Theorem 20 by recourse to Theorem 13. We have already established that f is adaptive submodular, in the proof of Theorem 19. It remains to show that f is strongly adaptive monotone and strongly adaptive submodular, that these instances are self-certifying, and that Q and η equal the corresponding terms in the statement of Theorem 13. We start with strong adaptive monotonicity. Fix ψ, e / ∈ dom(ψ), and o ∈ O. We must show Let V + (ψ) denote the active nodes after selecting dom(ψ) and observing ψ. By definition of the full adoption feedback model, V + (ψ) consists of precisely those nodes v for which there exists a path P uv from some u ∈ dom(ψ) to v via exclusively live edges. The edges whose status we observe consist of all edges exiting nodes in V + (ψ). It follows that every path from any u ∈ V + (ψ) to any v ∈ V \ V + (ψ) contains at least one edge which is observed by ψ to be dead. Hence, in every φ ∼ ψ, the set of nodes activated by selecting dom(ψ) . Note that once activated, nodes never become inactive. Hence, ) which implies Eq. ( 20) and strong adaptive monotonicity. Next we establish strong adaptive submodularity. Given that we have already established adaptive submodularity, it is sufficient to also prove pointwise submodularity. For a fixed realization φ we have a set of live edges {(u, v) : X uv = 1} which induce a set system in which u covers all nodes reachable from u via live edges. Let S u denote this set. It is straightforward to verify that a monotone submodular function f on nodes induces a monotone submodular function on sets of those nodes. That is, is submodular whenever f is. In particular, g is submodular if for every A and B we have g(A) + g(B) ≥ g(A ∩ B) + g(A ∪ B) however one can easily verify that this set of constraints is a subset of the corresponding submodularity constaints on f . Next we establish that these instances are self-certifying. Note that for every φ we have f (V, φ) = min Q, f (V ) = Q. From our earlier remarks, we know that f (dom(ψ), φ) = f (V + (ψ)) for every φ ∼ ψ. Hence for all ψ and φ, φ consistent with ψ, we have f (dom(ψ), φ) = f (dom(ψ), φ ) and so f (dom(ψ), φ) = Q if and only if f (dom(ψ), φ ) = Q, which proves that the instance is self-certifying. Finally we show that Q and η equal the corresponding terms in the statement of Theorem 13. As noted earlier, f (V, φ) = Q for all φ. We defined η as some value such that f (S) > Q -η implies f (S) ≥ Q for all S. Since range(f ) = min Q, f (S) : S ⊆ V , it follows that we cannot have f (S, φ) ∈ (Q -η, Q) for any S and φ, so that η satisfies the requirements of the corresponding term in Theorem 13. Hence we may apply Theorem 13 on this self-certifying instance with Q and η to obtain the claimed result. An important problem in AI is automated diagnosis. For example, suppose we have different hypotheses about the state of a patient, and can run medical tests to rule out inconsistent hypotheses. The goal is to adaptively choose tests to infer the state of the patient as quickly as possible. A similar problem arises in active learning. Obtaining labeled data to train a classifier is typically expensive, as it often involves asking an expert. In active learning (c.f. Cohn et al. (1996); McCallum and Nigam (1998)), the key idea is that some labels are more informative than others: labeling a few unlabeled examples can imply the labels of many other unlabeled examples, and thus the cost of obtaining the labels from an expert can be avoided. As is standard, we assume that we are given a set of hypotheses H, and a set of unlabeled data points X where each x ∈ X is independently drawn from some distribution D. Let L be the set of possible labels. Classical learning theory yields probably approximately correct (PAC) bounds, bounding the number n of examples drawn i.i.d. from D needed to output a hypothesis h that will have expected error at most ε with probability at least 1 -δ, for some fixed ε, δ > 0. That is, if h * is the target hypothesis (with zero error), and error(h The key question is how to request the labels for the pool to infer the remaining labels as quickly as possible. In the case of binary labels (or test outcomes) L = {-1, 1}, various authors have considered greedy policies which generalize binary search (Garey and Graham, 1974;Loveland, 1985;Arkin et al., 1993;Kosaraju et al., 1999;Dasgupta, 2004;Guillory and Bilmes, 2009;Nowak, 2009). The simplest of these, called generalized binary search (GBS) or the splitting algorithm, works as follows. Define the version space V to be the set of hypotheses consistent with the observed labels (here we assume that there is no label noise). In the worst-case setting, GBS selects a query x ∈ X that minimizes h∈V h(x) . In the Bayesian setting we assume we are given a prior p H over hypotheses; in this case GBS selects a query x ∈ X that minimizes h∈V p H (h) • h(x) . Intuitively these policies myopically attempt to shrink a measure of the version space (i.e., the cardinality or the probability mass) as quickly as possible. The former provides an O(log |H|)-approximation for the worst-case number of queries (Arkin et al., 1993), and the latter provides an O(log 1 min h p H (h) )-approximation for the expected number of queries (Kosaraju et al., 1999;Dasgupta, 2004) and a natural generalization of GBS obtains the same guarantees with a larger set of labels (Guillory and Bilmes, 2009). Kosaraju et al. also prove that running GBS on a modified prior p H (h) ∝ max p H (h), 1/|H| 2 log |H| is sufficient to obtain an O(log |H|)-approximation. Viewed from this perspective of the previous sections, shrinking the version space amounts to "covering" all false hypotheses with stochastic sets (i.e., queries), where query x covers all hypotheses that disagree with the target hypothesis h * at x. That is, x covers {h : h(x) = h * (x)}. As in §8, these sets may be correlated in complex ways determined by the set of possible hypotheses. As we will show, the reduction in version space mass is adaptive submodular, and this allows us to obtain a new analysis of GBS using adaptive submodularity, which has a weaker approximation guarantee than is optimal, but is arguably more amenable to extensions and generalizations than previous analyses. Theorem 21 In the Bayesian setting in which there is a prior p H on a finite set of hypotheses H, the generalized binary search algorithm makes OPT • ln 1 min h p H (h) + 1 2 queries in expectation to identify a hypothesis drawn from p H , where OPT is the minimum expected number of queries made by any policy. If min h p H (h) is sufficiently small, running the algorithm on a modified prior p H (h) ∝ max p H (h), 1/|H| 2 improves the approximation factor to O (ln |H|) 2 . We devote the better part of the remainder of this section to the proof of Theorem 21, which has several components. We first address the important special case of a uniform prior over hypotheses, i.e., p H (h) = 1/|H| for all h ∈ H, and then we reduce the case with a general prior to a uniform prior. We wish to appeal to Theorem 13, so we convert the problem into an Adaptive Stochastic Min Cost Cover problem. Define a realization φ h for each hypothesis h ∈ H. The ground set is E = X, and the outcomes are binary; we define O = {-1, 1} instead of using {0, 1} to be consistent with our earlier exposition. For all h ∈ H we set φ h ≡ h, meaning φ h (x) = h(x) for all x ∈ X. To define the objective function, we first need some notation. Given observed labels ψ ⊂ X × O, let V (ψ) denote the version space, i.e., the set of hypotheses for which h(x) = ψ(x) for all x ∈ dom(ψ). See Fig. 4 for an illustration of an active learning problem in the case of indicator hypotheses. For a set of hypotheses V , let p H (V ) := h∈V p H (h) denote their total prior probability. Finally, let ψ(S, h) = {(x, h(x)) : x ∈ S} be the function with domain S that agrees with h on S. We define the objective function by and use p (φ h ) = p H (h) = 1/|H| for all h. Let π * be an optimal policy for this Adaptive Stochastic Min Cost Cover instance. Note that there is an exact correspondence between policies for the original problem of finding the target hypothesis and our problem of covering the true realization; h * is identified as the target hypothesis if and only if the version space is reduced to {h * } which occurs if and only if φ h * is covered. Hence c avg (π * ) = OPT. Note that because we have assumed a uniform prior over hypotheses, we have f (X, φ h ) = 1 -1/|H| for all h. Furthermore, the instances are self-certifying. Lemma 22 The instances described above are self-certifying for arbitrary priors p H . Proof Intuitively, theses instances are self-certifying because to cover φ h * a policy must identify φ h * . More formally, these instances are self-certifying because for any φ h and ψ such that φ h ∼ ψ, we have that f (dom(ψ), φ h ) = f (X, φ h ) implies V (ψ) = {h}. This in turn means that φ h is the only realization consistent with ψ, which trivially implies that any realization φ ∼ ψ also has f (dom(ψ), φ ) = f (X, φ ); hence the instance is self-certifying. We next prove that the instances generated are adaptive submodular and strongly adaptive monotone under a uniform prior. Lemma 23 In the instances described above, f is strongly adaptive monotone and strongly adaptive submodular and with respect to a uniform prior p H . Proof Demonstrating strong adaptive monotonicity under a uniform prior amounts to proving that adding labels cannot grow the version space, which is clear in our model. That is, each query x eliminates some subset of hypotheses, and as more queries are performed, the subset of hypotheses eliminated by x cannot grow. Moving on to adaptive submodularity, consider the expected marginal contribution of x under two partial realizations ψ, ψ where ψ is a subrealization of ψ (i.e., ψ ⊂ ψ ), and x / ∈ dom(ψ ). Let ψ[x/o] be the partial realization with domain dom(ψ) ∪ {x} that agrees with ψ on its domain, and maps x to o. For each o ∈ O, let ). Since a hypothesis eliminated from the version space cannot later appear in the version space, we have a o ≥ b o for all o. Next, note the expected reduction in version space mass (and hence the expected marginal contribution) due to selecting x given partial realization ψ is The corresponding quantity for ψ has b o substituted for a o in Eq. ( 21 21). This is because ∂φ/∂z o ≥ 0 for each o implies that growing the version space in any manner cannot decrease the expected marginal benefit of query x, and hence shrinking it in any manner cannot increase the expected marginal benefit of x. It is indeed the case that ∂φ/∂z o ≥ 0 for each o. More specifically, it holds that which can be derived through elementary calculus. To further show strong adaptive submodularity, we prove pointwise submodularity of the objective. For any fixed realization, h, the objective S → f (S, φ) amounts to a weighted set cover problem on the incorrect hypotheses (plus a constant), and is thus submodular. Hence we can apply Theorem 13 to this self-certifying instance with maximum reward threshold Q = 1 -1/|H|, and minimum gap η = 1/|H|, to obtain an upper bound of OPT (ln (|H| -1) + 1) 2 on the number of queries made by the generalized binary search algorithm (which corresponds exactly to the greedy policy for Adaptive Stochastic Min Cost Cover) under the assumption of a uniform prior over H. Now consider general priors over H. We construct the Adaptive Stochastic Min Cost Cover instance as before, only we change the objective function to First note that the instances remain self-certifying. The proof of Lemma 22 goes through completely unchanged by the modification of f . We proceed to show adaptive submodularity and strong adaptive monotonicity. Lemma 24 The objective function f as described in Eq. ( 22) is strongly adaptive monotone and strongly adaptive submodular with respect to arbitrary priors p H . Proof The modified objective is still adaptive submodular, because (S, φ h ) → p H (h) is clearly so, and because adaptive submodularity is defined via linear inequalities it is preserved under taking nonnegative linear combinations. Note that f (X, φ h ) = 1 for all φ h . It is still strongly adaptive submodular, because it is still pointwise submodular. Showing f is strongly adaptive monotone requires slightly more work than before. Fix ψ, x / ∈ dom(ψ), and o ∈ O. We must show Plugging in the definition of f , the inequality we wish to prove may be simplified to where Φ is the random realization of the hypothesis, and p H (φ h ) = p H (h) for all h. Let V elim := V (ψ) -V (ψ[x/o]) be the set of hypotheses eliminated from the version space by the observation h(x) = o. Rewriting Eq. ( 24), we get Let LHS 25 denote the left hand side of Eq. ( 25). We prove Eq. ( 25) as follows. We conclude that f is adaptive submodular and strongly adaptive monotone. Hence we can apply Theorem 13 to this self-certifying instance with maximum reward threshold Q = 1, and minimum gap η = 1/ min h p H (h). As a result we obtain an upper bound of OPT (ln (1/ min h p H (h)) + 1) 2 on the number of queries made by generalized binary search for arbitrary priors, completing the proof of Theorem 21. To improve this to an O (log |H|) 2 -approximation in the event that min h p H (h) is extremely small using the observation of Kosaraju et al. (1999) The positive contributions must come from rare hypotheses. However, the total probability mass of these under p H is at most 1/|H|, and since π is progressive c(π, h) ≤ |H| for all h, hence the difference in costs is at most one. Let 2 be the approximation factor for generalized binary search when run on p H . Let π be the policy of generalized binary search, and let π * be an optimal policy under prior p H . Then Thus for a general prior a simple modification of GBS yields an O (log |H|) 2 -approximation. This result easily generalizes to handle the setting of multiple classes / test outcomes (i.e., |O| ≥ 2), and α-approximate greedy policies, where we lose a factor of α in the approximation factor. As we describe in the Appendix, we can generalize adaptive submodularity to incorporate costs on items, which allows us to extend this result to handle query costs as well. Recently, Gupta et al. (2010) showed how to simultaneously remove the dependence on both costs and probabilities from the approximation ratio. Specifically, within the context of studying an adaptive travelling salesman problem they investigated the Optimal Decision Tree problem, which is equivalent to the active learning problem we consider here. Using a clever, more complex algorithm than adaptive greedy, they achieve an O (log |H|)-approximation in the case of non-uniform costs and general priors. Theorem 21 and the extensions mentioned so far are in the noise free case, i.e., the result of query x and observes h * (x), where h * is the target hypothesis. Many practical problems may have noisy observations. Nowak (2009) considered the case in which the outcomes are binary, i.e., O = {-1, 1}, the same query may be asked multiple times, and for each instance of each query the noise is independent. In this case he gives performance guarantees for generalized binary search. While this setting may be appropriate if the noise is due to measurement error, in some applications the noise is persistent, i.e., if query x is asked several times, the observation is always the same. Recently, Golovin et al. (2010) and Bellala and Scott (2010) have used the adaptive submodularity framework to obtain the first algorithms with provable (squared-logarithmic) approximation guarantees for active learning with persistent noise. Greedy algorithms are often straightforward to develop and implement, which explains their popular use in practical applications, such as Bayesian experimental design and Active Learning, as discussed in §9 (also see the excellent introduction of Nowak ( 2009)) and Adaptive Stochastic Set Cover, e.g., for filter design in streaming databases as discussed in §7. Besides allowing us to prove approximation guarantees for such algorithms, adaptive submodularity provides the following immediate practical benefits: 1. The ability to use lazy evaluations to speed up its execution. 2. The ability to generate data-dependent bounds on the optimal value. In this section, we empirically evaluate their benefits within a sensor selection application, in a setting similar to the one described by Deshpande et al. (2004). In this application, we have deployed a network V of wireless sensors, e.g., to monitor temperature in a building or traffic in a road network. Since sensors are battery constrained, we must adaptively select k sensors, and then, given those sensor readings, predict, e.g., the temperature at all remaining locations. This prediction is possible since temperature measurements will typically be correlated across space. Here, we will consider the case where sensors can fail to report measurements due to hardware failures, environmental conditions or interference. More formally, we imagine every location v ∈ V is associated with a random variable X v describing the temperature at that location, and there is a joint probability distribution p (x V ) := P [X V = x V ] that models the correlation between temperature values. Here, X V = [X 1 , . . . , X n ] is the random vector over all temperature values. We follow Deshpande et al. (2004) and assume that the joint distribution of the sensors is multivariate Gaussian. A sensor v can make a noisy observation Y v = X v + ε v , where ε v is zero mean Gaussian noise with known variance σ 2 . If some measurements Y A = y A are obtained at a subset of locations, then the conditional distribution p allows predictions at the unobserved locations, e.g., by predicting E[X V | Y A = y A ] (which minimizes the mean squared error). Furthermore, this conditional distribution quantifies the uncertainty in the prediction: Intuitively, we would like to select sensors that minimize the predictive uncertainty. One way to quantify the predictive uncertainty is to use the remaining Shannon entropy We would like to adaptively select k sensors, to maximize the expected reduction in Shannon entropy (c.f. Sebastiani and Wynn (2000); Krause and Guestrin (2009a)). However, in practice, sensors are often unreliable, and might fail to report their measurements. We assume that after selecting a sensor, we find out whether it has failed or not before deciding which sensor to select next. We suppose that each sensor has an associated probability p fail (v) of failure, in which case no reading is reported, and that sensor failures are independent of each other and of the ambient temperature at v. Thus we have an instance of the Stochastic Maximization problem with E := V , O := {working, failed}, and For multivariate normal distributions, the entropy is given as where for sets A and B, Σ AB denotes the covariance (matrix) between random vectors X A and X B . Note that the predictive covariance does not depend on the actual observations y A , only on the set A of chosen locations. Thus, where as usual, As Krause and Guestrin (2005) show, the function is monotone submodular, whenever the observations Y V are conditionally independent given X V . This insight allows us to apply the result of §6 to show that the objective f defined in Eq. ( 26) is adaptive monotone submodular, using f (S) := g({v : (v, working) ∈ S}) for any S ⊆ E × O. Our first data set consists of temperature measurements from the network of 46 sensors deployed at Intel Research Berkeley, which were sampled at 30 second intervals for 5 consecutive days (starting Feb. 28 th , 2004). We define our objective function with respect to the empirical covariance estimated from the data. We also use data from traffic sensors deployed along the highway I-880 South in California. We use traffic speed data for all working days from 6 AM to 11 AM for one month, from 357 sensors. The goal is to predict the speed on all 357 road segments. We again estimate the empirical covariance matrix. For both data sets, we run the adaptive greedy algorithm, using both the naive implementation (Algorithm 1) and the accelerated version using lazy evaluations (Algorithm 2). We vary the probability of sensor failure, and evaluate the execution time and the number of evaluations of the function g (defined in Eq. ( 27)) each algorithm makes. Figures 5(a) and 5(b) plot execution time given a 50% sensor failure rate, on a computer with a 2.26 GHz dual core processor and 4 GB RAM. In these applications, function evaluations are the bottleneck in the computation, so the number of them serves as a machine-independent proxy for the running time. Figures 5(c) and 5(d) show the performance ratio in terms of this proxy. On the temperature data set, lazy evaluations speed up the computation by a factor of between roughly 3.5 and 7, depending on the failure probability. On the larger traffic data set, we obtain speedup factors between 30 and 38. We find that the benefit of the lazy evaluations increases with the problem size and with the failure probability. The dependence on problem size must ultimately be explained in terms of structural properties of the instances, which also benefit the nonadaptive accelerated greedy algorithm. The dependence on failure probability has a simpler explanation. Note that in these applications, if the accelerated greedy algorithm selects v, which then fails, then it does not need to make any additional function evaluations to select the next sensor. Contrast this with the naive greedy algorithm, which makes a function evaluation for each sensor that has not been selected so far. While adaptive submodularity allows us to prove worst-case performance guarantees for the adaptive greedy algorithm, in many practical applications it can be expected that these bounds are quite loose. For our sensor selection application, we use the data dependent bounds of Lemma 6 to compute an upper bound β avg on max π f avg (π [k] ) as described below, and compare it with the performance guarantee of Theorem 5. For the accelerated greedy algorithm, we use the upper bounds on the marginal benefits stored in the priority queue instead of recomputing the marginal benefits, and thus expect somewhat looser bounds. We find that for our application, the bounds are tighter than the worst case bounds. We also find that the "lazy" data dependent bounds are almost as tight as the "eager" bounds using the eagerly recomputed marginal benefits ∆(e | ψ) for the latest and greatest ψ, though the former have slightly higher variance. Figures 5(e) and 5(f) show the performance of the greedy algorithm as well as the three bounds on the optimal value. Two subtleties arise when using the data-dependent bounds to bound max π f avg (π [k] ). The first is that , whereas we would like to bound the difference between the optimal reward and the algorithm's current expected reward, conditioned on seeing However, in our applications f is strongly adaptive monotone, and strong adaptive monotonicity implies that for any π * we have Hence, if we let OPT(ψ) The second subtlety is that we obtain a sequence of bounds from Eq. ( 29). If we consider the (random) sequence of partial realizations observed by the adaptive greedy algorithm, ∅ = ψ 0 ⊂ ψ 1 ⊂ • • • ⊂ ψ k , we obtain k+1 bounds β 0 , . . . , β k , where Taking the expectation over Φ, note that for any π, and any i, Therefore for any 0 ≤ i ≤ k , β i is a random variable whose expectation is an upper bound on the optimal expected reward of any policy. At this point we may be tempted to use the minimum of these, i.e., β min := min i {β i } as our ultimate bound. However, a collection of random variables X 0 , . . . , X k with E [X i ] ≥ τ for all i does not, in general, satisfy min i {X i } ≥ τ . While it is possible in our case, with its independent sensor failures, to use concentration inequalities to bound min i {β i } -min i {E [β i ]} with high probability, and thus add an appropriate term to obtain a true upper bound from β min , we take a different approach; we simply use the average bound β avg := 1 k+1 k i=0 β i . Of course, depending on the application, a particular bound β i (chosen independently of the sequence ψ 0 , ψ 1 , . . . , ψ k ) may be superior. For example, if g is modular, then β 0 is best, whereas if g exhibits strong diminishing returns, then bounds β i with larger values of i may be significantly tighter. An important question in adaptive optimization is how much better adaptive policies can perform when compared to non-adaptive policies. This is quantified by the adaptivity gap, which is the worst-case ratio, over problem instances, of the performance of the optimal adaptive policy to the optimal non-adaptive solution. Asadpour et al. (2008) show that in the Stochastic Submodular Maximization problem with independent failures (as considered in §6), the expected value of the optimal non-adaptive policy is at most a constant factor 1 -1/e worse than the expected value of the optimal adaptive policy. While we currently do not have lower bounds for the adaptivity gap of the general Adaptive Stochastic Maximization problem (1), we can show that even in the case of adaptive submodular functions, the min-cost cover and min-sum cover versions have large adaptivity gaps, and thus there is a large benefit of using adaptive algorithms. In these cases, the adaptivity gap is defined as the worst-case ratio of the expected cost of the optimal non-adaptive policy divided by the expected cost of the optimal adaptive policy. For the Adaptive Stochastic Minimum Cost Coverage problem (2), Goemans and Vondrák (2006) show the special case of Stochastic Set Coverage without multiplicities has an adaptivity gap of Ω(|E|). Below we exhibit an adaptive stochastic optimization instance with adaptivity gap of Ω(|E|/ log |E|) for the Adaptive Stochastic Min-Sum Cover problem (3), which also happens to have the same adaptivity gap for Adaptive Stochastic Minimum Cost Coverage. Theorem 25 Even for adaptive submodular functions, the adaptivity gap of Adaptive Stochastic Min-Sum Cover is Ω(n/ log n), where n = |E|. Proof Suppose E = {1, . . . , n}. Consider the active learning problem where our hypotheses h : E → {-1, 1} are threshold functions, i.e., h(e) = 1 if e ≥ and h(e) = -1 if e < for some threshold . There is a uniform distribution over thresholds ∈ {1, . . . , n + 1}. In order to identify the correct hypothesis with threshold , our policy must observe at least one of -1 or (and both of them if 1 < ≤ n). Let π be an optimal non-adaptive policy for this problem. Note that π can be represented as a permutation of E, because observing an element multiple times can only increase the cost while providing no benefit over observing it once, and each element must eventually be selected to guarantee coverage. For the min-sum cover objective, consider playing π for n/4 time steps. Then P [ observed in n/4 steps] = n/4(n + 1). Likewise P [ -1 observed in n/4 steps] = n/4(n + 1). Since at least one of these events must occur to identify the correct hypothesis, by a union bound Thus a lower bound on the expected cost of π is n/8, since for n/4 time steps a cost of at least 1/2 is incurred. Thus, for both the min-cost and min-sum cover objectives the cost of the optimal non-adaptive policy is Ω(n). As an example adaptive policy, we can implement a natural binary search strategy, which is guaranteed to identify the correct hypothesis after O(log n) steps, thus incurring cost O(log n), proving an adaptivity gap of Ω(n/ log n). In this paper, we have developed the notion of adaptive submodularity, which characterizes when certain adaptive stochastic optimization problems are well-behaved in the sense that a simple greedy policy obtains a constant factor or polylogarithmic factor approximation to the best policy. In contrast, we can also show that without adaptive submodularity, the adaptive stochastic optimization problems (1), (2), and (3) are extremely inapproximable, even with (pointwise) modular objective functions (i.e., those where for each φ, f : 2 E × O E → R is modular/linear in the first argument): We cannot hope to achieve an O(|E| 1-ε ) approximation ratio for these problems, unless the polynomial hierarchy collapses down to Σ P 2 . Theorem 26 For all (possibly non-constant) β ≥ 1, no polynomial time algorithm for Adaptive Stochastic Maximization with a budget of βk items can approximate the reward of an optimal policy with a budget of only k items to within a multiplicative factor of O(|E| 1-ε /β) for any ε > 0, unless PH = Σ P 2 . This holds even for pointwise modular f . We provide the proof of Theorem 26 in Appendix 15.7. Note that by setting β = 1, we obtain O(|E| 1-ε ) hardness for Adaptive Stochastic Maximization. It turns out that in the instance distribution we construct in the proof of Theorem 26 the optimal policy covers every realization (i.e., always finds the treasure) using a budget of k = O(|E| ε/2 ) items. Hence if PH = Σ P 2 then any randomized polynomial time algorithm wishing to cover this instance must have a budget β = Ω(|E| 1-ε ) times larger than the optimal policy, in order to ensure the ratio of rewards, which is Ω(|E| 1-ε /β), equals one. This yields the following corollary. Corollary 27 No polynomial time algorithm for Adaptive Stochastic Min Cost Coverage can approximate the cost of an optimal policy to within a multiplicative factor of O(|E| 1-ε ) for any ε > 0, unless PH = Σ P 2 . This holds even for pointwise modular f . Furthermore, since in the instance distribution we construct the optimal policy π * covers every realization using a budget of k, it has c Σ (π * ) ≤ k. Moreover, since we have shown that under our complexity theoretic assumptions, any polynomial time randomized policy π with budget βk achieves at most o(β/|E| 1-ε ) of the (unit) value obtained by the optimal policy with budget k, it follows that c Σ (π) = Ω(βk). Since we require β = Ω(|E| 1-ε ) to cover any set of realizations constituting, e.g., half of the probability mass, we obtain the following corollary. Corollary 28 No polynomial time algorithm for Adaptive Stochastic Min-Sum Cover can approximate the cost of an optimal policy to within a multiplicative factor of O(|E| 1-ε ) for any ε > 0, unless PH = Σ P 2 . This holds even for pointwise modular f . There is a large literature on adaptive optimization under partial observability which relates to adaptive submodularity, which can be broadly organized into several different categories. Here, we only review relevant related work that is not already discussed elsewhere in the manuscript. Many approaches consider stochastic generalizations of specific classic non-adaptive optimization problems, such as Set Cover (Goemans and Vondrák, 2006;Liu et al., 2008), Knapsack (Dean et al., 2008(Dean et al., , 2005) ) and Traveling Salesman (Gupta et al., 2010). In contrast, in this paper our goal is to introduce a general problem structure -adaptive submodularity -that unifies a number of adaptive optimization problems. This is similar to how the classic notion of submodularity unifies various optimization problems such as Set Cover, Facility Location, nonadaptive Bayesian Experimental Design, etc. Another active area of research in sequential optimization is the study of competitive online algorithms. A particularly relevant example is Online Set Cover Alon et al. (2009), where there is a known set system, an arbitrary sequence of elements is presented to the algorithm, and the algorithm must irrevocably select sets to purchase such that at all times the purchased sets cover all elements which have appeared so far. Alon et al. (2009) obtain a polylogarithmic approximation to this problem, via an online primal-dual framework which has been profitably applied to many other problems. Buchbinder and Naor (2009) provide a detailed treatment of this framework. Note that competitive analysis focuses on worst-case scenarios. In contrast, we assume probabilistic information about the world and optimize for the average case. Recent work by Guillory andBilmes (2010, 2011) considers a class of adaptive optimization problems over a family of monotone submodular objectives {f h : h ∈ H}. In their problem, one must cover a monotone submodular objective f h * which depends on the (initially unknown) target hypothesis h * ∈ H, by adaptively issuing queries and getting responses. Unlike traditional pool-based active learning, each query may generate a response from a set of valid responses depending on the target hypothesis. The reward is calculated by evaluating f h * on the set of (query, response) pairs observed, and the goal is to obtain some threshold Q of objective value at minimum total query cost, where queries may have nonuniform costs. In the noisy variant of the problem (Guillory and Bilmes, 2011), the set of (query, response) pairs observed need not be consistent with any hypothesis in H, and the goal is to obtain Q of value for all hypotheses that are "close" to being consistent with the observations. For both variants, Guillory and Bilmes consider the worst-case policy cost, and provide greedy algorithms optimizing clever hybrid objective functions. They prove an approximation guarantee of ln(Q|H|) + 1 for integer valued objective functions {f h } h∈H in the noise-free case, and similar logarithmic approximation guarantees for the noisy case. While similar in spirit to this work, there are several significant differences between the two. Guillory and Bilmes focus on worst-case policy cost, while we focus mainly on average-case policy cost. The structure of adaptive submodularity depends on the prior p (φ), whereas there is no such dependence in Interactive Submodular Set Cover. This dependence in turn allows us to obtain results, such as Theorem 13 for selfcertifying instances, whose approximation guarantee does not depend on the number of realizations in the way that the guarantees for Interactive Submodular Set Cover depend on |H|. As Guillory and Bilmes prove, the latter dependence is fundamental under reasonable complexity-theoretic assumptions 6 . An interesting open problem within the adaptive submodularity framework that is highlighted by the work on Interactive Submodular Set Cover is to identify useful instance-specific properties that are sufficient to improve upon the worst-case approximation guarantee of Theorem 14. The paper that is perhaps closest in spirit to this work is the one on Stochastic Depletion problems by Chan and Farias (2009), who also identify a general class of adaptive optimization problems than can be near-optimally solved using greedy algorithms (which in their setting give a factor 2 approximation). However, the similarity is mainly on a conceptual level: The problems and approaches, as well as example applications considered, are quite different. A class of adaptive optimization problems studied extensively in operations research since Dantzig (1955) is the area of stochastic optimization with recourse. Here, an optimization problem, such as Set Cover, Steiner Tree or Facility Location, is presented in multiple stages. At each stage, more information is revealed, but costs of actions increase. A key difference to the problems studied in this paper is that in these problems, information gets revealed independently of the actions taken by the algorithm. There are general efficient, sampling based (approximate) reductions of multi-stage optimization to the deterministic setting; see, e.g., Gupta et al. (2005). Adaptive Stochastic Optimization is also related to the problem of Bayesian Global Optimization (for a recent survey of the area, c.f. Brochu et al. (2009)). In Bayesian Global Optimization, the goal is to adaptively select inputs in order to maximize an unknown function that is expensive to evaluate (and can possibly only be evaluated using noisy observations). A common approach that has been successful in many applications (for a recent application in machine learning, c.f. Lizotte et al. ( 2007)), is to assume a prior distribution, such as a Gaussian process, over the unknown objective function. Several criteria for selecting inputs have been developed, such as the Expected Improvement (Jones et al., 1998) criterion. However, while recently performance guarantees where obtained in the no-regret setting (Grünewälder et al., 2010;Srinivas et al., 2010), we are not aware of any approximation guarantees for Bayesian Global Optimization. 6. They reduce to Set Cover and use the result of Feige (1998), which requires the assumption NP DTIME(n O(log log n) ), but it suffices to assume only P = NP using the Set Cover approximation hardness result of Raz and Safra (1997) instead. The problem of decision making under partial observability has also been extensively studied in stochastic optimal control. In particular, Partially Observable Markov Decision Processes (Smallwood and Sondik, 1973), abbreviated as POMDPs, are a general framework that captures many adaptive optimization problems under partial observability. Unfortunately, solving POMDPs is PSPACE hard (Papadimitriou and Tsitsiklis, 1987), thus typically heuristic algorithms with no approximation guarantees are applied (Pineau et al., 2006;Ross et al., 2008). For some special instances of POMDPs related to Multi-armed Bandit problems, (near-)optimal policies can be found. These include the (optimal) Gittins-index policy for the classic Multi-armed Bandit problem (Gittins and Jones, 1979) and approximate policies for the Multi-armed Bandit problem with metric switching costs (Guha and Munagala, 2009) and special cases of the Restless Bandit problem (Guha et al., 2009). The problems considered in this paper can be formalized as POMDPs, albeit with exponentially large state space (where the world state represents the selected items and state/outcome of each item). Thus our results can be interpreted as widening the class of partially observable planning problems that can be efficiently approximately solved. This manuscript is an extended version of a paper that appeared in the Conference on Learning Theory Golovin and Krause (2010). More recently, Golovin and Krause (2011b) proved performance guarantees for the greedy policy for the problem of maximizing the expected value of a policy under constraints more complex than simply selecting at most k items. These include matroid constraints, where a policy can only select independent sets of items and the greedy policy obtains a 1/2-approximation for adaptive monotone submodular objectives, and more generally p-independence system constraints, where the greedy policy obtains a 1/(p + 1)-approximation. Golovin et al. (2010) and, shortly thereafter, Bellala and Scott (2010), used the adaptive submodularity framework to obtain the first algorithms with provable (squared logarithmic) approximation guarantees for the difficult and fundamental problem of active learning with persistent noise. Finally, Golovin et al. (2011) used adaptive submodularity in the context of a dynamic conservation planning, and obtain competitiveness guarantees for an ecological reserve design problem. Planning under partial observability is a central but notoriously difficult problem in artificial intelligence. In this paper, we identified a novel, general class of adaptive optimization problems under uncertainty that are amenable to efficient, greedy (approximate) solution. In particular, we introduced the concept of adaptive submodularity, generalizing submodular set functions to adaptive policies. Our generalization is based on a natural adaptive analog of the diminishing returns property well understood for set functions. In the special case of deterministic distributions, adaptive submodularity reduces to the classical notion of submodular set functions. We proved that several guarantees carried by the non-adaptive greedy algorithm for submodular set functions generalize to a natural adaptive greedy algorithm in the case of adaptive submodular functions, for constrained maximization and certain natural coverage problems with both minimum cost and minimum sum objectives. We also showed how the adaptive greedy algorithm can be accelerated using lazy evaluations, and how one can compute data-dependent bounds on the optimal solution. We illustrated the usefulness of the concept by giving several examples of adaptive submodular objectives arising in diverse AI applications including sensor placement, viral marketing, automated diagnosis and pool-based active learning. Proving adaptive submodularity for these problems allowed us to recover existing results in these applications as special cases and lead to natural generalizations. Our experiments on real data indicate that adaptive submodularity can provide practical benefits, such as significant speed ups and tighter data-dependent bounds. We believe that our results provide an interesting step in the direction of exploiting structure to solve complex stochastic optimization and planning problems under partial observability. In this appendix we provide all of the proofs omitted from the main text. For the results of §5, we do so by first explaining how our results generalize to the case where items have costs, and then proving generalizations which incorporate item costs. In this section we provide the preliminaries required to define and analyze the versions of our problems with non-uniform item costs. We suppose each item e ∈ E has a cost c(e), and the cost of a set S ⊆ E is given by the modular function c(S) = e∈S c(e). We define the generalizations of problems ( 1), (2), and (3) in §15.3, §15.4, and §15.5, respectively. Our results are with respect to the greedy policy π greedy and α-approximate greedy policies. With costs, the greedy policy selects an item maximizing ∆(e | ψ) /c(e), where ψ is the current partial realization. Definition 29 (Approximate Greedy Policy with Costs) A policy π is an α-approximate greedy policy if for all ψ such that there exists e ∈ E with ∆(e | ψ) > 0, and π terminates upon observing any ψ such that ∆(e | ψ) ≤ 0 for all e ∈ E. That is, an α-approximate greedy policy always obtains at least (1/α) of the maximum possible ratio of conditional expected marginal benefit to cost, and terminates when no more benefit can be obtained in expectation. A greedy policy is any 1-approximate greedy policy. It will be convenient to imagine the policy executing over time, such that when a policy π selects an item e, it starts to run e, and finishes running e after c(e) units of time. We next generalize the definition of policy truncation. Actually we require three such generalizations, which are all equivalent in the unit cost case. ) in its domain independently with probability t -e∈dom(ψ) c(e) /c(π(ψ)). In the proofs that follow, we will need a notion of the conditional expected cost of a policy, as well as an alternate characterization of adaptive monotonicity, based on a notion of policy concatenation. We prove the equivalence of our two adaptive monotonicity conditions in Lemma 36. Definition 33 (Conditional Policy Cost) The conditional policy cost of π conditioned on ψ, denoted c (π | ψ), is the expected cost of the items π selects under p Definition 34 (Policy Concatenation) Given two policies π 1 and π 2 define π 1 @π 2 as the policy obtained by running π 1 to completion, and then running policy π 2 as if from a fresh start, ignoring the information gathered 7 during the running of π 1 . Definition 35 (Adaptive Monotonicity (Alternate Version)) A function f : 2 E × O E → R ≥0 is adaptive monotone with respect to distribution p (φ) if for all policies π and π , it holds that f avg (π) ≤ f avg (π @π), where f avg (π) := E [f (E(π, Φ), Φ)] is defined w.r.t. p (φ). Lemma 36 (Adaptive Monotonicity Equivalence) Fix a function f : 2 E × O E → R ≥0 . Then ∆(e | ψ) ≥ 0 for all ψ with P [Φ ∼ ψ] > 0 and all e ∈ E if and only if for all policies π and π , f avg (π) ≤ f avg (π @π). Proof Fix policies π and π . We begin by proving f avg (π @π) = f avg (π@π ). Fix any φ and note that E(π @π, φ) = E(π , φ) ∪ E(π, φ) = E(π@π , φ). Hence Therefore f avg (π) ≤ f avg (π @π) holds if and only if f avg (π) ≤ f avg (π@π ). We first prove the forward direction. Suppose ∆(e | ψ) ≥ 0 for all ψ and all e ∈ E. Note the expression f avg (π@π ) -f avg (π) can be written as a conical combination of (nonnegative) ∆(e | ψ) terms, i.e., for some α ≥ 0, f avg (π@π ) -f avg (π) = ψ,e α (ψ,e) ∆(e | ψ). Hence f avg (π@π ) -f avg (π) ≥ 0 and so f avg (π) ≤ f avg (π@π ) = f avg (π @π). We next prove the backward direction, in contrapositive form. Suppose ∆(e | ψ) < 0 for some ψ with P [Φ ∼ ψ] > 0 and e ∈ E. Let e 1 , . . . , e r be the items in dom(ψ) and define policies π and π as follows. For i = 1, 2, . . . , r, both π and π select e i and observe Φ(e i ). If either policy observes Φ(e i ) = ψ(e i ) it immediately terminates, otherwise it continues. If π succeeds in selecting all of dom(ψ) then it terminates. If π succeeds in selecting all of dom(ψ) then it selects e and then terminates. We claim f avg (π@π ) -f avg (π) < 0. Note that E(π@π , φ) = E(π, φ) unless φ ∼ ψ, and if φ ∼ ψ then E(π@π , φ) = E(π, φ) ∪ {e} and also E(π, φ) = dom(ψ). Hence The last term is negative, as P [Φ ∼ ψ] > 0 and ∆(e | ψ) < 0 by assumption. Therefore f avg (π) > f avg (π@π ) = f avg (π @π), which completes the proof. 7. Technically, if under any realization φ policy π 2 selects an item that π 1 previously selected, then π 1 @π 2 cannot be written as a function from a set of partial realizations to E, i.e., it is not a policy. This can be amended by allowing partial realizations to be multisets over elements of E × O, so that, e.g., if e is played twice then (e, ψ(e)) appears twice in ψ. However, in the interest of readability we will avoid this more cumbersome multiset formalism, and abuse notation slightly by calling π 1 @π 2 a policy. This issue arises whenever we run some policy and then run another from a fresh start. The adaptive data dependent bound has the following generalization with costs. Lemma 37 (The Adaptive Data Dependent Bound with Costs) Suppose we have made observations ψ after selecting dom(ψ). Let π * be any policy. Then for adaptive monotone submodular f : A simple greedy algorithm can be used to compute Z; we provide pseudocode for it in Algorithm 3. The correctness of this algorithm is more readily discerned upon rewriting the linear program using variables x e = c(e)w e to obtain Z = max With item costs, the Adaptive Stochastic Maximization problem becomes one of finding some where k is a budget on the cost of selected items, and we define f avg (π) for a randomized policy π to be f avg (π) := E [f (E(π, Φ), Φ)] as before, where the expectation is now over both Φ and the internal randomness of π which determines E(π, φ) for each φ. We prove the following generalization of Theorem 5. Theorem 38 Fix any α ≥ 1 and item costs c : E → N. If f is adaptive monotone and adaptive submodular with respect to the distribution p (φ), and π is an α-approximate greedy policy, then for all policies π * and positive integers and k f Proof The proof goes along the lines of the performance analysis of the greedy algorithm for maximizing a submodular function subject to a cardinality constraint of Nemhauser et al. (1978). An extension of that analysis to α-approximate greedy algorithms, which is analogous to ours but for the nonadaptive case, is shown by Goundan and Schulz (2007). For brevity, we will assume without loss of generality that π The first inequality is due to the adaptive monotonicity of f and Lemma 36, from which we may infer f avg (π 2 ) ≤ f avg (π 1 @π 2 ) for any π 1 and π 2 . The second inequality may be obtained as a corollary of Lemma 37 as follows. Fix any partial realization ψ of the form (e, φ(e)) : e ∈ E(π [i] , φ) for some φ. Consider ∆(π * | ψ), which equals the expected marginal benefit of the π * portion of π [i] @π * conditioned on Φ ∼ ψ. Lemma 37 allows us to bound it as where the expectations are taken over the internal randomness of π * , if there is any. Note that since π * has the form π A simple rearrangement of terms then yields the second inequality in (32). Now define ), from which we infer ∆ 0 , where for this last inequality we have used the fact that 1 -x < e -x for all x > 0. Thus In this section, we provide arbitrary item cost generalizations of Theorem 13 and Theorem 14. With item costs the Adaptive Stochastic Minimum Cost Cover problem becomes one of finding, for some quota on utility Q, Hereby, recall that π covers , where the expectation is over any internal randomness of π. We will consider only Problem (34) for the remainder. We also consider the worstcase variant of this problem, where we replace the expected cost c avg (π) objective with the worst-case cost c wc (π) := max φ c(E(π, φ)). The definition of coverage (Definition 7 in §5.2 on page 13) requires no modification to handle item costs. Note, however, that coverage is all-or-nothing in the sense that covering a realization φ with probability less than one does not count as covering it. A corollary of this is that only items whose runs have finished help with coverage, whereas currently running items do not. For a simple example, consider the case where E = {e}, c(e) = 2, f (A, φ) = |A|, and policy π that selects e and then terminates. Then π [1] is a randomized policy which is π with probability 1 2 , and is the empty policy with probability 1 2 , so Hence, even though half the time π [1] covers all realizations, it is counted as not covering any. We begin with a claim relating pointwise submodularity to strong adaptive submodularity. Lemma 39 If f is adaptive submodular with respect to p (φ) and f is pointwise submodular meaning S → f (S, φ) is submodular for all φ, then f is strongly adaptive submodular with respect to p (φ). Proof By assumption f is adaptive submodular with respect to p (φ), so it is sufficient to prove that Eq. ( 16) holds, i.e., ∆(e | ψ; ψ ) ≥ ∆(e | ψ ). Fix any ψ ⊆ ψ and e ∈ E. Let δ φ (e, S) := f (S ∪ {e} , φ) -f (S, φ). Further let U x be the sources of edges in A x and V x be the targets. That is, U x := {ψ : ∃ψ , (ψ, ψ ) ∈ A x } and V x := {ψ : ∃ψ, (ψ, ψ ) ∈ A x }. Finally, the successors S x (ψ) of some ψ ∈ U x are the sources ψ that are minimal supersets of ψ in U x when interpreting partial realizations as set of (item, observation) pairs. Formally, S x (ψ) := {ψ : ψ ∈ U x , ψ , ψ ⊂ ψ ⊂ ψ } Definition 44 (Concatenative Pseudopolicies) Given two policies π and π * and some nonnegative x ∈ R, we define the concatenative pseudopolicy π[x]@π * as the stochastic process that, on each realization φ, runs policy π until it is just about to achieve x expected reward (formally, until reaching ψ (φ, x)), and then runs π * as if from scratch. Hence on realization φ it will play dom(ψ (φ, x)) ∪ E(π * , φ) where ψ (φ, x) is defined relative to π. Note that π[x]@π * isn't a proper policy, because it decides when to switch to executing π * based on information about φ that it cannot infer from its current partial realization. Specifically, upon reaching ψ ∈ U x it will select π * (∅) if there exists ψ such that (ψ, ψ ) ∈ A x and φ ∼ ψ , and will select π(ψ) otherwise. Our overall strategy will be to bound the expected cost c avg (π) of π by bounding the price θ it pays per unit of expected reward gained as it runs (measured as E [f (dom(ψ), Φ) | Φ ∼ ψ] where ψ is the current partial realization) and then integrating over the run. To bound the price π pays, we define an alternative cost scheme for policies, bound the price π pays in terms of the alternative cost of an optimal policy π * avg , and finally bound the alternative cost of π * avg in terms of its actual cost. First, observe that This holds because f (ψ) < x by definition of A x , and f (E(π * avg , φ), φ) = Q for all φ since π * avg covers every realization. Since Q is the maximum possible reward, if ∆ π * avg | ψ; ψ < Q -x then we can generate a violation of strong adaptive monotonicity by fixing some φ ∼ ψ , selecting E(π * avg , φ ), and then selecting dom(ψ) to reduce the expected reward. The bound in Eq. ( 37) is useful when x is far from Q, however we will require a stronger bound for x very close to Q. We will use slightly different analyses for general instances and for self-certifying instances. We begin with general instances. For these, we will prove a stronger bound for obtaining the final δη reward, i.e., for x ∈ Fix ψ ∈ dom(π) and any φ ∼ ψ. We say ψ covers φ if π covers φ by the time it observes ψ. By definition of δ and η, if some φ ∼ ψ is not covered by ψ then Q -f (dom(ψ), ψ ) ≥ δη. Hence the last item that π selects, say upon observing ψ, must increase its conditional expected value from f (dom(ψ), ψ ) ≤ Q -δη to Q for all φ ∼ ψ . For self-certifying instances we show that We first argue that the last item that π selects must increase its conditional expected value from at most Q -η to Q. For suppose π currently observes ψ, and has not achieved conditional value Q, i.e., f (ψ) < Q. Then some φ ∼ ψ is uncovered. Since the instance is self-certifying, every φ with φ ∼ ψ is uncovered, and has f (dom(ψ), φ) < f (E, φ) = Q. By definition of η, for each φ with φ ∼ ψ we then have f (dom(ψ), φ) ≤ Q -η, which implies f (dom(ψ), ψ ) ≤ Q -η. Next we upper-bound the price π pays in terms of an alternative pricing scheme, ĉ, for π * avg . Note the cost of π * avg under ĉ may be higher than c avg (π * avg ). We will eventually bound the alternative cost in terms of the true cost. Fix x and any e that may be played by π * avg under some realization φ. Consider any (ψ, ψ ) ∈ A x with φ ∼ ψ . Fix some ψ e ∈ dom(π * avg ) such that e = π * avg (ψ e ). We charge e a price of c(e)/∆(e | ψ ∪ ψ e ) per-unit reward in this case. Hence, under realization φ, the alternative cost is Taking the expectation over φ such that φ ∼ ψ and φ ∼ ψ e (which is equivalent to φ ∼ ψ ∪ ψ e ) yields an expected alternative cost of For brevity, we will define ∆ a,e := ∆(e | ψ ∪ ψ e ; ψ ∪ ψ e ) for arc a = (ψ, ψ ) ∈ A x and ψ e . Note we could have multiple ψ e such that π * avg (ψ e ) = e, however for notational convenience we can imagine creating separate copies of e for each such partial realization and assume without loss of generality that for each item there is a unique partial realization that π * avg plays it under; That is, for each e there is at most one ψ e such that π * avg (ψ e ) = e. Next, for arc a = (ψ, ψ ) define w a,e to be the probability that π * avg plays e conditioned on ψ : Next, consider the cost ĉ π * avg | ψ; ψ which is the expected alternative cost paid by π * avg starting from dom(ψ) and conditioned on ψ for some fixed x and a = (ψ, ψ ) ∈ A x . Note that since the alternative cost depends on x and a, this alternative cost does as well. Formally, Next, note that V x := {ψ : ∃ψ, (ψ, ψ ) ∈ A x } partitions the set of realizations. We define the alternative cost of π and prove ĉ(e)/c(e) ≤ ln Q δη + 1 for general instances, and ĉ(e)/c(e) ≤ ln Q η + 1 for selfcertifying instances. To simplify the exposition we define some additional notation. Fix π, π * avg , x, and e. Let q[ψ] : be the conditional expected benefit of selecting e in the concatenative pseudopolicy π[x]@π * immediately upon seeing ψ (see definition 44). By construction, {ψ : ψ ∈ V x } partitions the set of all realizations (since for each φ there is a unique edge in the execution arborescence where the policy passes x expected reward in the trace τ (φ) as the policy runs). Furthermore, conditioned on ∃(ψ, ψ Next we need a subclaim about how ∆[ψ] relates to to ∆[ψ ] for ψ ⊂ ψ . Recall the definition of S x (ψ) (from definition 43). Lemma 45 Fix π, π * avg , x, and e and let p i , p i , ∆[ψ], and δ[ψ] be as above. For any Proof From Lemma 45, we have Next consider the random sequence of partial realizations in U x ∩τ (Φ), which we will denote as ψ 1 , ψ 2 , . . . , ψ k . We assume without loss of generality that no policy selects an item that will give it zero reward in expectation. Let Y i (Φ) be a random variable such that By adaptive submodularity, Y i ≥ 0 for all i, and also E [Y i | Φ ∼ ψ e ] = ε(ψ i ) for all i from Eq. ( 61). Next, note that because the maximum possible reward is Q and the minimum increment is η, and the minimum probability of any realization is δ, we have for all ψ ∆[ψ] ≤ Q, and For self-certifying instances, the argument is similar, except that we have ∆[ψ] > 0 ⇒ ∆[ψ] ≥ η because the the minimum increment is η, and the reward is determined by the current partial realization. Hence ∆[ψ k ] ≥ η, and so We show how to obtain the result for general instances. The result for self-certifying instances is strictly analogous. From Eq. ( 37) and Eq. ( 38), we have that for all (ψ, ψ ) ∈ A x ∆ π * avg | ψ; ψ ≥ max (Q -x, δη) . (68) From Eq. ( 46) and the fact that π is an α-approximate greedy policy by assumption we have that for all (ψ, ψ ) ∈ A ≤ αc avg (π * avg ) ln ≤ αc avg (π * avg ) ln This completes the proof for general instances (as mentioned, the result for self-certifying instances is strictly analogous, merely with different bounds used to relate ĉavg (π * avg ) to c avg (π * avg )), aside from proving Lemma 47. The elegant proof that we present here is due to Feldman and Vondrák (2012). Lemma 47 Suppose f : 2 E × O E → R ≥0 is adaptive submodular and strongly adaptive monotone with respect to p (φ) and there exists Q such that f (E, φ) = Q for all φ. Fix any policy π that covers every realization, and let θ(x) be defined as in the proof of Theorem 40. Then It is straightforward to show that E [C e (ψ, ψ ∪ {(e, Φ(e))}) | Φ ∼ ψ] = c(e), using the basic fact that if event A is the union of finitely many disjoint events B 1 , . . . , B m and X is any random variable, then Hence, in expectation this charging scheme charges the policy exactly c avg (π). Moreover, the rate at which it charges π upon observing ψ and selecting e = π(ψ) is exactly c(e)/∆(e | ψ) per unit expected gain. Let θ(ψ) := c(π(ψ))/∆(π(ψ) | ψ). Under this charging scheme the policy pays θ(ψ)dx to obtain an additional dx expected reward, given it already has x, the true realization is φ, and (ψ, ψ ) ∈ A x for some ψ with φ ∼ ψ . Let ψ (φ, x) be the partial realization in the execution trace of π under φ immediately before it obtains x reward in expectation. Formally ψ (φ, x) is the maximal element of dom(π) such that φ ∼ ψ and f (ψ) < x, where f (ψ) is defined as in Eq. ( 36). The total alternative cost of π under φ is then where the expectation is taken with respect to Φ. We can then obtain the claimed equality by exchanging the integral and the expectation operators: We can justify this operator exchange in several ways, for example we can apply Tonelli's theorem since the prices θ are always non-negative. Next we consider the worst-case cost. We generalize Theorem 14 by incorporating arbitrary item costs. Theorem 48 Suppose f : 2 E × O E → R ≥0 is adaptive monotone and adaptive submodular with respect to p (φ), and let η be any value such that f (S, φ) > f (E, φ) -η implies f (S, φ) = f (E, φ) for all S and φ. Let δ = min φ p (φ) be the minimum probability of any realization. Let π * wc be the optimal policy minimizing the worst-case cost c wc (•) while guaranteeing that every realization is covered. Let π be an α-approximate greedy policy with respect to the item costs. Finally, let Q := E [f (E, Φ)] be the maximum possible expected reward. Then We next claim that π [ →] has worst-case cost at most + αk. It is sufficient to show that the final item executed by π [ →] has cost at most αk for any realization. As we will prove, this follows from the facts that π is an α-approximate greedy policy and π * wc covers every realization at cost at most k. The data dependent bound, Lemma 37 on page 43, guarantees that Proving Eq. ( 83) is quite straightforward if f is strongly adaptive monotone. Given that f is only adaptive monotone, it requires some additional effort. So fix A ⊂ E and let π A be a non-adaptive policy that selects all Using Lemma 51, together with a geometric argument developed by Feige et al. (2004), we now prove Theorem 15. Proof of Theorem 15: Let Q := E [f (E, Φ)] be the maximum possible expected reward, where the expectation is taken w.r.t. p (φ). Let π be an α-approximate greedy policy. Define R i := Q -f avg π [i] and define P i := Q -f avg π [←i] . Let x i := Pi 2si , let y i := Ri 2 , and let h(x) ). We claim f avg π [←i] ≤ f avg π [i] and so P i ≥ R i . This clearly holds if π [←i] is the empty policy, and otherwise π can always select an item that contributes zero marginal benefit, namely an item it has already played previously. Hence an α-approximate greedy policy π can never select items with negative expected marginal benefit, and so f avg π [←i] ≤ f avg π [i] . By Lemma 51, f avg π * [xi] ≤ f avg π [←i] + x i s i . Therefore For similar reasons that f avg π [←i] ≤ f avg π [i] , we have f avg π [i-1] ≤ f avg π [i] , and so the sequence y 1 , y 2 , . . . is non-increasing. The adaptive monotonicity and adaptive submodularity of f imply that h(x) is non-increasing. Informally, this is because otherwise, if f avg (π * [x] ) > f avg (π * [x+1] ) for some x, then the optimal policy must be sacrificing immediate rewards at time x in exchange for greater returns later, and it can be shown that if such a strategy is optimal, then adaptive submodularity cannot hold. Eq. ( 91) and the monotonicity of h and i → y i imply that ∞ x=0 h(x)dx ≥ i≥0 x i (y i -y i+1 ) (see Figure 6). The left hand side is a lower bound for c Σ (π * ), and because s i = α (R i -R i+1 ) the right hand side simplifies to 1 4α We now provide the proof of Theorem 26 whose statement appears on page 33 in §12. Proof of Theorem 26: We construct a hard instance based on the following intuition. We make the algorithm go "treasure hunting". There is a set of t locations {0, 1, , . . . , t -1}, there is a treasure at one of these locations, and the algorithm gets unit reward if it finds it, and zero reward otherwise. There are m "maps," each consisting of a cluster of s bits, and each purporting to indicate where the treasure is, and each map is stored in a (weak) secret-sharing way, so that querying few bits of a map reveals nothing about where it says the treasure is. Moreover, all but one of the maps are fake, and there is a puzzle indicating which map is the correct one indicating the treasure's location. Formally, a fake map is one which is probabilistically independent of the location of the treasure, conditioned on the puzzle. The conditional distribution on realizations: p(φ | ψ) A policy, which maps partial realizations to items. The set of all items selected by π when run under realization φ. The The indicator for proposition P , which equals one if P is true and zero if P is false. , call a policy π progressive if it eliminates at least one hypothesis from its version space in each query. Let p H (h) = max p H (h), 1/|H| 2 /Z be the modified prior, where Z := h max p H (h ), 1/|H| 2 is the normalizing constant. Let c(π, h) be the cost (i.e., # of queries) of π under target h. Then c avg (π, p) := h c(π, h)p(h) is the expected cost of π under prior p. We will show that c avg (π, p H ) is a good approximation to c avg (π, p H ). Call h rare if p H (h) < 1/|H| 2 , and common otherwise. First, note that h max p H (h ), 1/|H| 2 ≤ 1 + 1/|H|, and so p H (h) ≥ |H| |H|+1 p H (h), for all h. In some situations, we may not have exact knowledge of the prior p (φ). Obtaining algorithms that are robust to incorrect priors remains an interesting source of open problems. We briefly discuss some robustness guarantees of our algorithm in §4 on page 9. For a more formal definition of u(π, t), see §15.5 on page 56. We provide a well-known example in active learning that illustrates this phenomenon crisply in §9; see Fig.4on page 24. We consider the general question of the magnitude of the potential benefits of adaptivity in §11 on page 31 .