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R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^1)
Reinforcement Learning
Slides from Sutton and Barto
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^2)
The Agent-Environment Interface
Agent
Environment
action a s (^) t t
reward r t r (^) t+ 1 s (^) t+ 1
state
Agent and environment interact at discrete time steps : t = 0,1, 2, K Agent observes state at step t : s (^) t ∈ S produces action at step t : at ∈ A ( st ) gets resulting reward : rt + 1 ∈ℜ and resulting next state: s (^) t + 1
t
... (^) s t (^) a
rt +1 (^) s t +1 (^) at +
rt +2 (^) s t +2 (^) at (^) +
rt +3 (^) s t +3 (^) a t +...
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^3)
Policy at step t , π t :
a mapping from states to action probabilities
π t ( s , a ) = probability that at = a when st = s
The Agent Learns a Policy
Reinforcement learning methods specify how the agent changes its policy as a result of experience. Roughly, the agent’s goal is to get as much reward as it can over the long run.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^4)
Returns
Suppose the sequence of rewards after step t is : rt + 1 , rt + 2 , rt + 3 , K What do we want to maximize?
In general, we want to maximize the expected return , E R { } t , for each step t.
Episodic tasks : interaction breaks naturally into episodes, e.g., plays of a game, trips through a maze. Rt = rt + 1 + rt + 2 + L + rT ,
where T is a final time step at which a terminal state is reached, ending an episode.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^7)
Another Example
Get to the top of the hill as quickly as possible.
reward = −1 for each step where not at top of hill ⇒ return = − number of steps before reaching top of hill
Return is maximized by minimizing number of steps reach the top of the hill.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^8)
A Unified Notation
In episodic tasks, we number the time steps of each episode starting from zero. We usually do not have distinguish between episodes, so we write instead of for the state at step t of episode j. Think of each episode as ending in an absorbing state that always produces reward of zero:
We can cover all cases by writing
st s (^) t , j
s^ r^1 = + 0 s 1 r 2 = +1 (^) s 2 r 3 = +1 r r 4 = 0 5 = 0
Rt = γ k^ rt + k + 1 , k = 0
∞ ∑
where γ can be 1 only if a zero reward absorbing state is always reached.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^9)
The Markov Property
By “the state” at step t , the book means whatever information is available to the agent at step t about its environment. The state can include immediate “sensations,” highly processed sensations, and structures built up over time from sequences of sensations. Ideally, a state should summarize past sensations so as to retain all “essential” information, i.e., it should have the Markov Property :
Pr (^) { s (^) t + 1 = s ′ , rt + 1 = r st , at , rt , st − 1 , at − 1 ,K, r 1 , s 0 , a 0 }=
Pr (^) { s (^) t + 1 = s ′ , rt + 1 = r st , a (^) t } for all s ′, r , and histories st , a (^) t , rt , st − 1 , at − 1 ,K, r 1 , s 0 , a 0.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^10)
Markov Decision Processes
If a reinforcement learning task has the Markov Property, it is basically a Markov Decision Process (MDP). If state and action sets are finite, it is a finite MDP. To define a finite MDP, you need to give: state and action sets one-step “dynamics” defined by transition probabilities :
reward probabilities :
Ps sa ′^ = Pr (^) { s (^) (^) t + 1 = s ′ st = s , a (^) t = a } for all s , s ′ ∈ S , a ∈ A ( s ).
Rsa s^ ′= E r { t (^) + 1 st = s , at = a , s (^) t + 1 = s ′} for all s , s ′ ∈ S , a ∈ A ( s ).
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^13)
Value Functions
State - value function for policy π :
V^ π^ ( s ) = E π { R (^) t st = s }= E π γ k^ rt + k + 1 st = s k = 0
∞ ∑
⎧ ⎨ ⎩
⎫ ⎬ ⎭
Action - value function for policy π :
Q^ π^ ( s , a ) = E π { R (^) t s (^) t = s , at = a }= E π γ k^ rt + k + 1 s (^) t = s , at = a k = 0
∞ ∑
⎧ ⎨ ⎩
⎫ ⎬ ⎭
The value of a state is the expected return starting from that state; depends on the agent’s policy:
The value of taking an action in a state under policy π
is the expected return starting from that state, taking that
action, and thereafter following π :
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^14)
Bellman Equation for a Policy π
Rt = rt + 1 + γ rt + 2 + γ 2 rt + 3 + γ 3 rt + 4 L
= rt + 1 +γ (^) ( r (^) t + 2 +γ rt + 3 +γ 2 rt + 4 L)
= rt + 1 +γ Rt + 1
The basic idea:
So: V^ π^ ( s ) = E π { R (^) t st = s }
= E π { rt (^) + 1 +γ V s ( (^) t + 1 ) s (^) t = s }
Or, without the expectation operator:
V^ π^ ( s ) = π ( s , a ) Ps sa ′^ [ R (^) sa s^ ′+γ V^ π^ ( s ′ )] s ′
∑ a
∑
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^15)
More on the Bellman Equation
V^ π^ ( s ) = π ( s , a ) Ps sa ′^ [ R (^) sa s^ ′+γ V^ π^ ( s ′ )] s ′
∑ a
∑
This is a set of equations (in fact, linear), one for each state.
The value function for π is its unique solution.
Backup diagrams : s s,a
a
s'
r a'
s'
r
(a) (b)
for V π (^) for Q^ π
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^16)
π ≥ π ′ if and only if V
π ( s ) ≥ V π ′ ( s ) for all s ∈ S
Optimal Value Functions
For finite MDPs, policies can be partially ordered :
There is always at least one (and possibly many) policies that is better than or equal to all the others. This is an optimal
policy. We denote them all π *.
Optimal policies share the same optimal state-value function :
Optimal policies also share the same optimal action-value function :
V
∗ ( s ) = max π
V
π ( s ) for all s ∈ S
Q ∗( s , a ) = max π Q^ π^ ( s , a ) for all s ∈ S and a ∈ A ( s )
This is the expected return for taking action a in state s and thereafter following an optimal policy.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^19)
Why Optimal State-Value Functions are Useful
a) gridworld (^) b) V * (^) c) π*
22.0 24.4 22.0 19.4 17. 19.8 22.0 19.8 17.8 16. 17.8 19.8 17.8 16.0 14. 16.0 17.8 16.0 14.4 13. 14.4 16.0 14.4 13.0 11.
A B
A'
V ∗
V
∗
Any policy that is greedy with respect to is an optimal policy.
Therefore, given , one-step-ahead search produces the long-term optimal actions.
E.g., back to the gridworld:
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^20)
What About Optimal Action-Value Functions?
Given , the agent does not even have to do a one-step-ahead search:
Q
π ∗ ( s ) = arg max a ∈ A ( s )
Q ∗ ( s , a )
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^21)
Solving the Bellman Optimality Equation
Finding an optimal policy by solving the Bellman Optimality Equation requires the following: accurate knowledge of environment dynamics; we have enough space an time to do the computation; the Markov Property. How much space and time do we need? polynomial in number of states (via dynamic programming methods; Chapter 4), BUT, number of states is often huge (e.g., backgammon has about 10**20 states). We usually have to settle for approximations. Many RL methods can be understood as approximately solving the Bellman Optimality Equation.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^22)
Policy Evaluation
State - value function for policy π :
V^ π^ ( s ) = E π { R (^) t st = s }= E π γ k^ rt + k + 1 st = s k = 0
∞ ∑
⎧ ⎨ ⎩
⎫ ⎬ ⎭
Bellman equation for V^ π^ : V^ π^ ( s ) = π ( s , a ) Ps sa ′^ [ R (^) sa s^ ′+γ V^ π^ ( s ′ )] s ′
a
— a system of S simultaneous linear equations
Policy Evaluation : for a given policy π, compute the
state-value function V π
Recall:
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^25)
Policy Improvement
Suppose we have computed V for a deterministic policy π.
π
For a given state s , would it be better to do an action a^ ≠^ π( s )?
Q^ π^ ( s , a ) = E π { rt (^) + 1 + γ V^ π^ ( st + 1 ) s (^) t = s , at = a }
= Psa s ′
s ′
∑ Rs s ′
a +γ V π ( s ′)
[ ]
The value of doing a in state s is :
It is better to switch to action a for state s if and only if
Q^ π^ ( s , a ) > V^ π^ ( s )
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^26)
Policy Improvement Cont.
π ′( s ) = argmax
a
Q^ π^ ( s , a )
= argmax
a
Pssa ′
s ′
∑ Rss ′
a +γ V π ( s ′)
[ ]
Do this for all states to get a new policy π ′that is
greedy with respect to V^ π^ :
Then V π^ ′≥ V^ π
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^27)
Policy Improvement Cont.
What if V π^ ′= V^ π^? i.e., for all s ∈ S , V π^ ′( s ) = max a Ps sa ′ s ′
∑ Rs s ′
a (^) +γ V π (^) ( s ′) [ ]?
But this is the Bellman Optimality Equation. So V π^ ′= V ∗^ and both π and π ′are optimal policies.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^28)
Policy Iteration
π 0 → V
π (^0) →π 1 → V
π 1 → L π
→ V
→π
policy evaluation policy improvement “greedification”
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^31)
Value Iteration Cont.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^32)
Asynchronous DP
All the DP methods described so far require exhaustive sweeps of the entire state set. Asynchronous DP does not use sweeps. Instead it works like this: Repeat until convergence criterion is met:
- Pick a state at random and apply the appropriate backup Still need lots of computation, but does not get locked into hopelessly long sweeps Can you select states to backup intelligently? YES: an agent’s experience can act as a guide.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^33)
Generalized Policy Iteration
π (^) V
evaluation
improvement
V → V π
π→greedy( V )
π* V^ *
starting V π
V (^) = (^) V π
π^ =^ g re e
d y^ (^ V^ )
V * π*
Generalized Policy Iteration (GPI): any interaction of policy evaluation and policy improvement, independent of their granularity.
A geometric metaphor for convergence of GPI:
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^34)
Efficiency of DP
To find an optimal policy is polynomial in the number of states… BUT, the number of states is often astronomical, e.g., often growing exponentially with the number of state variables (what Bellman called “the curse of dimensionality”). In practice, classical DP can be applied to problems with a few millions of states. Asynchronous DP can be applied to larger problems, and appropriate for parallel computation. It is surprisingly easy to come up with MDPs for which DP methods are not practical.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^37)
Simplest TD Method
T T T T T
T T^ T T T
st + 1
rt + 1
st
V ( st ) ← V ( s (^) t ) +α (^) [ r (^) t + 1 + γ V ( s (^) t + 1 ) − V ( st )]
T T T T^ T
T (^) T T T T
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^38)
cf. Dynamic Programming
V ( st ) ← E π { rt (^) + 1 + γ V ( st )}
T
T T^ T
st
rt + 1
st + 1
T
T T
T
T T
T
T
T
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^39)
TD Bootstraps and Samples
Bootstrapping: update involves an estimate
MC does not bootstrap DP bootstraps TD bootstraps
Sampling: update does not involve an
expected value
MC samples DP does not sample TD samples
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction (^40)
Advantages of TD Learning
TD methods do not require a model of the environment, only experience TD, but not MC, methods can be fully incremental You can learn before knowing the final outcome
- Less memory
- Less peak computation You can learn without the final outcome
- From incomplete sequences Both MC and TD converge (under certain assumptions to be detailed later), but which is faster?
