TThesis Proposal: Algorithmic Social Intervention
Bryan WilderDepartment of Computer Science and Center for Artificial Intelligence in SocietyUniversity of Southern [email protected]
Abstract
Social and behavioral interventions are a critical tool for governments and communities totackle deep-rooted societal challenges such as homelessness, disease, and poverty. However,real-world interventions are almost always plagued by limited resources and limited data, whichcreates a computational challenge: how can we use algorithmic techniques to enhance the tar-geting and delivery of social and behavioral interventions? The goal of my thesis is to providea unified study of such questions, collectively considered under the name “algorithmic socialintervention”. This proposal introduces algorithmic social intervention as a distinct area withcharacteristic technical challenges, presents my published research in the context of these chal-lenges, and outlines open problems for future work. A common technical theme is decisionmaking under uncertainty: how can we find actions which will impact a social system in de-sirable ways under limitations of knowledge and resources? The primary application area formy work thus far is public health, e.g. HIV or tuberculosis prevention. For instance, I have de-veloped a series of algorithms which optimize social network interventions for HIV prevention.Two of these algorithms have been pilot-tested in collaboration with LA-area service providersfor homeless youth, with preliminary results showing substantial improvement over status-quoapproaches. My work also spans other topics in infectious disease prevention and underlyingalgorithmic questions in robust and risk-aware submodular optimization.
My research examines how techniques in artificial intelligence (including optimization, machinelearning, game theory, and social network analysis) can be used to intervene in social and be-havioral systems. Societies around the world face challenges of enormous scale: preventing andtreating disease, confronting poverty, and a range of other issues impacting billions of people. Inresponse, governments and communities deploy interventions addressing these problems (e.g., out-reach campaigns to enroll patients in treatment or educational programs to raise awareness aboutpreventative strategies). However, these interventions are always subject to limited resources andare deployed under considerable uncertainty about properties of the system; deciding manually onthe best way to deploy an intervention is extremely difficult.Motivated by such challenges, the goal of this thesis is to establish a set of algorithmic techniqueswhich confront underlying challenges in the delivery of social and behavioral interventions (acrossboth public health and other areas) and to field-test these techniques in socially impactful problemsettings. We refer to this domain as algorithmic social intervention . Social intervention domainsmotivate a range of common technical challenges (see Figure 1 and Section 2 for more details). Mypublished work spans all of these areas, though many interesting open problems remain. Specifically,I have studied information gathering [51, 52], optimization under uncertainty [55, 49, 50, 54, 53] andadaptive sequential decision making [55, 52]. Together with partners in social work and nonprofit1 a r X i v : . [ c s . A I] M a r igure 1: Technical components of algorithmic social intervention and related publications.agencies, I have empirically evaluated two of the resulting algorithms, with pilot tests showingsubstantial improvements over status-quo techniques [58, 52].Specifically, my research thus far has focused on algorithmic approaches to target and enhanceinterventions in public health settings. One line of work focuses on HIV prevention among homelessyouth, where information about HIV is spread through the youths’ social network. The challenge isselecting influential peer leaders who will be able to maximize the spread of the resulting diffusion. Ihave developed a set of algorithms for selecting peer leaders under uncertainty about the structure ofthe social network and how information propagates [56, 51, 52]. Two of these algorithms have beenpilot-tested with LA-area drop in centers serving homeless youth. These studies show substantialimprovement over the status-quo heuristic used to select peer leaders [52, 58]. Another area isinfectious disease prevention, where the challenge is to target limited intervention resources (e.g.,outreach campaigns to improve treatment uptake) to the population groups which will have thelargest impact on overall disease rates. I developed an algorithm to near-optimally target suchinterventions, with a particular focus on the problem of reducing tuberculosis spread in India [54].In simulation, this algorithm averts over 8,000 cases of tuberculosis per year compared to thestatus-quo policy. Underlying these applications are a number of fundamental technical challenges,related to decision making under uncertainty. Endemic to public health interventions is a lack ofinformation about the system: where the problems lie, how agents interact, and ultimately whatoutcome an intervention will have. Similar challenges arise in social and behavioral interventionsacross numerous contexts. Much of my work formalizes underlying challenges in decision makingunder uncertainty which are motivated by such applications, develops algorithmic solutions, andproves theoretical guarantees on their performance. The ultimate objective is algorithms that2ome with both rigorous theoretical analysis and field-tested practical performance. Towards thisend, I have developed algorithms for robust [49] and risk-averse [50] submodular optimization.Submodularity formalizes a natural diminishing returns property which occurs across many settings(including the HIV and tuberculosis prevention settings discussed above), making submodularoptimization under uncertainty an important and natural algorithmic challenge.The remainder of this proposal is organized as follows. Section 2 defines the area of algorith-mic social intervention in greater detail and expands on the technical challenges common in suchdomains. Sections 3, 4, and 5 survey completed research, divided by focus area: Section 3 coversinfluence maximization in the field, Section 4 covers submodular optimization under uncertainty,and Section 5 covers infectious disease prevention. Lastly, Section 6 discusses proposed future work. The goal for this thesis is to establish a unified study of algorithmic social intervention: compu-tational approaches for optimally targeting and enhancing social and behavioral interventions toachieve policy or community-level goals. The aim is to bridge algorithm design, optimization, andmachine learning with practice, field deployments, and social impact. Relevant domains are oftencharacterized by the following goals and challenges (though not all may be present in a singledomain): • Interventions are delivered in a preexisting social context composed of many agents with theirown goals and behaviors. The interactions of these agents collectively produce the system’sbehavior. • Agents’ behaviors are not totally determined by the intervention: particular incentives, ser-vices, or rules may be introduced, but then agents make their own decisions in response. • Agents are not perfectly rational, requiring the use of models and techniques from the socialand behavioral sciences to describe behavior. • There are many unknowns: the dynamics and interactions between agents are complex andare not fully specified by the available data. • Applications often focus on vulnerable or underserved populations.Figure 1 divides the underlying technical challenges of such domains into several stages. Eachstage also lists associated publications. The first stage is information gathering . Here, the challengeis to acquire the data needed to optimize the intervention in an efficient manner. For instance,in a social network intervention, it may be necessary to minimize the number of nodes who aresurveyed to obtain edges. The second stage is optimization under uncertainty . Since the availabledata is rarely enough to fully specify the objective function, methods such as robust, stochastic,or risk-aware optimization are necessary. The third stage is adaptive sequential decision making .Once an intervention is in progress, the algorithmic system has the opportunity to interact withthe world, observe the consequences of its decisions, and adjust accordingly. Lastly, a critical partof algorithmic social intervention is to evaluate field impact . While typical means of assessment(theoretical analysis, simulation experiments) are important tools, it is critical to validate thealgorithm in a field experiment, ideally in comparison to alternate heuristics and algorithms.3
Influence maximization in the field
Influence maximization is a crucial technique used in preventative health interventions, such as HIVprevention amongst homeless youth. Drop-in centers for homeless youth train a subset of youthas peer leaders who will disseminate information about HIV through their social networks. Thechallenge is to find a small set of peer leaders who will have the greatest possible influence. Whilemany previous algorithms have been proposed for influence maximization [23, 13, 21, 44], nonefully address the challenges of influence maximization in a field setting. Across public health (andother) settings, agencies will be uncertain about the structure of the social network and how infor-mation propagates. Accordingly, it is necessary to develop algorithms which gather only the mostparsimonious amount of information required to locate influential seeds and incorporate remainingunknowns into the optimization process. Moreoever, practical algorithms must also handle thecontingencies of real-world deployments; for instance, youth invited to attend an intervention maysimply fail to show up. This line of work develops a series of influence maximization algorithmswhich address such challenges.
The DOSIM algorithm, developed in [56] performs robust optimization under uncertainty abouthow influence spreads through the social network. We first formalize the influence maximizationproblem as follows. The youth have a social network represented as a graph G = ( V, E ). Each youthis initially inactive, meaning that they have not received information about HIV prevention. Oncenodes are activated by the intervention, they have a chance to influence their peers. We model thisprocess through a variant on the classical independent cascade model (ICM) which has been usedby previous work on HIV prevention and better reflects realistic time dynamics [57, 55, 58]. Theprocess unfolds over discrete time steps t = 1 ...T , where T is a time horizon. There is a propagationprobability p e for each edge e . When a node becomes active, it attempts to activate each of itsneighbors. Each attempt succeeds independently with probability p e . Activation attempts aremade at each time step until either the neighbor is influenced or the time horizon is reached. Theobjective is to select a set of K seed nodes at each time step t so that the expected total influencespread is maximized.The key challenge is that the propagation probabilities p e are not known. We model this as azero-sum game between the influencer, who selects the seed nodes, and an adversary (nature) whoselects the true p e . The goal of the influencer is to find a strategy which performs near-optimallyregardless of the unknown parameters, that is, their payoff in the game is the ratio of the expectedinfluence spread resulting from their chosen seed set to the optimal influence spread achievable ifthe true parameters were known in advance.The algorithmic challenge is to compute equilibria in this game. However, it is not apparenthow to do so, since both players have extremely large strategy spaces. In particular, nature has an infinite strategy space consisting of all (continuous) choices for the unknown parameters subject tointerval uncertainty, while the influencer can choose from all possible subsets of seed nodes. In orderto resolve this dilemma, two key technical approaches are used in [55]. First, to handle nature’sinfinite strategy space, it is proved that the strategy space may be discretized to a polynomialnumber of points with only arbitrarily small loss. Second, to handle the influencer’s exponentiallylarge number of actions, a double oracle approach is employed. Double oracle is an approachfor solving large zero-sum games which incrementally builds an equilibrium starting from a smallnumber of strategies. The algorithm proceeds over a series of iterations. At the first iteration, each4layer is restricted to a small number of pure strategies. We compute a minimax equilibrium inthis restricted game (e.g., via linear programming) and then find each player’s (approximate) bestresponse to current mixed strategy of their opponent. This best response is added to the player’scurrent strategy set, and the algorithm continues to iterate. Convergence to an equilibrium isguaranteed when the best response of each player is already contained in their current strategyset. In [55], we show in simulation that DOSIM results in substantially more robust solutions thatsimply planning based on a set of nominal parameters. In Section 3.2, we also give field resultsfrom [58] showing that DOSIM is empirically successful at finding high-quality seed sets in a realworld pilot study. Previous algorithms for influence maximization assume that the social network is given explicitlyas input. However, in many real-world domains, the network is not initially known and must begathered via laborious field observations. For example, collecting network data from vulnerablepopulations such as homeless youth, while crucial for health interventions, requires significant timespent gathering field observations [36]. Social media data is often unavailable when access totechnology is limited, for instance in developing countries or with vulnerable populations. Evenwhen such data is available, it often includes many weak links which are not effective at spreadinginfluence [6]. For instance, a person may have hundreds of Facebook friends whom they barely know.In principle, the entire network could be reconstructed via surveys, and then existing influencemaximization algorithms applied. However, exhaustive surveys are very labor-intensive and oftenconsidered impractical [45]. For influence maximization to be relevant to many real-world problems,it must contend with limited information about the network, not just limited computation .The major informational restriction is the number of nodes which may be surveyed to explorethe network. Thus, a key question is: how can we find influential nodes with a small number ofqueries?
We formalize this problem as exploratory influence maximization and seek a principledalgorithmic solution, i.e., an algorithm which makes a small number of queries and returns a setof seed nodes which are approximately as influential as the globally optimal seed set. Each querytargets a given node in the graph and reveals all of that node’s edges. At each step, the algorithmmay either query the neighbor of a previously queried node, or query a uniformly random node inthe graph. Existing field work uses heuristics, such as sampling some percentage of the nodes andasking them to nominate influencers [45]. To our knowledge, no previous work directly addressesthis question from an algorithmic perspective.We show that for general graphs, any algorithm for exploratory influence maximization may per-form arbitrarily badly unless it examines almost the entire network. However, real world networksoften have strong community structure, where nodes form tightly connected subgroups which areonly weakly connected to the rest of the network [30]. Consequently, influence mostly propagateslocally. Community structure has been used to develop computationally efficient influence max-imization algorithms [47, 14]. Here, we use it to design a highly information-efficient algorithm.We make four contributions.
First , we introduce exploratory influence maximization and showthat it is intractable for general graphs.
Second , we present the ARISEN algorithm, which exploitscommunity structure to find influential nodes.
Third , we show that ARISEN has strong empiricalperformance on an array of real world social networks.
Fourth , we formally analyze ARISEN ongraphs drawn from the Stochastic Block Model (SBM) [16], a widely studied model of communitystructure. We prove that it approximates the optimal influence if the entire network were knownby querying only a polylogarithmic number of nodes in the network size.We give the main idea behind the algorithm here and defer a formal description to the main5aper [51]. In a graph with community structure, a reasonable strategy is to try and select one seednode from each community (or the K largest communities if we have a budget of K seed nodes).The rationale is that we expect influence to propagate widely within a given community but onlyto a limited extent between communities. So, multiple seed nodes within a given communitywould be redundant compared to seeding another community entirely. The goal of the ARISENalgorithm is to use a small number of queries to choose seed nodes which are likely to lie in the K largest communities. The underlying approach is as follows. First, we sample a set of prospectiveseed nodes uniformly at random. Then, we use queries to simulate a random walk around eachprospective seed; it can be shown that this random walk will stay (with high probability) within thestarting community. The nodes encountered on this random walk are used to estimate the averagedegree of the community, which is in turn used to estimate the community’s size. Using theseestimates, ARISEN constructs a probability distribution over the prospective seeds and sampleseach actual seed independently at random from this distribution. The main challenge is that wecannot in general tell whether two prospective seed nodes lie in the same community, and so wemust construct a distribution which implicitly leads to seed nodes in different communities beingselected. The number of times a given community is sampled as a prospective seed is proportionalto that community’s size. Hence, ARISEN’s probability distribution assigns each prospective seednode weight inversely proportional to its community’s estimated size. This evens out the samplingbias towards large communities and ensures that (in expectation) each of the largest K communitiesis seeded exactly once.We analyze ARISEN theoretically on graphs which are drawn from the Stochastic Block Model(SBM), a common model of community structure. The SBM originated in sociology [16] and latelyhas been intensively studied in computer science and statistics (see e.g. [1, 27, 33]). In the SBM, thenetwork is partitioned into disjoint communities C ....C L . Each within-community edge is presentindependently with probability p w and each between-community edge is present independently withprobability p b . Recall that the Erd˝os-R´enyi random graph G ( n, p ) is the graph on n nodes whereevery edge is independently present with probability p . In the SBM, community C i is internallydrawn as G ( | C i | , p w ) with additional random edges to other communities. While the SBM is asimplified model, our experimental results show that ARISEN also performs well on real-worldgraphs. ARISEN takes as input the parameters n, p w , and p b , but is not given any prior informationabout the realized draw of the network. It is reasonable to assume that the model parameters areknown since they can be estimated using existing network data from a similar population (in ourexperiments, we show that this approach works well). For instance in HIV prevention, homelessyouth social networks have been shown to exhibit community structure and several studies havegathered networks from which to infer p w and p b [57, 36].We state here a simplified version of our main theoretical result which captures the intuition.Suppose that the top K communities each have equal size µ , and occupy a linear portion of thenetwork – for concreteness, µK ≥ . n . We have Theorem 1 (Simplified case) . Under the above conditions, ARISEN can be implemented withapproximation ratio (1 − e . − . − K − o (1)) β ( µ ) using O (log n ) queries. Here, β ( µ ) is a constant which depends on p w and q . We a defer a detailed explanation to thepaper [51] and just note here that β ( µ ) is the fraction of nodes contained in the giant connectedcomponent of an Erd˝os-R´enyi random graph G ( µK, p w · q ). The query cost is chosen so that therandom walk based estimates of each community’s size are accurate with high probability. We em-phasize that only a polylogarithmic number of nodes need be queried, an exponential improvementover exhaustive surveys. The first term in the approximation ratio is nearly 1 − /e , up to errorterms which decrease as n and K become large. We show that each of the top K communities is6eeded with probability close to 1 − /e . The second term, β ( µ ), is the fraction of each of the top K communities which can be influenced by a seed node.In the full paper, simulation results bear out the main conclusion, that ARISEN is able tofind influential seed nodes with a small number of queries. Experimentally, ARISEN outperformsa range of heuristics and is often able to closely approximate the optimal influence spread whilequerying 15-20% of the network. Thus far, algorithms for influence maximization in the field have a high barrier to entry: theyrequire a great deal of time to gather the complete social network of the youth, expertise to selectappropriate parameters, and computational power to run the algorithms. None of these are likelyavailable to resource-strained service providers who will ultimately be the ones to deploy influencemaximization. This paper [52] presents CHANGE, a novel system for influence maximization toameliorate these difficulties. CHANGE draws on the insights used to develop DOSIM and ARISEN,but is tailored to the constraints of a field deployment in public health settings. Specifically,CHANGE is designed to avoid DOSIM’s high computational cost and circumvent some practicaldifficulties in deploying ARISEN. Specifically, it is difficult to use ARISEN’s random walk basedprocedure with homeless youth, because it is often infeasible to locate a sequence of youth at theagency (youth may not be at the agency that day or be otherwise unreachable). Hence, CHANGEshould be thought of as a streamlined, field-ready system which draws on a series of insights intoinfluence maximization among homeless youth but which lacks theoretical guarantees for somecomponents of the system.CHANGE is easy to deploy, but this simplicity is crucially enabled by a series of insights intothe social structure of homeless youth (which may be useful for other vulnerable populations). Weconducted a pilot test of CHANGE’s performance in a real deployment by a drop-in center servinghomeless youth in a major U.S. city. CHANGE was used to plan a series of interventions designedto spread HIV awareness among the youth.
CHANGE obtained comparable influence spread tostate of the art algorithms while surveying only 18% of nodes for network data , a finding which isbacked by additional simulation results.Overall, CHANGE offers a practical, field-tested vehicle for deployed influence maximizationwhich drastically lowers the barrier to entry.
To our knowledge, this is the first real-world pilotstudy of a network sampling algorithm for influence maximization and only the second ever fieldtest of any influence maximization algorithm.
Overview of algorithmic contributions:
We now summarize how CHANGE handles thechallenges above. A diagram of the agent can be found in Figure 2.First, to address the data gathering challenge, we present an easily deployable sampling protocolwhich randomly selects a small set of youth to interview. For each of these youth, a randomly chosenneighbor is also interviewed. We show that this procedure gathers enough of the network to enablehigh-quality influence maximization even though it surveys only a small number of nodes directly.Second, to address computational power challenge (which in turn stems from unknown pa-rameters), we present a heuristic for selecting influence maximization solutions which are robustto uncertainty in the probability p that influence will spread. We show that this heuristic findssolutions which obtain approximately 90% of the maximum possible influence spread under any value for p . Importantly, this heuristic runs in minutes on a laptop, while DOSIM (the previouslyproposed algorithm for this problem) requires hours or even days of time on a high performancecomputing cluster. 7 ctions CHANGE agent
Edges from sampled nodesPeer leaders present/absent Sample node and random neighborSelect peer leaders
Network sampling
Robust parameter choice
Peer leader selection
Observations
Figure 2: The CHANGE agent.Third, we integrate these components with an adaptive greedy algorithm for planning interven-tions and prove the first theoretical guarantee for influence maximization under execution errors.The challenge is that some youth selected as peer leaders may not attend the intervention [55, 58].Our algorithm selects its action with such uncertainties in mind, observes which youth do attend,and then plans the next round using this observation. We prove that it obtains a constant-factorapproximation to the optimal adaptive policy.A detailed presentation of the CHANGE agent can be found in the full paper [52]. However,the next section presents field results for both DOSIM and CHANGE from pilot studies carriedout in collaboration with LA-area drop in centers serving homeless youth.
In the pilot tests, trained social workers delivered the
Have You Heard intervention, previouslypublished in the public health literature [36]. The social workers conducted a day-long class withthe selected youth, covering HIV awareness and prevention, and training the youth as peer leadersto communicate with other youth at the agency. Four pilot tests have been conducted so far, eachusing a different algorithm to select the peer leaders. Each pilot test used a distinct populationwith its own social network. The four algorithms were DOSIM and CHANGE, introduced above,along with HEALER [57] (a previously developed algorithm for the probem) and degree centrality(DC). DC is the status-quo heuristic used by agencies, and simply picks the highest degree nodesto seed.CHANGE first queried a subset of 18% of nodes for network data, while DOSIM, HEALER,and DC received the full network in advance. Three sets of peer leaders were selected by eachalgorithm, with approximately 4 peer leaders in each set. Peer leaders were paid $60. One monthafter the start of the study, we conducted a follow up survey with all of the youth who initiallyenrolled. We asked the youth whether they had received information about HIV prevention from apeer who was part of the study. Youth were paid $20 to respond to the follow up survey. The fieldresults are reported across two separate papers ([58] for HEALER, DOSIM, and DC, and [52] forCHANGE), but both studies used identical protocols. 60-70 participants were recruited for eachstudy.Figure 3 presents the main result: the amount of influence spread generated by each algorithm.Specifically, we used the follow up survey to examine the percentage of youth who were not peerleaders who reported that they received information about HIV prevention. We see that the8
HANGE HEALER DOSIM DC020406080100 % n o n - P L r e a c h e d
80 70 71 27
Figure 3: Percentage of non-peer leaders who reported receiving information about HIV in the pilotstudy corresponding to each algorithm.AI-based algorithms (CHANGE, HEALER, DOSIM) do fairly well, reaching 70-80% of non-peerleaders. However, DC performs poorly, reaching only 27% of non-peer leaders. While these resultsare preliminary, they show that there is promise in using algorithmic techniques to enhance influencemaximization interventions. We also note that CHANGE performs just as well as HEALER andDOSIM despite querying only 18% of the network for links. We do not claim that CHANGE actuallyoutperforms the other two algorithms (the difference could be caused by small sample sizes or otherexternal factors); however, the close results indicate that it may be possible to find influential nodesusing only a limited amount of network data. More detailed analysis of these results can be foundin [58, 52], where we examine the robustness of these results through simulation, give more detailedexplanations for the differences between algorithms, and formulate general insights and lessonslearned about influence maximization in a field setting.
Inspired by the challenges of influence maximization with limited data, this next section considersthe general problem of optimizing a monotone submodular function under uncertainty about thetrue objective. We study the problem from two perspectives (presented in [49] and [50] respectively).First, robust optimization , where the goal is to maximize the worst case from a set of possibleobjectives. Second, risk-averse optimization , where we aim to avoid disastrous outcomes instead ofsimply maximizing expected utility. In both cases, we substantially improve the existing state ofthe art, claims that are borne out both by theoretical guarantees and experimental results.
Let X be a set of items with | X | = n . A function f : 2 X → R is submodular if for any A ⊆ B and i ∈ X \ B , f ( A ∪ { i } ) − f ( A ) ≥ f ( B ∪ { i } ) − f ( B ). We restrict our attention to functions that are monotone , i.e., f ( A ∪ { i } ) − f ( A ) ≥ i ∈ X, A ⊂ X . Without loss of generality, we assumethat f ( ∅ ) = 0 and hence f ( S ) ≥ ∀ S . Let I be a collection of subsets of X . For instance, we couldhave I = { S ⊆ X : | S |≤ k } . In general, we will allow I to be any matroid. The objective is to finda utility-maximizing element of I .We consider the robust optimization setting where the true objective to be optimized is notknown exactly. Instead, it belongs to an uncertainty set which gives the set of possibilities consistentwith prior knowledge. Let F = { f ...f m } be a finite set of submodular functions on the ground set9 . We are promised that the true objective belongs to F but do not know which element of F it is.Accordingly, we aim to maximize the minimum value, max S ∈I min f i ∈F f i ( S ). The total number ofobjective functions m may be very large, potentially exponentially large in the size of the groundset n .Since the robust submodular optimization problem is in general inapproximable [25], we considera common relaxation of it to a zero sum game [26, 12]. We would like to find a minimax equilibriumof the game where the maximizing player’s pure strategies are the subsets in I , and the minimizingplayer’s pure strategies are the functions in F . The payoff to the strategies S ∈ I and f i ∈ F is f i ( S ). We call a game in this form a submodular best response (SBR) game. For the maximizingplayer, computing the minimax equilibrium is equivalent to solvingmax p ∈ ∆( I ) min f ∈F E S ∼ p [ f ( S )] (1)where ∆( I ) is the set of all distributions over the elements of I . Oftentimes, we will workwith independent distributions over X , which can be fully specified by a vector x ∈ R n + . x i givesthe marginal probability that item i is chosen. Denote by p I x the independent distribution withmarginals x .The equilibrium computation problem has been studied by Krause et al. [26] and Chen et al.[12] using very similar techniques: both iterate dynamics where the adversary plays a no-regretlearning algorithm and the decision maker plays a greedy best response. This algorithm maintainsa variable for every function in F and so is only computationally tractable when F is small. Bycontrast, we deal with the setting where F is exponentially large, with the objective function arisingfrom an underlying combinatorial structure. In [49], we explore two applications falling into thissetting: a robust budget allocation problem, and security games played on networks. In both cases,our framework leads to the first sub-exponential time algorithm for the problem. Here, we juststate the main algorithmic result.We solve Problem 1 under the assumption that there is a best response oracle available forthe adversary, which computes the minimizing function for a given distribution of the maximizingplayer. However, we require only a weaker oracle, which we call an best response to independentdistributions oracle (BRI). A BRI oracle is only required to compute a best response to mixedstrategies which are independent distributions, represented as the marginal probability that eachitem in X appears. Given a vector x ∈ R n + , where x i is the probability that element i ∈ X is chosen,a BRI oracle computes arg min f i ∈F E S ∼ p I x [ f i ( S )]. We use S ∼ x to denote that S is drawn fromthe independent distribution with marginals x . In some domains (e.g., network security games), aBRI oracle is readily available even when the full best response is NP-hard.Our main technical contribution is the EQUATOR algorithm, which computes a (1 − /e ) -approximation to Problem 1, modulo an additive loss of (cid:15) . Crucially, EQUATOR makes onlypolynomially many calls to the BRI, with no direct runtime dependence on |F | . Specifically,EQUATOR takes time polynomial in n , (cid:15) , and M , where M is an upper bound on the value ofany single item ( M ≥ max f i ∈F ,j ∈ S f i ( { j } )). In general, this results in a pseudopolynomial timealgorithm (since there is polynomial dependence on M ), though M is constant in many cases ofinterest.Since the pure strategy sets can be exponentially large, it is unclear what it even means tocompute an equilibrium: representing a mixed strategy may require exponential space. Our solutionto this dilemma is to show how to efficiently sample pure strategies from an approximate equilibriummixed strategy. This suffices for the maximizing player to implement their strategy. Alternatively,we can build an approximate mixed strategy with sparse support by drawing a polynomial numberof samples and outputing the uniform distribution over the samples. In order to generate these10
200 400 600 800 1000 n W o r s t c a s e p r o fi t a EQUATORDOGreedy n − R un t i m e ( s ) b EQUATORDOGreedy | L | W o r s t c a s e p r o fi t c EQUATORDOGreedy n R un t i m e ( s ) d EQUATORDOGreedy
Figure 4: Experimental results for budget allocation.samples, EQUATOR first solves a continuous optimization problem. This continuous relaxationuses the multilinear relaxations of the the functions in F (we refer the reader to [9] for more detailson the multilinear relaxation). Essentially, the multilinear extension of a submodular function f defines a continuous function over the hypercube [0 , n which agrees with f at the vertices.EQUATOR optimizes the pointwise minimum of the multilinear extensions of the functions in F and then uses known techniques (see [11]) to round the resulting fractional point to a distributionover integral sets. This continuous optimization problem is non-convex and nonsmooth. We designa novel stochastic Frank-Wolfe algorithm which obtains a (1 − /e )-approximation to the continuousproblem. After the rounding step, we have the following guarantee: Theorem 2.
EQUATOR outputs a set S ∈ I such that min i E [ f i ( S )] ≥ (1 − e ) OP T − (cid:15) withprobability at least − δ . Its runtime is ˜ O (cid:16) T M k n(cid:15) + T k M n(cid:15) log δ (cid:17) where T is the time toperform linear optimization over the convex hull of I and T is the time to compute a gradient. We remark that T is small ( O ( n log n )) in cases of interest such as the k -cardinality constraint,while T is typically dominated by the runtime of the BRI . This theoretical result substantiallyimproves over the current state of the art; no-regret learning based algorithms proposed for thisproblem [26, 12] work only when F is small, while the “double oracle” algorithms often used inpractice [18, 19] may take exponential runtime in the worst case.Figure 4 shows experimental results for a robust budget allocation problem. Budget allocationmodels an advertiser’s choice of how to divide a finite budget B between a set of advertising channels[41, 42, 3]. Each channel is a vertex on the left hand side L of a bipartite graph. The right hand R consists of customers. Each customer v ∈ R has a value w v which is the advertiser’s expected profitfrom reaching v . In the robust problem, the profits w are not known exactly, instead belonging toan uncertainty set (e.g., based on historical data).We compare EQUATOR to the state of the art double oracle algorithm [8, 18] (DO), whichcomputes a (1 − /e )-approximate solution but takes exponential time in the worst case. We also The ˜ O notation hides logarithmic terms w is chosen as the worst case in the uncertaintyset, with n increasing on the x axis. Figure 4(b) plots the average runtime for each n . We see thatdouble oracle produces highly robust solutions. However, for even n = 500, its execution was haltedafter 10 hours. Greedy is highly scalable, but produces solutions that are approximately 40% lessrobust than double oracle. EQUATOR produces solution quality within 7% of double oracle andruns in less than 30 seconds with n = 1000. In Figure 4(c), we see that both double oracle andEQUATOR find highly robust solutions, with EQUATOR’s solution value within 8% of that ofdouble oracle. By contrast, greedy obtains no profit in the worst case for | L | >
20, validating theimportance of robust solutions on real problems. In Figure 4(d), we observe that double oracle wasterminated after 10 hours for n = 500 while EQUATOR scales to n = 1000 in under 40 seconds. Weconclude that EQUATOR is empirically successful at finding highly robust solutions in an efficientmanner, complementing its theoretical guarantees. Decision-making under uncertainty is an ubiquitous problem. Suppose we want to maximize afunction F ( x , y ), where x is a vector of decision variables and y a random variable drawn froma distribution D . A natural approach is to maximize E y [ F ( x , y )], i.e., to maximize the expectedvalue of the chosen decision. However, decision makers are often risk-averse : they would ratherminimize the chance of having a very low reward than focus purely on the average. This is arational behavior when failure can have large consequences. For instance, if a corporation suffersa disastrous loss, they may simply go out of business. Or in many cases, low performance entailssafety issues. For instance, if a sensor network for water contamination detects problems instantlyin 80% cases, but fails entirely in 20%, the population will inevitably be exposed to an unacceptablehealth risk. It is much better to have a sensor network which always detects contaminants, even ifit requires somewhat more time on average.Hence, it is natural to move beyond average-case analysis and optimize a risk-aware objectivefunction. One widespread choice is the conditional value at risk (CVaR). CVaR takes a tunableparameter α . Roughly, it measures the performance of a decision in the worst α fraction of scenarios.It is known that when the objective F is a concave function, then CVaR can be optimized viaa concave program as well. However, many natural objective functions are not concave, andno general algorithms are known for nonconcave functions. We focus on submodular functions.Submodularity captures diminishing returns and appears in application domains ranging from viralmarketing [23], to machine learning [28], to auction theory [46]. We analyze submodular functionsin two settings: Continuous:
Continuous submodularity, which has lately received increasing attention [4, 5, 42]generalizes the notion of a submodular set function to continuous domains. Many well-knowndiscrete problems (e.g., sensor placement, influence maximization, or facility location) admit naturalextensions where resources are divided in a continuous manner. Continuous submodular functionshave also been extensively studied in economics as a model of diminishing returns or strategicsubstitutes [24, 40]. Our main result is a (1 − e )-approximation algorithm for maximizing theCVaR of any monotone, continuous submodular function. No algorithm was previously known forthis problem. Portfolio of discrete sets:
Our results for continuous submodular functions also transfer to setfunctions. We study a setting where the algorithm can select a distribution over feasible sets, which12s of interest when the aim is to select a portfolio of sets to hedge against risk [35]. This is a similarrelaxation as in the robust setting studied above. We give a black-box reduction from the discreteportfolio problem to CVaR optimization of continuous submodular functions, allowing us to applyour algorithm for the continuous problem. The state of the art for the discrete portfolio setting is analgorithm by Ohsaka and Yoshida [35] for CVaR influence maximization. Our results are strongerin two ways: (i) they apply to any submodular function and (ii) give stronger approximationguarantee. Allowing the algorithm to select a convex combination of sets is provably necessary:Maehara [32] proved that restricted to single sets, it is NP-hard to compute any multiplicativeapproximation to the CVaR of a submodular set function.In this overview, we focus on the continuous setting; details on the reduction from the discreteportfolio problem to continuous submodular optimization can be found in the full paper [50].Our main contribution is the RASCAL algorithm, which computes a (1 − /e )-approximation tooptimizing the CVaR of a smooth, continuous submodular function (up to an additive loss of (cid:15) ).RASCAL jointly exploits properties of both submodularity and the CVaR to provably approximatethe non-concave maximization problem. We start out by formalizing the problem. Continuous submodularity:
Let X = (cid:81) ni =1 X i be a subset of R n , where each X i is a compactsubset of R . A twice-differentiable function F : X → R is diminishing returns submodular (DR-submodular) if for all x ∈ X and all i, j = 1 ...n , ∂ F ( x ) ∂x i ∂x j ≤ F only shrinks as x grows, just as the marginal gains of a submodular set function only decreaseas items are added. Continuous submodular functions need not be convex or concave (concavityrequires that the Hessian is negative semi-definite, not that the individual entries are nonpositive).We consider monotone functions, where F ( x ) ≤ F ( y ) ∀ x (cid:22) y ( (cid:22) denotes element-wise inequality).We assume that F lies in [0 , M ] for some constant M . Without loss of generality, we assume F (0) = 0 (normalization).In our setting F is a function of both the decision variables x and a random parameter y .Specifically, we consider functions F ( x , y ) where F ( · , y ) is continuous submodular in x for eachfixed y . We allow any DR-submodular F which satisfies some standard smoothness conditions.First, we assume that F is L -Lipschitz for some constant L (for concreteness, with respect to the (cid:96) norm ). Second, we assume that F is twice differentiable with L -Lipschitz gradient. Third, weassume that F has bounded gradients, ||∇ F || ≤ G . Only the last condition is strictly necessary;our approach can be extended to any F with bounded gradients via known techniques [15]. Conditional value at risk:
Intuitively, the CVaR measures performance in the α worstfraction of cases. First, we define the value at risk at level α ∈ [0 , α ( x ) = inf { τ ∈ R : Pr y [ F ( x , y ) ≤ τ ] ≥ α } . That is, VaR α ( x ) is the α -quantile of the random variable F ( x , y ). CVaR is the expectation of F ( x , y ), conditioned on it falling into this set of α -worst cases:CVaR α ( x ) = E y [ F ( x , y ) | F ( x , y ) ≤ VaR α ( x )] . CVaR is a more popular risk measure than VaR both because it counts the impact of the entire α -tail of the distribution and because it has better mathematical properties [38]. Optimization problem:
We consider the problem of maximizing CVaR α ( x ) over x belongingto some feasible set P . We allow P to be any downward closed polytope. A polytope is downward We use the (cid:96) norm for concreteness. However, our arguments easily generalize to any (cid:96) p norm. (cid:96) such that x (cid:23) (cid:96) ∀ x ∈ P and for any y ∈ P , (cid:96) (cid:22) x (cid:22) y impliesthat x ∈ P . Without loss of generality, we assume that P is entirely nonnegative with (cid:96) = 0.Otherwise, we can define the translated set P (cid:48) = { x − (cid:96) : x ∈ P} and corresponding function F (cid:48) ( x , y ) = F ( x − (cid:96) , y ). Let d = max x , y ∈P || x − y || be the diameter of P .We want to solve the problem max x ∈P CVaR α ( x ). It is important to note that CVaR α ( x )need not be a smooth DR-submodular function in x . However, we would like to leverage the niceproperties of the underling F . Towards this end, we note that the above problem can be rewrittenin a more useful form [38]. Let [ t ] + = max( t, α ( x ) is equivalent to solvingmax x ∈P ,τ ∈ [0 ,M ] H ( x , τ ) = τ − α E (cid:2) [ τ − F ( x , y )] + (cid:3) (2)where τ is an auxiliary parameter. For any fixed x , the optimal value of τ is VaR α ( x ) [38]. Itis known that when F ( · , y ) is concave in x , this is a concave optimization problem. However, littleis known when F may be nonconcave.We now introduce the RASCAL (Risk Averse Submodular optimization via Conditional vALueat risk) algorithm for continuous submodular CVaR optimization. RASCAL solves Problem 2,which is a function of both the decision variables x and the auxiliary parameter τ . Roughly, τ should be understood as a threshold maintained by the algorithm for what constitutes a “bad”scenario: at each iteration, RASCAL tries to increase F ( x , y ) for those scenarios y such that F ( x , y ) ≤ τ .More formally, RASCAL is a coordinate ascend style algorithm. Each iteration first makes aFrank-Wolfe style update to x . Recall that Frank-Wolfe is a gradient-based algorithm originallydeveloped for concave optimization. However, it can be modified to maximize continuous submod-ular functions [5]. RASCAL then sets τ to its optimal value given the current x . This approach ismotivated by the unique properties of the CVaR objective H . It can be shown that H is jointlyup-concave in the variable ( x , τ ). However, H is not monotone in τ . Indeed, H is decreasing in τ for τ > VaR α ( x ). The Frank-Wolfe algorithm relies crucially on monotonicity; nonmonotonicity ismuch more difficult to handle.Instead, we exploit a unique form of structure in H . Specifically, H is monotone in x , but onlyup-concave (not fully concave). Conversely, while H is nonmonotone in τ , we can easily solve theone-dimensional problem max τ ∈ [0 ,M ] H ( x , τ ) for any fixed x (see the full paper for details). Ourapproach makes use of both properties: the Frank-Wolfe update leverages monotone up-concavityin x , while the update to τ leverages easy solvability of the one-dimensional subproblem.In order to make this approach work, two ingredients are necessary. First, we need access to thegradient of H in order to implement the Frank-Wolfe update for x . Unfortunately, H is not evendifferentiable everywhere. We instead present a smoothed estimator SmoothGrad which restoresdifferentiability at the cost of introducing a controlled amount of bias. Second, we need to solvethe one-dimensional problem of finding the optimal value of τ . We in fact introduce a subroutine SmoothTau which solves a smoothed version of the optimal τ problem. In the end, we obtain thefollowing theoretical guarantee: Theorem 3.
For any (cid:15) > , RASCAL outputs a solution x ∈ P satisfying CVaR α ( x ) ≥ (1 − /e ) OP T − (cid:15) with probability at least − δ . There are K = O (cid:16) L d α(cid:15) + L Gd α (cid:15) (cid:17) iterations, requiring O ( sK ) total evaluations of F , O ( sK ) evaluations of ∇ F , and K calls to a linear optimizationoracle for P . Here, s = O (cid:16) nM (cid:15) log δ log L (cid:15) (cid:17) is the number of samples taken from the underlyingdistribution. Infectious disease prevention
Treatable infectious diseases cause hundreds of thousands of cases of disability and death worldwide.Often, this burden is caused by long-term diseases which are continuously present in the population,as opposed to short-term epidemics like influenza. For instance, tuberculosis (TB) deaths in Indianumbered over 480,000 in 2014 [48], and even developed nations like the U.S. have observed over395,000 cases of gonorrhea in 2015 [10]. In both cases, many individuals remain undiagnosedalthough treatment is available. Outreach efforts to increase screening can lower disease burden;e.g., the Indian government conducts advertising campaigns for TB awareness. Limited resourcesrequire these campaigns to be carefully targeted at the most effective groups for reducing disease.Targeting is complicated by changing population dynamics, as individuals age and migrate overtime, as well as by uncertainty around disease transmission rates. Officials currently make suchdecisions by hand as no algorithmic assistance is available.To remedy this situation, we design an algorithm to divide a limited outreach budget betweendemographic groups in order to minimize long term disease prevalence under uncertain populationdynamics. Our approach contrasts with existing algorithms for disease control, which often considerdisease spread between nodes on a static graph [39, 7]. This is a sensible model of short term diseasespread but is less suitable for long-term planning in diseases such as TB or gonorrhea, where peopleare born, die, age, and move [31]. Accounting for changes in the underlying agents is particularlysalient for a policymaker who must divide resources between demographic groups over many yearsto maximize societal long-term health. For instance, India produces 5 year plans to combat TB[37]. Our approach also contrasts with previous work on agent-based disease models [20, 29].Such models may include realistic behaviors, but their complexity usually precludes algorithmicapproaches to finding the optimal policy in an entire feasible set.An additional challenge, largely unexplored in previous algorithmic work, is that of uncertainty.Data is always limited; policymakers are never sure of exactly how many people are infected in eachgroup, or of the contact patterns between them. In order to impact real world policy, algorithmsfor resource allocation must account for such uncertainties.We introduce a model which both captures underlying agent dynamics and can be solved usingan algorithmic approach in a stochastic setting. We make four main contributions, which are ex-plored in detail in the full paper [54].
First , we present the MCF-SIS model (Multiagent ContinuousFlow-SIS) where disease spreads in a multiagent system with birth, death, and movement. Thesystem evolves according to SIS (susceptible-infected-susceptible) dynamics and is stratified acrossage groups. This introduces a new problem in multiagent systems: computing the optimal resourceallocation under MFS-SIS, as in the case where an outreach campaign must decide how to dividelimited advertising dollars (or rupees) between the groups.MCF-SIS introduces a continuous, nonconvex, highly nonlinear optimization problem whichcannot be solved by existing methods. Many factors must be accounted for. E.g., between-groupdisease transmission makes focusing on the groups with the most infected agents suboptimal. More-over, agents in a targeted group are not cured instantaneously, so, e.g., to reduce prevalence in agegroup 30, we may need to start targeting resources at age 27. Lastly, we consider a stochasticsetting where parts of the model (contact patterns between agents, the number of infected agentsin each group, etc.) are not known exactly but are drawn from a distribution.Our second contribution shows that optimal allocation in MCF-SIS is a continuous submodular problem. This opens up a novel set of optimization techniques which have not previously been usedin disease prevention. Continuous submodularity generalizes submodular set functions to contin-uous domains. Intuitively, infections averted by spending one unit of treatment resources can nolonger be averted by additional spending, creating diminishing returns.
Continuous submodularity s deliberately enabled by our modeling choices, in particular our shift from the discrete, graph-basedsetting common in previous work [39, 7] to a continuous, population-based model. Our third contribution is a new algorithm called DOMO (Disease Outreach via MultiagentOptimization), which obtains an efficient (1 − /e )-approximation to the optimal allocation. Ouralgorithm builds on a recent theoretical framework for submodular optimization [5]. DOMO’sgeneralization of this framework to the stochastic setting may be of independent interest.Our fourth contribution is to instantiate MCF-SIS in two domains using empirical data whichtakes into account behavioral, demographic, and epidemic trends: first, TB spread in India, andsecond, gonorrhea in the United States. DOMO averts 8,000 annual person-years of TB and 20,000person-years of gonorrhea compared to current policy. There are many promising future questions related to algorithmic social intervention. Here, I detailtwo directions in progress.
Oftentimes, finding the best intervention amounts to performing optimization in a complex model.For instance, epidemiologists have build enormously complicated models of disease spread, whichsimulate the (stochastic) interactions of millions of agents and account for a range of factors. Suchmodels are very faithful to what are believed to be the real-world processes of diseases spread,but suffer from very high computational cost and are poorly understood from an optimizationperspective. Hence, researchers interested in optimization (including my work [54]) seek simplerand more tractable models. It is hoped that these simpler models are sufficiently faithful to realityto yield useful insights, but they will clearly not be as accurate as more detailed simulations.Hence, a natural direction is to pursue better methods for multi-fidelity optimization: usinga simpler model as a guide, or surrogate, to optimize a more complex one. Such methods haverecently attracted interest in machine learning for use in hyperparameter optimization [22], andhave previously been studied in several engineering disciplines [43, 17, 2]. However, previous mod-els suffer from a variety of shortcomings in how they treat both the high-fidelity and surrogatemodels. For instance, most do not incorporate stochasticity in the high-fidelity model, where onlynoisy observations of the ground truth are available. This can easily become problematic becauserandomness is ubiquitous in modeling, especially in noisy domains like human interaction. Withrespect to the low-fidelity model, previous work usually assumes black-box access. However, thisneglects the potential advantage that can be gained through exploiting known structure in thesurrogate. For instance, if we were to use the MCF-SIS model introduced in [54] as a surrogatefor a complex disease model, the DOMO algorithm can be used to find provably good approximatesolutions.Accordingly, the purpose of this project is to remedy such shortcomings by proposing multi-fidelity optimization methods which naturally incorporate stochasticity and leverage known struc-ture in the surrogate model. The immediate application for such techniques is optimizing policiesfor preventing disease spread, but many other application areas are possible.
In this project, we consider a more nuanced treatment of uncertainty in submodular optimization,which yields improved properties in learning and optimizing from limited data. Suppose that16e wish to maximize a submodular function which is not known exactly. For instance, we mayhave a finite collection of samples from an unknown distribution and wish to maximize expectedperformance over that distribution. Or, we may have a probabilistic model (e.g., a model ofinfluence spread) but do not believe that this model is exactly correct. We can draw samples fromthis model and optimize empirical performance over the samples (the de facto approach in influencemaximization), but such a process will not incorporate our uncertainty about the true distributionthat the objective is drawn from.In both settings (limited data and model uncertainty), is there a better approach than maxi-mizing empirical performance on the samples? One attractive alternative is distributionally robust optimization. Let the empirical distribution on sample objective functions f ...f n be denoted by ˆ p n .Let D ( p || ˆ p n ) be a divergence measure between another distribution p and the empirical distributionˆ p n (e.g., the χ divergence). The distributionally robust optimization problem is to solvemax S min p : D ( p || ˆ p n ) ≤ ρ E f ∼ p [ f ( S )] . That is, we aim to maximize our worst-case expected performance over all distributions that are“close” to the observed distribution ˆ p n . One advantage of this formulation is that it can be seen asmaximizing a high-probability bound on expected performance. Let D be the unknown distributiongenerating the objective. Given n samples from D , classical arguments (e.g., the Bernstsein bound)show that E f ∼D [ f ( S )] ≥ E f ∼ ˆ p n [ f ( S )] − C (cid:114) Var D [ f ( S )] n where C is a constant (e.g., depending on the probability with which we want the bound tohold). Hence, when the variance is large, we can do better by optimizing the entire term on the right-hand side instead of just empirical performance on the samples (the first term). It has recently beenshown [34] that (under some conditions) the distributionally robust problem corresponds exactlyto such a variance-regularized objective. This has led to improved generalization for convex lossfunctions, where distributionally robust optimization remains a convex optimization problem. Thepurpose of this project is to extend distributionally robust techniques to submodular optimization.This will entail the development of new algorithmic tools to deal with the (natively combinatorial)nonconvex problem. However, such development is a very relevant direction for algorithmic socialintervention since objectives in many problems are inferred from limited data or uncertain models. References [1] Emmanuel Abbe and Colin Sandon. Community detection in general stochastic block models:Fundamental limits and efficient algorithms for recovery. In
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