Graphical Multiagent Models Quang Duong Computer Science and Engineering Chair: Michael P. Wellman 1 Example: Election In The City Of AA May, political analyst Political discussion Vote • • • • Phone surveys Demographic information Party registration … 2 Modeling Objectives Vote Republican or Democrat? Construct a model that takes into account people (agent) interactions (graph edges) in: – Representing joint probability of all vote outcomes* – Computing marginal and conditional probabilities 3 Modeling Objectives (cont.) Generate predictions: – Individual actions, dynamic behavior induced by individual decisions – Detailed or aggregate 4 More Applications Of Modeling Multiagent Behavior Computer Network/ Internet Financial Institutions Social Network 5 Challenges: Uncertainty from the system modeler’s perspective 1a. Agent choice Vote for personal favorite or conform with others? 1b. Correlation Will the historic district of AA unanimously pick one candidate to support? 1c. Interdependence May does not know all friendship relations in AA 6 Challenges: Complexity 2a. Representation and inference Number of all action configurations (all vote outcomes) is exponential in the number of agents (people). 2b. Historical information People may change their minds about whom to vote for after discussions. 7 Existing Approaches That This Work Builds On Game-theory Approach: • Assume game structure/perfect rationality Statistical Modeling Approach: • Aggregate statistical measures/ make simplifying assumptions 8 Approach Outline Graphical Multiagent Models (GMMs) are probabilistic graphical models designed to • Facilitate expressions of different knowledge sources about agent reasoning • Capture correlated behaviors uncertainty while • Exploiting dependence structure complexity 9 Roadmap (Ch. 2) Background (Ch. 3) GMM (static) (Ch. 5) Learning Dependence Graph Structure (Ch. 4) HistoryDependent GMM (Ch. 6) Application: Information Diffusion 10 Multiagent Systems • n agents {1,…,i,…,n} • Agent i chooses action ai, joint action (action configuration) of the system: a = (a1,…, an) • In dynamic settings: – time period t, time horizon T. – history Ht of history horizon h, Ht = (at-h,…,at-1) 11 Game Theory Each player (agent i) chooses a strategy (action ai). Strategy profile (joint action a) of all players. Payoff function: ui(ai,a-i) Player i‘s regret εi(a): maximum gain if player i chooses strategy ai’, instead of strategy ai, given than everyone else fixes their strategies. a* is a Nash equilibrium (NE) if for every player i, regret εi(a) = 0. 12 Graphical Representations of Multiagent Systems 1. Graphical Game Models [Kearns et al. ‘01] An agent’s payoff depends on strategy chosen by itself and its neighbors Ji Payoff/utility: ui(ai,aJi) Similar approaches: Multiagent influence diagrams (MAIDs) [Koller & Milch ’03] Networks of Influence Diagrams [Gal & Pfeffer ’08] Action-graph games [Jiang et al ‘11]. 13 Graphical Representations (cont.) 2. Probabilistic graphical models Markov random field (static) [Kindermann & Laurie ’80, KinKoller & Friedman ‘09] Dynamic Bayesian Networks [Kanazawa & Dean ’89, Ghahramani ’98] 14 This Work Building on incorporating Probabilistic Graphical Models Game Models demonstrate and examine the benefits of applying probabilistic graphical models to the problem of modeling multiagent behavior in scenarios with different sets of assumptions and information available to the system modeler. 15 Roadmap (Ch. 2) Background 1. Overview 2. Examples 3. Knowledge Combination 4. Empirical Study (Ch. 3) GMM (static) (Ch. 5) Learning Dependence Graph Structure (Ch. 4) HistoryDependent GMM (Ch. 6) Application: Information Diffusion 16 Graphical Multiagent Models (GMMs) [Duong, Wellman & Singh ‘08] • Nodes: agents. Edges: dependencies among agent actions • Dependence neighborhood Ni 2 7 4 6 3 1 1 5 17 GMMs Joint probability distribution of system’s actions Pr(a) ∝ Πi πi(aNi) potential of neighborhood’s joint actions Factor joint probability distribution into neighborhood potentials. (Markov random field for graphical games [Daskalakis & Papadimitriou ’06]) 18 Example GMMs • Markov Random Field for computing pure strategy Nash equilibrium • Markov Random Field for computing correlated equilibrium • Information diffusion GMMs [Ch. 6] • Regret GMMs [Ch. 3] 19 Examples: Regret potential Assume a graphical game Regret ε(aNi) πi(aNi) = exp(-λ εi(aNi)) Illustration: Assume: prefers Republican to Democrat (fixing others’ choices) Near zero λ: picks randomly Larger λ: more likely to pick Republican 20 Flexibility: Knowledge Combination • Assume known graph structures, given GMMs G1 and G2 that represent 2 different knowledge sources Regret GMM GMM1 reG Knowledge Combination Heuristic Rule-based GMM 2 GMM hG 1. Direct update 2. Opinion pool 3. Mixing data Final GMM finalG 21 Empirical Study 1.6 performance score ratio 1.4 1.2 1 ratio > 1: combined model performs better than input model 0.8 0.6 Mixing data GMM vs. regret GMM 0.4 0.2 0 example 1 example 1 example 2 example 2 Mixing data GMM vs. heuristic GMM • Combining knowledge sources in one GMM improves predictions • Combined models fail to improve on input models when input does not capture any underlying behavior 23 Summary Of Contributions (Ch. 3) (I.A) GMMs accommodate expressions of different knowledge sources (I.B) This flexibility allows the combination of models for improved predictions 26 Roadmap (Ch. 2) Background (Ch. 3) GMM (static) (Ch. 5) Learning Dependence Graph Structure (Ch. 4) HistoryDependent GMM 1. Consensus Dynamics 2. Description 3. Joint vs. individual behavior 4. Empirical study (Ch. 6) Application: Information Diffusion 27 Example: Consensus Dynamics [Kearns et al. ’09] abstracted version of the AA mayor election example Examine the ability to make collective decisions with limited communication and observation Observation graph Agent 1’s perspective 2 5 Agent 3 1 6 4 Blue Red neither consensus consensus 1 1.0 0.5 0 2 0.5 1.0 0 28 time Network structure here plays a large role in determining the outcomes 29 Modeling Multiagent Behavior In Consensus Dynamics Scenario time Time series action data + observation graph 1. Predict detailed actions 2. Predict aggregate measures or History-Dependent Graphical Multiagent Models (hGMMs) [Duong, Wellman, Singh & Vorobeychik ’10] We condition actions on abstracted history Ht Note: dependence graphs can be different from observation graphs. 1 t-1 1 t 1 t+1 31 hGMMs 1 t-1 1 t 1 t+1 (Undirected) within-time edges: dependencies between agent actions in the same time period, and define dependence neighborhood Ni for each agent i. A GMM at every time t 32 hGMMs 1 t-1 1 1 t+1 (Directed) across-time edges: dependencies of agent i’s action on some abstraction of prior actions by agents in i’s conditioning set Γi Example: frequency function. 33 hGMMs Joint probability distribution of system’s actions at time t potential of neighborhood’s joint actions at t Pr(at | H) ∝ Πi πi(atNi | HtΓi) history of the conditioning set 34 Challenge: Dependence 2 1 t-2 1 t-1 2 2 t • Conditional independence 1 2 1 t-2 t-1 • Dependence induced byt history abstraction/summarization (*) 35 Individual vs. Joint Behavior Models Given complete history, autonomous agents’ behaviors are conditionally independent Individual behavior models: πi(ati | HtΓi,complete) Joint behavior models allow specifying any action dependence within one’s within-time neighborhood, given some (abstracted) history πi(atNi | HtΓi,abstracted) 36 Empirical Study: Summary Evaluation: compares joint behavior and individual behavior models by likelihood of testing data (time-series votes) * Observation graph defines both dependence neighborhoods N and conditioning sets Γ 1. Joint behavior outperform individual behavior models for shorter history lengths, which induce more action dependence. 1. Approximation does not deteriorate performance 37 Summary Of Contributions (Ch. 4) (II.A) hGMMs support inference about system dynamics (II.B) hGMMs allow the specification of action dependence emerging from history abstraction 38 Roadmap (Ch. 2) Background 1. Learning Graphical Game Models (Ch. 3) GMM (static) (Ch. 5) Learning Dependence Graph structure (Ch. 4) HistoryDependent GMM 2. Learning hGMMs (Ch. 6) Application: Information Diffusion 39 Learning History-Dependent Graphical Multiagent Models Objective Given action data + observation graph, build a model that predicts: – Detailed actions in next period – Aggregate measures of actions in the more distant future Challenge: Learn dependence graph – (Within-time) Dependence graph ≠ observation graph – Complexity of the dependence graph 42 Consensus Dynamics Joint Behavior Model Extended Joint Behavior hGMM (eJCM) πi(aNi | HtΓi) = ri(aNi) f(ai , HtΓi)γ Ι(ai , Hti)β 1 2 3 1. ri(aNi) = reward for action ai, discounted by the number of dissenting neighbors in Ni 1. 2. frequency of ai chosen previously by agents in the conditioning set Γi inertia proportional to how long i has maintained its most recent action 43 Consensus Dynamics Individual Behavior Models 1. Extended Individual Behavior hGMM (eICM): similar to eJCM but assumes that Ni contains i only πi(ai | HtΓi) = Pr(ai | HtΓi) ∝ ri(ai) f(ai , HtΓi)γ Ι(ai , Hti)β 2. Proportional Response Model (PRM): only incorporates the most recent time period [Kearns et al., ‘09]: Pr(ai | HtΓi) ∝ ri(ai) f(ai , HtΓi) 3. Sticky Proportional Response Model (sPRM) 44 Learning hGMMS Input: • <action observations (time series)> Search space: 1.Model parameters γ, β Output: hGMM 2.Within-time edges • observation graph Objective: likelihood of data Constraint: max node degree 45 Greedy Learning Initialize the graph with no edges Repeat: Add edges that generate the biggest increase (>0) in the training data’s likelihood Until no edge can be added without violating the maximum node degree constraint 46 Empirical Study: Learning from human-subject data Use asynchronous human-subject data Vary the following environment parameters: • Discretization intervals, delta (0.5 and 1.5 seconds) • History lengths, h • Graph structures/payoff functions: coER_2, coPA_2, & power22 (strongly connected minority) Goal: evaluate eJCM, eICM, PRM, and sPRM using 2 metrics • Negative likelihood of agents’ actions • Convergence rates/outcomes 47 Predicting Dynamic Behavior eJCMs and eICMs outperform the existing PRMs/sPRMs eJCMs predict actions in the next time period noticeably more accurately than PRMs and sPRMs, and (statistically significantly) more accurate than eICMs 48 Predicting Consensus Outcomes power22, delta=0.5 eICM power22, delta=0.5 power22, delta=1.5 consensus probability 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 PRM experiment power22, delta=1 1.0 0.0 eJCM eJCM eICM eICM PRM PRM experiment experiment eJCM eICM eJCMs have comparable prediction performance with other models in 2 settings: coER_2 and coPA_2. In power22, eJCM predict consensus probability and colors much more accurately. 49 PR Graph Analysis intra red intra blue inter In learned graphs, intra edges >> inter edges. 0.8 In power22, a large majority of edges are intra red identify the presence of a strongly connected red minority 0.6 0.4 0.2 w ith in −t im e (c oP A _2 ) (c oP A _2 ) _2 ) rv at io n ob se tim e in − w ith rv at io n (c oE R (c oE R ow er 2 ob se w ith in −t im e (p (p ow er er va t io n _2 ) 2) 22 ) 0.0 ob s proportion of edges 1.0 50 Summary Of Contributions (Ch. 5.2) (II.B) [revisit] This study highlights the importance of joint behavior modeling (III.C) It is feasible to learn both dependence graph structure and model parameters (III.D) Learned dependence graphs can be substantially different from observation graphs 51 Modeling Multiagent Systems: Step By Step Given as input Dependence graph structure Observation graph structure Learn from data GMM hGMM Potential function Intuition, background information Approximation 52 Roadmap (Ch. 2) Background (Ch. 3) GMM (static) (Ch. 5) Learning Dependence Graph structure (Ch. 4) HistoryDependent GMM 1. Definition 2. Joint behavior modeling 3. Learning missing edges 4. Experiments (Ch. 6) Application: Information Diffusion 53 Networks with Unobserved Links True network G* • Links facilitate how information diffuses from one node to another • Real-world nodes have links unobserved by third parties Observed Network G 54 Problem [Duong, Wellman & Singh ‘11] Given: a network (with missing links) and snapshots of the network states over time. 1. Network G 2. Diffusion traces (on G*) Objective: model information diffusions on this network 55 Approach 1: Structure Learning Recover missing edges • Learn network G’ • Learn parameters of an individual behavior model built on G’ • Learning algorithms: NetInf [Gomez-Rodriguez et al. ’10] and MaxInf 56 Approach 2: Potential Learning Construct an hGMM on G without recovering missing links • hGMMs allow capturing state correlations between neighbors who appear disconnected in the input network • Theoretical evidence [6.3.2] • Empirical illustrations: hGMMs outperform individual behavior models on learned graph – random graph with sufficient training data – preferential attachment graph (varying amounts of data) 57 Summary of Contributions (Ch. 6) (II.C) Joint behavior hGMM, can capture state dependence caused by missing edges 58 Conclusions 1. The machinery of probabilistic graphical models helps to improve modeling in multiagent systems by: • allowing the representation and combination of different knowledge sources of agent reasoning • relaxing assumptions about action dependence (which may be a result of history abstraction or missing edges) 2. One can learn from action data both: (i) model parameters, and (ii) dependence graph structure, which can be different from interaction/observation graph structure 59 Conclusions (cont.) 3. The GMM framework contributes to the integration of: • strategic behavior modeling techniques from AI and economics • probabilistic models from statistics that can efficiently extract behavior patterns from massive amount of data for the goal of understanding fast-changing and complex multiagent systems. 60 Summary • Graphical multiagent models: flexibility to represent different knowledge sources and combine them [UAI ’08] • History-dependent GMM: capture dependence in dynamic settings [AAMAS ’10, AAMAS ’12] • Learning graphical game models [AAAI ’09] • Learning hGMM dependence graph, distinguishing observation/interactions graphs and probabilistic dependence graphs [AAMAS ‘12] • Modeling information diffusion in networks with unobserved links [SocialCom ‘11] 61 Acknowledgments • Advisor: Professor Michael P. Wellman • Committee members: Prof. Satinder Singh Baveja, Prof. Edmund H. Durfee, and Asst. Prof. Long Nguyen • Research collaborators: Yevgeniy Vorobeychik (Sandia Labs), Michael Kearns (U Penn), Gregory Frazier (Apogee Research), David Pennock and others (Yahoo/Microsoft Research) • Undergraduate advisor: David Parkes. • Family • Friends • CSE staff 62 THANK YOU! 63