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Decision Making in Robots and Autonomous Agents Preferences and Elicitation (Based on material from C. Boutilier et al. ). Subramanian Ramamoorthy School of Informatics 22 March, 2013. Preference Elicitation in AI. Shopping for a Car:. Luggage Capacity? Two Door? Cost? Engine Size? - PowerPoint PPT Presentation
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Decision MakingDecision Makingin Robots and Autonomous in Robots and Autonomous
AgentsAgents
Preferences and Elicitation(Based on material from C. Boutilier et al.)
Subramanian RamamoorthySchool of Informatics
22 March, 2013
Preference Elicitation in AILuggage Capacity?Two Door? Cost?Engine Size?Color? Options?
Shopping for a Car:
22/03/2013 2
Intelligence
Readiness
Objective conditions
Political situation
Budget limitations
Campaign-level goals
Experience
Beliefs
Strategic and tactical intentions and preferences
Outcome evaluation for alternative decisions
Proposals of preferable courses of actions
Better decision
Beliefs and preference refinement through
incremental questioning
Monitor and filter information
Adapt presentation
22/03/2013 3[Source: R. Brafman]
The Preference Bottleneck
• Preference elicitation: the process of determining a user’s preferences/utilities to the extent necessary to make a decision on her behalf
• Why a bottleneck?– preferences vary widely– large (multiattribute) outcome spaces– quantitative utilities (the “numbers”) difficult to assess
22/03/2013 4
Shirt
Trousers
::::
black black red whiteblack white white redwhite black white redwhite white red white
Jacketblack white black white
22/03/2013 5
worst
best22/03/2013 6
Automated Preference Elicitation
• The interesting questions:– decomposition of preferences– what preference info is relevant to the task at hand?– when is the elicitation effort worth the improvement it
offers in terms of decision quality?– what decision criterion to use given partial utility info?
22/03/2013 7
Overview
• Preference Elicitation in A.I.
– Constraint-based Optimization
– Factored Utility Models
– Types of Uncertainty
– Types of Queries
• Single User Elicitation
22/03/2013 8
Constraint-based Decision Problems
• Constraint-based optimization (CBO):
– outcomes over variables X = {X1 … Xn}
– constraints C over X spell out feasible decisions
– generally compact structure, e.g., X1 & X2 ¬ X3
– add a utility function u: Dom(X) → R – preferences over configurations
22/03/2013 9
Constraint-based Decision Problems
• Must express u compactly like C– generalized additive independence (GAI)
• model proposed by Fishburn (1967) • nice generalization of additive linear models
– given by graphical model capturing independence
22/03/2013 10
11
Factored Utilities: GAI Models• Set of K factors fk over subset of vars X[k]
– “local” utility for each local configuration
–
• [Fishburn67] u in this form exists iff– lotteries p and q are equally preferred whenever p and q
have the same marginals over each X[k]
Kkk kfu )xx) ][(( A Bf1(A)
a: 3a: 1
f2(B)b: 3b: 1
Cf3(BC) bc: 12 bc: 2…
u(abc) = f1(a)+ f2(b)+ f3(bc)
22/03/2013
12
Optimization with GAI Models
• Optimize using simple Integer Program– number of variables linear in size of GAI model
Kk
kfu k )xx) ][((A B
f1(A)a: 3a: 1
f2(B)b: 3b: 1
C f3(BC) bc: 12 bc: 2…
CAIukDomkkik
kkXI , tosubj. max
])[(][][][][},{
Xxxxx
22/03/2013
Difficulties in CBO
• Utility elicitation: how do we assess individual user preferences?– need to elicit GAI model structure (independence)– need to elicit (constraints on) GAI parameters– need to make decisions with imprecise parameters
22/03/2013 13
Strict Utility Function Uncertainty
• User’s actual utility u unknown• Assume feasible set F U = [0,1]n
– allows for unquantified or “strict” uncertainty– e.g., F a set of linear constraints on GAI terms
• How should one make a decision? elicit information?
u(red,2door,280hp) > 0.4u(red,2door,280hp) > u(blue,2door,280hp)
22/03/2013 14
f2(L,N)l,n: [2,4]l,n: [1,2]
Strict Uncertainty Representation
L
NP
Utility Function
f1(L)l: [7,11]l: [2,5]
22/03/2013 15
Bayesian Utility Function Uncertainty
• User’s actual utility u unknown• Assume density P over U = [0,1]n
• Given belief state P, EU of decision x is:EU(x , P) = U (px . u) P( u ) du
• Decision making is easy, but elicitation harder?– must assess expected value of information in query
22/03/2013 16
f2(L,N)l,n: l,n: …
Bayesian Representation
L
NP
Utility Function
f1(L)l: l:
22/03/2013 17
Query Types
• Comparison queries (is x preferred to x’ ?)– impose linear constraints on parameters
• k fk(x[k]) > k fk(x’[k]) – Interpretation is straightforward
U
22/03/2013 18
Query Types
• Bound queries (is fk(x[k]) > v ?)– response tightens bound on specific utility parameter– can be phrased as a local standard gamble query
U
22/03/2013 19
Overview
• Preference Elicitation in A.I.
• Single User Elicitation
– Foundations of Local queries
– Bayesian Elicitation
– Regret-based Elicitation
22/03/2013 20
Difficulties with Bound Queries• Bound queries focus on local factors
– but values cannot be fixed without reference to others!– seemingly “different” local prefs correspond to same u
u(Color,Doors,Power) = u1(Color,Doors) + u2(Doors,Power)
u(red,2door,280hp) = u1(red,2door) + u2(2door,280hp)
u(red,4door,280hp) = u1(red,4door) + u2(4door,280hp)
10 6 4
6 3 3
1 9
22/03/2013 21
Local Queries
• We wish to avoid queries on whole outcomes– can’t ask purely local outcomes– but can condition on a subset of default values
• Conditioning set C(f) for factor fi(Xi) :– variables that share factors with Xi
– setting default outcomes on C(f) renders Xi independent of remaining variables
– enables local calibration of factor values
22/03/2013 22
Local Standard Gamble Queries
• Local std. gamble queries– use “best” and “worst” (anchor) local outcomes
-- conditioned on default values of conditioning set– bound queries on other parameters relative to these– gives local value function v(x[i]) (e.g., v(ABC) )
• Hence we can legitimately ask local queries:
• But local Value Functions not enough: – must calibrate: requires global scaling
22/03/2013 23
Global Scaling• Elicit utilities of anchor outcomes wrt global best and worst
outcomes
– the 2*m “best” and “worst” outcomes for each factor
– these require global std gamble queries (note: same is true for pure additive models)
22/03/2013 24
Bound Query Strategies
• Identify conditioning sets Ci for each factor fi
• Decide on “default” outcome• For each fi identify top and bottom anchors
– e.g., the most and least preferred values of factor i– given default values of Ci
• Queries available:– local std gambles: use anchors for each factor, C-sets– global std gambles: gives bounds on anchor utilities
22/03/2013 25
Overview
• Preference Elicitation in A.I.
• Single User Elicitation
– Foundations of Local queries
– Bayesian Elicitation
– Regret-based Elicitation
22/03/2013 26
Partial preference informationBayesian uncertainty
• Probability distribution p over utility functions• Maximize expected (expected) utility
MEU decision x* = arg maxx Ep [u(x)]• Consider:
– elicitation costs– values of possible decisions– optimal tradeoffs between elicitation effort and
improvement in decision quality
22/03/2013 27
Query Selection
• At each step of elicitation process, we can– obtain more preference information– make or recommend a terminal decision
22/03/2013 28
Bayesian ApproachMyopic EVOI
MEU(p)
q1q2
...
r1,1 r2,1r1,2 r2,2
...
MEU(p1,1) MEU(p1,2) MEU(p2,1) MEU(p2,2)22/03/2013 29
Expected value of information
• MEU(p) = Ep [u(x*)]
• Expected posterior utility: EPU(q,p) = Er|q,p [MEU(pr)]• Expected value of information of query q:
EVOI(q) = EPU(q,p) – MEU(p)
MEU(p)
q1q2 ...
r1,1 r2,1r1,2 r2,2
...MEU(p1,1) MEU(p1,2) MEU(p2,1) MEU(p2,2)
22/03/2013 30
Bayesian ApproachMyopic EVOI
• Ask query with highest EVOI - cost• [Chajewska et al ’00]
– Global standard gamble queries (SGQ) “Is u(oi) > l?”– Multivariate Gaussian distributions over utilities
• [Braziunas and Boutilier ’05]– Local SGQ over utility factors– Mixture of uniforms distributions over utilities
22/03/2013 31
Local elicitation in GAI models [Braziunas and Boutilier ’05]
• Local elicitation procedure– Bayesian uncertainty over local factors– Myopic EVOI query selection
• Local comparison query “Is local value of factor setting xi greater than l”?– Binary comparison query– Requires yes/no response– query point l can be optimized analytically
22/03/2013 32
Experiments
• Car rental domain: 378 parameters [Boutilier et al. ’03]– 26 variables, 2-9 values each, 13 factors
• 2 strategies– Semi-random query
• Query factor and local configuration chosen at random• Query point set to the mean of local value function
– EVOI query• Search through factors and local configurations• Query point optimized analytically
22/03/2013 33
Experiments
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8
10
12
14
16
Mixture of UniformsUniformGaussian
No. of queries
Percentage utility error (w.r.t. truemax utility)
22/03/2013 34
Overview
• Single User Elicitation
– Foundations of Local queries
– Bayesian Elicitation
– Regret-based Elicitation
• why MiniMax Regret (MMR) ?
• Decision making with MMR
• Elicitation with MMR
22/03/2013 35
Minimax Regret: Utility Uncertainty
• Regret of x w.r.t. u:
• Max regret of x w.r.t. F:
• Decision with minimax regret w.r.t. F:
),(),(),( * uxEUuxEUuxR u
),(max),( uxRFxMRFu
),()(;),(minarg *
)(
* FxMRFMMRFxMRx FXFeasx
F
22/03/2013 36
Why Minimax Regret?*
• Appealing decision criterion for strict uncertainty– contrast maximin, etc.– not often used for utility uncertainty
x
x’
x’x
x
x’x’x
xx’
x
x’
Bett
er
u1 u2 u3 u4 u5 u6
22/03/2013 37
Why Minimax Regret?
• Minimizes regret in presence of adversary– provides bound worst-case loss– robustness in the face of utility function uncertainty
• In contrast to Bayesian methods:– useful when priors not readily available– can be more tractable– effective elicitation even if priors available
22/03/2013 38
Overview
• Single User Elicitation
– Foundations of Local queries
– Bayesian Elicitation
– Regret-based Elicitation
• why MiniMax Regret (MMR) ?
• Decision making with MMR
• Elicitation with MMR
22/03/2013 39
Computing Max Regret
• Max regret MR(x,F) computed as an IP– number of variables linear in GAI model size– number of (precomputed) constants (i.e., local regret
terms) quadratic in GAI model size
– r( x[k] , x’[k] ) = uT (x’[k] ) – u (x[k] )
CAIrk k
kkkXI ik , tosubj. max
]['][']['][}',{ ]['
x
xxxx
22/03/2013 40
Minimax Regret in Graphical Models
• We convert minimax to min (standard trick)– obtain a MIP with one constraint per feasible config– linearly many vars (in utility model size)
• Key question: can we avoid enumerating all x’ ?
22/03/2013 41
Constraint Generation
• Very few constraints will be active in solution
• Iterative approach: – solve relaxed IP (using a subset of constraints)– Solve for maximally violated constraint– if any add it and repeat; else terminate
22/03/2013 42
Constraint Generation Performance• Key properties:
– aim: graphical structure permits practical solution– convergence (usually very fast, few constraints)– very nice anytime properties – considerable scope for approximation– produces solution x* as well as witness xw
22/03/2013 43
Overview
• Single User Elicitation
– Foundations of Local queries
– Bayesian Elicitation
– Regret-based Elicitation
• why MiniMax Regret (MMR) ?
• Decision making with MMR
• Elicitation with MMR
22/03/2013 44
Regret-based Elicitation[Boutilier, Patrascu, Poupart, Schuurmans IJCAI05; AIJ 06]
• Minimax optimal solution may not be satisfactory• Improve quality by asking queries
– new bounds on utility model parameters• Which queries to ask?
– what will reduce regret most quickly?– myopically? sequentially?
• Closed form solution seems infeasible– to date people have looked only at heuristic elicitation
22/03/2013 45
Elicitation Strategies I• Halve Largest Gap (HLG)
– ask if parameter with largest gap > midpoint– MMR(U) ≤ maxgap(U), hence nlog(maxgap(U)/)
queries needed to reduce regret to – bound is tight
f1(a,b) f1(a,b) f1(a,b) f1(a,b) f2(b,c) f2(b,c) f2(b,c) f2(b,c)22/03/2013 46
Elicitation Strategies II• Current Solution (CS)
– only ask about parameters of optimal solution x* or regret-maximizing witness xw
– intuition: focus on parameters that contribute to regret• reducing u.b. on xw or increasing l.b. on x* helps
– use early stopping to get regret bounds (CS-5sec)
f1(a,b) f1(a,b) f1(a,b) f1(a,b) f2(b,c) f2(b,c) f2(b,c) f2(b,c)22/03/2013 47
Elicitation Strategies III• Optimistic-pessimistic (OP)
– query largest-gap parameter in one of:• optimistic solution xo
• pessimistic solution xp
• Computation:– CS needs minimax optimization– OP needs standard optimization– HLG needs no optimization
• Termination:– CS easy– Others ?
22/03/2013 48
Results (Small Random)
10vars; < 5 vals
10 factors, at most 3 vars
Avg 45 trials
22/03/2013 49
Results (Car Rental, Unif)
26 vars; 61 billion configs
36 factors, at most 5 vars; 150 parameters
Avg 45 trials
22/03/2013 50
Results (Real Estate, Unif)
20 vars; 47 million configs
29 factors, at most 5 vars; 100 parameters
Avg 45 trials
22/03/2013 51
Results (Large Rand, Unif)
25 vars; < 5 vals
20 factors, at most 3 vars
A 45 trials
22/03/2013 52
Summary of Results
• CS works best on test problems– time bounds (CS-5): little impact on query quality– always know max regret (or bound) on solution– time bound adjustable (use bounds, not time)
• OP competitive on most problems– computationally faster (e.g., 0.1s vs 14s on RealEst)– no regret computed so termination decisions harder
• HLG much less promising
22/03/2013 53
Interpretation
• HLG: – provable regret reduced very quickly
• But:– true regret faster (often to optimality)– OP and CS restricted to feasible decisions– CS focuses on relevant parameters
22/03/2013 54
Concluding Remarks
• Local parameter elicitation– Theoretically sound– Computationally practical– Easier to answer
• Bayesian EVOI / Regret-based elicitation– Good guides for elicitation– Integrated in computationally tractable algorithms
• Questions for Future Work:– Sequential reasoning
22/03/2013 55