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Transfer in Variable - Reward
Hierarchical Reinforcement Learning
Hui Li
March 31, 2006
Overview
• Multi-criteria reinforcement learning
• Transfer in variable-reward hierarchical
reinforcement learning
• Results
• Conclusions
Multi-criteria reinforcement learning
Reinforcement Learning
• Definition
Reinforcement learning is the process by which the agent
learns an approximately optimal policy through trial and
error interactions with the environment.
Agent
Policy
state st reward rt
rt+1
st+1
s0 a0 : r0
Pss’(a)
Environment
s1 a1 : r1
Pss’(a)
s2 a2 : r2
Pss’(a)
...
• Goal
The agent’s goal is to maximize the cumulative amount of
rewards he receives over the long run.
• A new value function -- average adjusted sum of rewards (bias)
N
h ( s)  lim E ( ( rt ( s)    ))

N 

t 1
N
  ( s)  lim
E (  rt ( s))
N 
Average reward (gain) per
time step at given policy 

t 1
N
 
• Bellman equation
h ( s)  rimm ( s,  ( s))   Pss ' ( ( s)) h ( s' )  
s'
• H-learning:
model-based version of average reward reinforcement learning
h( s)  max {rimm ( s, a)   Pss ' (a)h( s' ) }
a
s'
  (1   )    (rimm ( s, a)  V ( s' )  V ( s)) learning rate, 0<<1
old
new
New observation rs(a)
• R-learning:
model-free version of average reward reinforcement learning
R( s, a )  (1   ) R( s, a )   ( rimm ( s, a )    max R( s' , a ' ))
a'
  (1   )    ( rimm ( s, a )  max R( s' , a ' )  max R( s, a ))
a'
a
Multi-criteria reinforcement learning
In many situations, it is nature to express the objective as
making some appropriate tradeoffs between different kinds of
rewards.
Goals:
• Eating food
• Guarding food
• Minimize the number
of steps it walks
Buridan’s donkey problem
Weighted optimization criterion:
weight, which represents the
r( s, a)  wi ri ( s, a) importance of each reward

i
If the weight vector w is static, never changes over time, then
the problem reduces to the reinforcement learning with a scalar
value of reward.
If the weight vector varies from time to time, learning policy
for each weight vector from scratch is very inefficient.
Since the MDP model is a liner transformation, the average
reward  and the average adjusted reward h(s) are linear
in the reward weights for a given policy .
 
   wi i  w  


i
 
h ( s)   wi hi ( s)  w  h ( s)


i
 
R ( s, a )   wi Ri ( s)  w  R ( s, a )


i
1
2
6
7
3
4
5
• Each line represents the
weighted average reward given a
 


policy k ,  
wi i  w  
i
• Solid lines represent those
active weighted average rewards
• Dot lines represent those
inactive weighted average rewards
• Dark line segments represent the
best average rewards for any
weight vectors
The key idea:
: the set of all stored policies.
Only those policies which have active average rewards are stored.
Update equations:
Variable-reward hierarchical reinforcement
learning
• The original MDP M is split into sub-SMDP {M0,… Mn},
each sub-SMDP representing a sub-task
• Solving the root task M0 solves the entire MDP M
• The task hierarchy is represented as a directed acyclic graph
known as the task graph
• A local policy i for the subtask Mi is a mapping from the
states to the child tasks of Mi
• A hierarchical policy  for the whole task is an assignment of
a local policy i to each subtask Mi
• The objective is to learn an optimal policy that optimizes the
policy for each subtask assuming that its children’s polices are
optimized
Forest
Home base
Enemy base
Goldmine
Peasants
Two kinds of subtask:
• Composite subtask :
Root:
the whole task
Harvest: the goal is to harvest wood or gold
Deposit: the goal is to deposit a resource into home base
Attack: the goal is to attack the enemy base
• Primitive subtask: primitive actions
north, south, east, west,
pick a resource, put a resource
attack the enemy base
idle
SMDP – semi-MDP
A SMDP is a tuple < S, A, P, r, t >
• S, A, P, r are defined the same as in MDP;
• t(s, a) is the execution tine for taking action a in state s
• Bellman equation of SMDP for average reward learning
h( s)  max {rimm ( s, a)   Pss ' (a)h( s' ) t ( s, a)}
a
s'
A sub-task is a tuple < Bi, Ai , Gi >
• Bi : state abstraction function which maps state s in the
original MDP into an abstract state in Mi
• Ai : The set of subtasks that can be called by Mi
• Gi : Termination predicate
The value function decomposition satisfied the following set of
Bellman equations:
where
At root, we only store the average adjusted reward
Results
• Learning curves for a test reward weight after having seen 0, 1,
2, …, 10 previous training weight vectors
• Negative transfer: learning based
on one previous weight is worse
than learning from scratch.
Transfer ratio: FY/FY/X
• FY is the area between the learning curve and its
optimal value for problem with no prior learning
experience on X.
• FY/X is the area between the learning curve and its
optimal value for problem given prior training on X.
Conclusions
• This paper showed that hierarchical task structure can
accelerate transfer across variable-reward MDPs more
than in the flat MDP
• This hierarchical task structure facilitates multi-agent
learning
References
[1] T. Dietterich, Hierarchical Reinforcement Learning with the MAXQ Value Function Decomposition.
Journal of Artificial Intelligence Research, 9:227–303, 2000.
[2] N. Mehta and P. Tadepalli, Multi-Agent Shared Hierarchy Reinforcement Learning. ICML Workshop
on Richer Representations in Reinforcement Learning, 2005.
[3] S. Natarajan and P. Tadepalli, Dynamic Preferences in Multi-Criteria Reinforcement Learning, in
Proceedings of ICML-05, 2005.
[4] N. Mehta, S. Natarajan, P. Tadepalli and A. Fern, Transfer in Variable-Reward Hierarchical
Reinforcement Learning, in NIPS Workshop on transfer learning, 2005.
[5] Barto, A., & Mahadevan, S. (2003). Recent Advances in Hierarchical Reinforcement Learning,
Discrete Event Systems.
[6] S. Mahadevan, Average Reward Reinforcement Learning: Foundations, Algorithms, and Empirical
Results, Machine Learning, 22, 169-196 (1996)
[7] P. Tadepalli and D. OK, Model-based Average Reward Reinforcement Learning, Artificial
intelligence 1998