Reinforcement Learning I:
The setting and classical
stochastic dynamic programming
algorithms
Tuomas Sandholm
Carnegie Mellon University
Computer Science Department
Reinforcement Learning
(Ch. 17.1-17.3, Ch. 20)
passive
Learner
active
Sequential decision problems
Approaches:
1. Learn values of states (or state histories) & try to maximize
utility of their outcomes.
• Need a model of the environment: what ops & what
states they lead to
2. Learn values of state-action pairs
• Does not require a model of the environment (except
legal moves)
• Cannot look ahead
Reinforcement Learning …
Deterministic transitions
Stochastic transitions
M ija is the probability to reaching state j when taking action
a in state i
A simple environment that
presents the agent with a
sequential decision problem:
3
+1
2
-1
Move cost = 0.04
1 start
1
2
3
4
(Temporal) credit assignment problem sparse reinforcement problem
Offline alg: action sequences determined ex ante
Online alg: action sequences is conditional on observations along the way;
Important in stochastic environment (e.g. jet flying)
Reinforcement Learning …
M = 0.8 in direction you want to go
0.1 left
0.2 in perpendicular
0.1 right
Policy: mapping from states to actions
An optimal
policy for
the
stochastic
environment:
utilities of states:
3
+1
3
0.812
2
-1
2
0.762
1
0.705
1
1
1
2
3
4
0.868
0.912
+1
0.660
-1
0.655
0.611
0.388
2
3
4
Observable (accessible): percept identifies the state
Environment
Partially observable
Markov property: Transition probabilities depend on state only, not on the
path to the state.
Markov decision problem (MDP).
Partially observable MDP (POMDP): percepts does not have enough info to
identify transition probabilities.
Partial observability in previous
figure
(2,1)
vs.
(2,3)
U(A)  0.8*U(A) in (2,1) + 0.2*U(A) in (2,3)
Have to factor in the value of new info obtained by
moving in the world
Observable MDPs
Assume additivity (almost always true in practice):
Uh ([S0,S1…Sn]) = R0 + Uh([S1,…Sn])
Utility function on histories
Policy*(i) = arg max
a
U(i) = R(i) + max
a
a
M
 ijU ( j )
j
a
M
 ijU ( j )
j
Classic Dynamic Programming
(DP)
Start from last step & move backward
Complexity of
Naïve search O(|A|n)
Actions per step
DP O(n|A||S|)
# possible states
Problem: n= if loops or otherwise infinite horizon
Does not require there to
exist a “last step” unlike
dynamic programming
The utility values for selected states at each iteration step in the application
of VALUE-ITERATION to the 4x3 world in our example
Thrm: As t, value iteration converges to exact U even if updates
are done asynchronously & i is picked randomly at every step.
When to stop value iteration?
Idea: Value determination (given a policy) is simpler than value iteration
Value Determination Algorithm
The VALUE-DETERMINATION algorithm can be implemented in one of
two ways. The first is a simplification of the VALUE-ITERATION
algorithm, replacing the equation (17.4) with
U t 1 (i )  R(i )   M ijPolicy(i )U t ( j )
j
and using the current utility estimates from policy iteration as the initial
values. (Here Policy(i) is the action suggested by the policy in state i)
While this can work well in some environments, it will often take a very
long time to converge in the early stages of policy iteration. This is because
the policy will be more or less random, so that many steps can be required to
reach terminal states
Value Determination Algorithm
The second approach is to solve for the utilities directly. Given a fixed policy
P, the utilities of states obey a set of equations of the form:
U (i )  R (i )   M ijP ( i )U t ( j )
j
For example, suppose P is the policy shown in Figure 17.2(a). Then using the
transition model M, we can construct the following set of equations:
U(1,1) = 0.8u(1,2) + 0.1u(1,1) + 0.1u(2,1)
U(1,2) = 0.8u(1,3) + 0.2u(1,2)
and so on. This gives a set of 11 linear equations in 11 unknowns, which can
be solved by linear algebra methods such as Gaussian elimination. For small
state spaces, value determination using exact solution methods is often the
most efficient approach.
Policy iteration converges to optimal policy, and policy improves
monotonically for all states.
Asynchronous version converges to optimal policy if all states are visited
infinitely often.
Discounting
Infinite horizon  Infinite U  Policy & value iteration fail to converge.
Also, what is rational:  vs. 
Solution: discounting
U (H )   vi R
i
Finite if
0  v 1
i
Reinforcement Learning II:
Reinforcement learning (RL)
algorithms
(we will focus solely on observable
environments in this lecture)
Tuomas Sandholm
Carnegie Mellon University
Computer Science Department
Passive learning
Epochs = training sequences:
(1,1)(1,2)(1,3)(1,2)(1,3)(1,2)(1,1)(1,2)(2,2)(3,2) –1
(1,1)(1,2)(1,3)(2,3)(2,2)(2,3)(3,3) +1
(1,1)(1,2)(1,1)(1,2)(1,1)(2,1)(2,2)(2,3)(3,3) +1
(1,1)(1,2)(2,2)(1,2)(1,3)(2,3)(1,3)(2,3)(3,3) +1
(1,1)(2,1)(2,2)(2,1)(1,1)(1,2)(1,3)(2,3)(2,2)(3,2) -1
(1,1)(2,1)(1,1)(1,2)(2,2)(3,2) -1
Passive learning …
(b)
3
+1
(a) 2
-1
1 start
1
3
.5
.5
.33
+1
.5
.5
.5
.33
.33
.33
.33
1.0
1.0
-1
2
-0.0380 0.0886
3
4
0.2152
+1
.5
.5.5
.5
.33
.5
.33
.33
.33
.5
.5
(a) A simple stochastic environment.
(c) 2
1
-0.1646
-0.4430
-1
-0.2911 -0.0380 -0.5443 -0.7722
1
2
3
4
(b) Each state transitions to a neighboring state with equal
probability among all neighboring states. State (4,2) is
terminal with reward –1, and state (4,3) is terminal with
reward +1.
(c) The exact utility values.
LMS – updating
[Widrow & Hoff 1960]
function LMS-UPDATE(U,e,percepts,M,N) returns an update U
if TERMINAL?[e] then reward-to-go  0
simple average
for each ei in percepts (starting at end) do
batch mode
reward-to-go  reward-to-go + REWARD[ei]
U[STATE[ei]]  RUNNING-AVERAGE (U[STATE[ei]], reward-to-go, N[STATE[ei]])
end
Average reward-to-go that state has gotten
Converges slowly to LMS
estimate or training set
But utilities of states are not
independent!
NEW
U=?
P=0.9
-1
P=0.1
+1
OLD
U = -0.8
An example where LMS does poorly. A new state is reached for the
first time, and then follows the path marked by the dashed lines,
reaching a terminal state with reward +1.
Adaptive DP (ADP)
Idea: use the constraints (state transition probabilities) between
states to speed learning.
Solve
U (i )  R (i )   M ijU ( j )
j
= value determination.
No maximization over actions because agent is
passive unlike in value iteration.
using DP
Large state space
e.g. Backgammon: 1050 equations in 1050 variables
Temporal Difference (TD)
Learning
Idea: Do ADP backups on a per move basis, not for the whole
state space.
U (i )  U (i )   [ R(i )  U ( j )  U (i )]
Thrm: Average value of U(i) converges to the correct value.
Thrm: If  is appropriately decreased as a function of times a
state is visited (=[N[i]]), then U(i) itself converges to the
correct value
Algorithm TD()
(not in Russell & Norvig book)
Idea: update from the whole epoch, not just on state transition.

U (i )  U (i )    m  k [ R(im )  U (im 1 )  U (im )]
mk
Special cases:
=1: LMS
=0: TD
Intermediate choice of  (between 0 and 1) is best.
Interplay with  …
Convergence of TD()
Thrm: Converges w.p. 1 under certain boundaries conditions.
Decrease i(t) s.t.
  i (t )  
t
i2 (t )  
t
In practice, often a fixed  is used for all i and t.
Passive learning in an unknown
environment
M
a
ij unknown
ADP does not work directly
LMS & TD() will operate unchanged
… Changes to ADP
a
Construct an environment model (of M ij ) based on
observations (state transitions) & run DP
Quick in # epochs, slow update per example
As the environment model approaches the correct model,
the utility estimates will converge to the correct utilities.
Passive learning in an unknown
environment
ADP: full backup
TD: one experience back up
As TD makes a single adjustment (to U) per observed
transitions, ADP makes as many (to U) as it needs to restore
consistency between U and M. Change to M is local, but
effects may need to be propagated throughout U.
Passive learning in an unknown
environment
TD can be viewed as a crude approximation of ADP
Adjustments in ADP can be viewed as pseudo experience in TD
A model for generating pseudo-experience can be used in TD
directly: DYNA [Sutton]
Cost of thinking vs. cost of acting
Approximating <value policy> iterations directly by restricting the backup
after each observed transition. Prioritized sweeping heuristic prefers to make
adjustments to states whose likely successors have just undergone large
adjustments in U(j)
- Learns roughly as fast as full ADP (#epochs)
- Several orders of magnitude less computation  allows doing problems
that are not solvable via ADP
- M is incorrect early on  minimum decreasing adjustment size before
recompute U(i)
Active learning in an unknown
environment
Agent considers what actions to take.
Algorithms for learning in the setting (action choice discussed later)
ADP:
U (i )  R(i )  max  M ijaU ( j )
a
j
Learn M ija instead of M ij as before
TD():
Unchanged!
Model-based (learn M)
Tradeoff
Model-free (e.g. Q-learning)
Which is better? open
Q-learning
Q (a,i)
U (i )  max Q(a, i )
a
Q(a, i )  R(i )   M ija max Q(a ' , j )
j
a'
Direct approach (ADP) would require learning a model M ija.
Q-learning does not:
Do this update after each state transition:
Q(a, i )  Q(a, i )   [ R(i )  max Q(a' , j )  Q(a, i )]
a'
Exploration
Tradeoff between exploitation (control) and exploration (identification)
Extremes: greedy vs. random acting
(n-armed bandit models)
Q-learning converges to optimal Q-values if
* Every state is visited infinitely often (due to exploration),
* The action selection becomes greedy as time approaches infinity, and
* The learning rate  is decreased fast enough but not too fast
(as we discussed in TD learning)
Common exploration methods
1.
E.g. in value iteration in an ADP agent:
Optimistic estimate of utility U+(i)
U  (i )  R(i )  max f [ M ijaU  ( j ), N (a, i )]
a
2.
3.
Exploration fn. e.g.
f (u, n)  
j
E.g. in TD() or Q-learning:
Choose best action w.p. p and a random action otherwise.
E.g. in TD() or Q-learning:
Boltzmann exploration
 M ija U ( j )
*
Pa* 
e

e
a
j
T

 M ijaU ( j )
j
T
R+ if n<N
u o.w.
Reinforcement Learning III:
Advanced topics
Tuomas Sandholm
Carnegie Mellon University
Computer Science Department
Generalization
With table lookup representation (of U,M,R,Q) up to 10,000
states or more
Chess ~ 10120
Industrial problems
Backgammon ~ 1050
Hard to represent & visit all states!
Implicit representation, e.g. U(i) = w1f1(i) + w2f2(i) + …+ wnfn(i)
Chess 10120 states  n weights
This compression does generalization
E.g. Backgammon: Observe 1/1044 state space and beat any
human.
Generalization …
Could use any supervised learning algorithm for the
generalization part:
input
sensation
generalization
estimate
(Q or U…)
update from RL
Convergence results do not apply with generalization.
Pseudo-experiments require predicting many steps ahead
(not supported by standard generalization methods)
Convergence results of Q-learning
tabular
function approximation
state aggregation
converges to Q*
converges to Q*
general
averagers
linear
converges to Q*
error in Q 
max
i, j in same class
Q d (i )  Q d ( j )
off-policy
on-policy
1 v
prediction
converges to Q
diverges
control
diverges
chatters, bound
unknown
Applications of RL
•
•
•
•
Checker’s [Samuel 59]
TD-Gammon [Tesauro 92]
World’s best downpeak elevator dispatcher [Crites at al ~95]
Inventory management [Bertsekas et al ~95]
– 10-15% better than industry standard
• Dynamic channel assignment [Singh & Bertsekas, Nie&Haykin ~95]
– Outperforms best heuristics in the literature
•
•
•
•
•
•
•
•
Cart-pole [Michie&Chambers 68-] with bang-bang control
Robotic manipulation [Grupen et al. 93-]
Path planning
Robot docking [Lin 93]
Parking
Football
Tetris
Multiagent RL [Tan 93, Sandholm&Crites 95, Sen 94-, Carmel&Markovitch 95-, lots
of work since]
• Combinatorial optimization: maintenance & repair
– Control of reasoning [Zhang & Dietterich IJCAI-95]
TD-Gammon
•
•
•
•
Q-learning & back propagation neural net
Start with random net
Learned by 1.5 million games against itself
As good as best human in the world
TD-Gammon (self-play)
Performance
against
Gammontool
Neurogammon (15,000
supervised learning
examples)
# hidden units
• Expert labeled examples are scarce, expensive & possibly wrong
• Self-play is cheap & teaches the real solution
• Hand-crafted features help
Multiagent RL
• Each agent as a Q-table entry e.g. in a communication network
• Each agent as an intentional entity
– Opponent’s behavior varies for a given sensation of the agent
• Opponent uses different sensation than agent, e.g. longer window or
different features (Stochasticity in steady state)
• Opponent learned: sensation  Q-values (Nonstationarity)
• Opponent’s exploration policy (Q-values  action probabilities)
changed.
• Opponent’s action selector chose different action. (Stochasticity)
Sensation at step n: < anme1 , anopponent
>,
1
reward from step n-1
Qcoop
Q-storage
deterministic
Qdef
p(coop)
Explorer
p(def)
Random
Process
an
Future research in RL
•
•
•
•
•
Function approximation (& convergence results)
On-line experience vs. simulated experience
Amount of search in action selection
Exploration method (safe?)
Kind of backups
– Full (DP) vs. sample backups (TD)
– Shallow (Monte Carlo) vs. deep (exhaustive)
•  controls this in TD()
• Macros
– Advantages
• Reduce complexity of learning by learning subgoals (macros) first
• Can be learned by TD()
– Problems
• Selection of macro action
• Learn models of macro actions (predict their outcome)
• How do you come up with subgoals