Reinforcement Learning I: The setting and classical

stochastic dynamic programming algorithms

Tuomas SandholmCarnegie Mellon University

Computer Science Department

Reinforcement Learning(Ch. 17.1-17.3, Ch. 20)

Learnerpassive

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 to2. 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 transitionsaijM is the probability to reaching state j when taking action

a in state i

start

3

2

1

1 2 3 4

+1

-1

A simple environment that presents the agent with a sequential decision problem:

Move cost = 0.04

(Temporal) credit assignment problem sparse reinforcement problem

Offline alg: action sequences determined ex anteOnline 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 0.2 in perpendicular

0.1 left0.1 right

Policy: mapping from states to actions

3

2

1

1 2 3 4

+1

-1

0.705

3

2

1

1 2 3 4

+1

-1

0.812

0.762

0.868 0.912

0.660

0.655 0.611 0.388

An optimal policy for the stochastic environment:

utilities of states:

EnvironmentObservable (accessible): percept identifies the statePartially 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) = j

aij

ajUM )(maxarg

U(i) = R(i) + j

aij

ajUM )(max

Classic Dynamic Programming (DP)

Start from last step & move backward

Complexity of Naïve search O(|A|n)

DP O(n|A||S|)Actions per step

# 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

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

j

tiPolicy

ijt jUMiRiU )()()( )(1

Value Determination AlgorithmThe 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:

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.

j

tiP

ij jUMiRiU )()()( )(

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

i

iiRvHU )(

Finite if 10 v

Reinforcement Learning II:Reinforcement learning (RL)

algorithms(we will focus solely on observable

environments in this lecture)

Tuomas SandholmCarnegie Mellon University

Computer Science Department

Passive learning

(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

Epochs = training sequences:

Passive learning …

start

3

2

1

1 2 3 4

+1

-1

.5

.5 .33

.5

.5 .5

.5 .5

.5

.5

.5 .33

.5

.33.33

.33 .33.33 .5

.33

.33

1.0

1.0+1

-1

-0.2911

3

2

1

1 2 3 4

+1

-1

-0.0380

-0.1646

0.0886 0.2152

-0.4430

-0.0380 -0.5443 -0.7722

(a)

(b)

(c)(a) A simple stochastic environment.

(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 for each ei in percepts (starting at end) do 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

simple average batch mode

Converges slowly to LMS estimate or training set

But utilities of states are not independent!

NEWU = ?

OLDU = -0.8

P=0.9

P=0.1

-1

+1

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

j

ij jUMiRiU )()()(

= value determination.No maximization over actions because agent is passive unlike in value iteration.

using DP

Large state spacee.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.

)]()()([)()( iUjUiRiUiU

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.

km

mmmkm iUiUiRiUiU )]()()([)()( 1

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.

)(

)(

2 t

t

ti

ti

In practice, often a fixed is used for all i and t.

Passive learning in an unknown environment

aijM unknown

ADP does not work directlyLMS & TD() will operate unchanged

… Changes to ADPConstruct an environment model (of ) based on observations (state transitions) & run DP

aijM

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 actingApproximating <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:

TD(): Unchanged!

j

aij

ajUMiRiU )(max)()(

aijM ijMLearn instead of as before

Model-based (learn M)

Model-free (e.g. Q-learning)

Which is better? open

Tradeoff

Q-learningQ (a,i)

),(max)( iaQiUa

j

a

aij jaQMiRiaQ ),'(max)(),(

'

Direct approach (ADP) would require learning a model .

Q-learning does not:

aijM

)],(),'(max)([),(),('

iaQjaQiRiaQiaQa

Do this update after each state transition:

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 a 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)

2. E.g. in TD() or Q-learning:Choose best action w.p. p and a random action otherwise.

3. E.g. in TD() or Q-learning:Boltzmann exploration

j

aij

aiaNjUMfiRiU )],(),([max)()( Exploration fn. e.g.

),( nufR+ if n<Nu o.w.

a

T

jUM

T

jUM

aj

aij

j

aij

e

eP

)(

)(

*

*

Reinforcement Learning III:Advanced topics

Tuomas SandholmCarnegie Mellon University

Computer Science Department

GeneralizationWith table lookup representation (of U,M,R,Q) up to 10,000 states or more

Chess ~ 10120 Backgammon ~ 1050Industrial problems

Hard to represent & visit all states!

Implicit representation, e.g. U(i) = w1f1(i) + w2f2(i) + …+ wnfn(i)Chess 10120 states n weightsThis 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

generalizationestimate

(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-learningtabular function approximation

state aggregationconverges to Q* general

diverges

averagersconverges to Q*

converges to Q*linear

on-policyoff-policy

converges to Q

prediction control

chatters, boundunknown

diverges

v

jQiQ dd

1

)()(maxQin error class samein ji,

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

• Expert labeled examples are scarce, expensive & possibly wrong• Self-play is cheap & teaches the real solution• Hand-crafted features help

Performance against Gammontool

# hidden units

TD-Gammon (self-play)

Neurogammon (15,000 supervised learning examples)

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)

Q-storage ExplorerRandomProcess

Qcoop

Qdef

deterministic

p(coop)

p(def)

an

Sensation at step n: < >, reward from step n-1

opponentn

men aa 11,

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