optimal nudging. presentation ud
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Optimal NudgingA new approach to solving SMDPs
Reinaldo Uribe MUniversidad de los Andes — Oita University
Colorado State University
Nov. 11, 2013
Snakes & Ladders
Player advances thenumber of steps indicatedby a die.
Landing on a snake’smouth sends the playerback to the tail.
Landing on a ladder’sbottom moves the playerforward to the top.
Goal: reaching state 100.
Snakes & Ladders
Boring!(No skill required, only luck.)
Player advances thenumber of steps indicatedby a die.
Landing on a snake’smouth sends the playerback to the tail.
Landing on a ladder’sbottom moves the playerforward to the top.
Goal: reaching state 100.
Variation: Decision Snakes and Ladders
Sets of “win” and“loss” terminal states.
Actions: either“advance” or “goback,” to be decidedbefore throwing the die.
Reinforcement Learning: Finding an optimal policy.
“Natural” Rewards: ±1on “win”/“lose”, 0othw.
Optimal policymaximizes totalexpected reward.
Dynamic programmingquickly finds theoptimal policy.
Probability of winning:pw = 0.97222 . . .
We know a lot!
Markov Decision Process: States, Actions, TransitionProbabilities, Rewards.
Policies and policy value.
Max winning probability 6= max earnings.
Taking an action costs (in units different from rewards.)
Different actions may have different costs.
Semi-Markov model with average rewards.
We know a lot!
Markov Decision Process: States, Actions, TransitionProbabilities, Rewards.
Policies and policy value.
Max winning probability 6= max earnings.
Taking an action costs (in units different from rewards.)
Different actions may have different costs.
Semi-Markov model with average rewards.
We know a lot!
Markov Decision Process: States, Actions, TransitionProbabilities, Rewards.
Policies and policy value.
Max winning probability 6= max earnings.
Taking an action costs (in units different from rewards.)
Different actions may have different costs.
Semi-Markov model with average rewards.
We know a lot!
Markov Decision Process: States, Actions, TransitionProbabilities, Rewards.
Policies and policy value.
Max winning probability 6= max earnings.
Taking an action costs (in units different from rewards.)
Different actions may have different costs.
Semi-Markov model with average rewards.
We know a lot!
Markov Decision Process: States, Actions, TransitionProbabilities, Rewards.
Policies and policy value.
Max winning probability 6= max earnings.
Taking an action costs (in units different from rewards.)
Different actions may have different costs.
Semi-Markov model with average rewards.
We know a lot!
Markov Decision Process: States, Actions, TransitionProbabilities, Rewards.
Policies and policy value.
Max winning probability 6= max earnings.
Taking an action costs (in units different from rewards.)
Different actions may have different costs.
Semi-Markov model with average rewards.
Better than optimal?
(Old optimal policy)
(Optimal policy) withaverage rewardρ = 0.08701
pw = 0.48673 (was0.97222 — 50.06%)
d = 11.17627 (was84.58333 — 13.21%)
This policy maximizes pwd
Better than optimal?
(Optimal policy) withaverage rewardρ = 0.08701
pw = 0.48673 (was0.97222 — 50.06%)
d = 11.17627 (was84.58333 — 13.21%)
This policy maximizes pwd
Better than optimal?
(Optimal policy) withaverage rewardρ = 0.08701
pw = 0.48673 (was0.97222 — 50.06%)
d = 11.17627 (was84.58333 — 13.21%)
This policy maximizes pwd
Better than optimal?
(Optimal policy) withaverage rewardρ = 0.08701
pw = 0.48673 (was0.97222 — 50.06%)
d = 11.17627 (was84.58333 — 13.21%)
This policy maximizes pwd
So, how are average-reward optimal policies found?
Algorithm 1 Generic SMDP solver
Initializerepeat forever
ActDo RL to find value of current π . Usually 1-step Q-learning
Update ρ.
Average-adjusted Q-learning:
Qt+1(st, at)← (1− γt)Qt(st, at) + γt
(rt+1 − ρtct+1 +max
aQt(st+1, a)
)
Generic Learning Algorithm
Table of algorithms. ARRL
Algorithm Gain update
AACJalali and Ferguson 1989 ρt+1 ←
t∑i=0
r(si, πi(si))
t+ 1
R–LearningSchwartz 1993
ρt+1 ← (1− α)ρt+α(rt+1 +max
aQt(st+1, a)−max
aQt(st, a)
)H–LearningTadepalli and Ok 1998
ρt+1 ← (1−αt)ρt+αt
(rt+1 −Ht(st) +Ht(st+1)
)αt+1 ←
αt
αt + 1
SSP Q-LearningAbounadi et al. 2001
ρt+1 ← ρt + αt minaQt(s, a)
HARGhavamzadeh and Mahadevan 2007 ρt+1 ←
t∑i=0
r(si, πi(si))
t+ 1
Generic Learning Algorithm
Table of algorithms. SMDPRL
Algorithm Gain update
SMARTDas et al. 1999
ρt+1 ←
t∑i=0
r(si, πi(si))
t∑i=0
c(si, πi(si))MAX-Q
Ghavamzadeh and Mahadevan 2001
Nudging
Algorithm 2 Nudged Learning
Initialize (π, ρ, Q)repeat
Set reward scheme to (r − ρc).Solve by any RL method.Update ρ
until Qπ(sI) = 0
Note: ‘by any RL method’ refers to a well-studied problem forwhich better algorithms (both practical and with theoreticalguarantees) exist.
ρ can (and will) be updated optimally.
Nudging
Algorithm 3 Nudged Learning
Initialize (π, ρ, Q)repeat
Set reward scheme to (r − ρc).Solve by any RL method.Update ρ
until Qπ(sI) = 0
Note: ‘by any RL method’ refers to a well-studied problem forwhich better algorithms (both practical and with theoreticalguarantees) exist.
ρ can (and will) be updated optimally.
The w − l space.Definition
(Policy π has expected average reward vπ and expected averagecost cπ. Let D be a bound on the absolute value of vπ)
wπ =D + vπ
2cπ, lπ =
D − vπ
2cπ.
D
D
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The w − l space.Value and Cost
(Policy π has expected average reward vπ and expected averagecost cπ. Let D be a bound on the absolute value of vπ)
wπ =D + vπ
2cπ, lπ =
D − vπ
2cπ.
D
D
l
w
−D
−0.5
D
0
0.5D
D
D
D
l
w
1
2
48
The w − l space.Nudged value
(Policy π has expected average reward vπ and expected averagecost cπ. Let D be a bound on the absolute value of vπ)
wπ =D + vπ
2cπ, lπ =
D − vπ
2cπ.
D
D
l
w
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−D/2
0
D/2
The w − l space.As a projective transformation.
The w − l space.As a projective transformation.
1
D
−D
Policy V
alu
e
Episode Length
The w − l space.As a projective transformation.
1
D
−D
Policy V
alu
e
Episode LengthD
−D
w
l
Sample task: two states, continuous actions
s1a1 ∈ [0, 1]r1 = 1 + (a1 − 0.5)2
c1 = 1 + a1
s2
a2 ∈ [0, 1]r2 = 1 + a2c2 = 1 + (a2 − 0.5)2
Sample task: two states, continuous actions
s1a1 ∈ [0, 1]r1 = 1 + (a1 − 0.5)2
c1 = 1 + a1
s2a2 ∈ [0, 1]r2 = 1 + a2c2 = 1 + (a2 − 0.5)2
Sample task: two states, continuous actions
Policy Space (Actions)
0
a2
1
0 a1 1
Sample task: two states, continuous actions
Policy Values and Costs
Policy v
alu
e
Policy cost
4
4
Sample task: two states, continuous actions
Policy Manifold in w − l
l
w
D/2
D/2
And the rest...
Neat geometry, linear problems in w − l.Easily exploited using straightforward algebra / calculus.
Updating average reward between iterations can be optimized.
Becomes finding the (or rather an) intersection between twoconics.
Which can be solved in O(1) time.
Worst case, uncertainty reduces in half.
Typically much better than that.
Little extra complexity added to already PAC methods.
And the rest...
Neat geometry, linear problems in w − l.Easily exploited using straightforward algebra / calculus.
Updating average reward between iterations can be optimized.
Becomes finding the (or rather an) intersection between twoconics.
Which can be solved in O(1) time.
Worst case, uncertainty reduces in half.
Typically much better than that.
Little extra complexity added to already PAC methods.
Thank you.r-uribe@uniandes.edu.co
Untitled by Li Wei, School of Design, Oita University, 2009.
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