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## Reinforcement Learning: A Tutorial

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Outline

Outline

- Core RL Theory
- Value functions
- Markov decision processes
- Temporal-Difference Learning
- Policy Iteration
- The Space of RL Methods
- Function Approximation
- Traces
- Kinds of Backups
- Lessons for Artificial Intelligence
- Concluding Remarks

RL is Learning from Interaction

• complete agent

• temporally situated

• continual learning & planning

• object is to affect environment

• environment stochastic & uncertain

Environment

action

state

reward

Agent

Ball-Ballancing Demo

• RL applied to a real physical system

• Illustrates online learning

• An easy problem...but learns as you watch!

• Courtesy of GTE Labs:

Hamid Benbrahim

Judy Franklin

John Doleac

Oliver Selfridge

• Same learning algorithm as early pole-balancer

Markov Decision Processes (MDPs)

In RL, the environment is a modeled as an MDP,

defined by

S – set of states of the environment

A(s) – set of actions possible in state s S

P(s,s',a) – probability of transition from s to s' given a

R(s,s',a)– expected reward on transition s to s' given a

– discount rate for delayed reward

discrete time, t = 0, 1, 2, . . .

r

r

r

. . .

. . .

t +1

t +2

s

s

t +3

s

s

t +1

t +2

t +3

a

a

a

a

t

t +1

t +2

t

t +3

The Objective is to Maximize Long-term Total Discounted Reward

Find a policy sS aA(s)(could be stochastic)

that maximizes thevalue (expected future reward) of each s :

and each s,apair:

V (s) = E{r+ r+r+ s =s, }

2

. . .

t +1t +2t +3 t

rewards

Q (s,a) = E{r+ r+r+ s =s, a =a, }

2

. . .

t +1t +2t +3 t t

These are called value functions - cf. evaluation functions

Optimal Value Functions and Policies

There exist optimal value functions:

And corresponding optimal policies:

* is the greedy policy with respect toQ*

1-Step Tabular Q-Learning

On each state transition:

Update:

a table

entry

TD Error

Optimal behavior found without a model of the environment!

(Watkins, 1989)

Assumes finite MDP

Summary of RL Theory:Generalized Policy Iteration

policy

evaluation

value

learning

Value

Function

Policy

policy

improvement

greedification

Outline

- Core RL Theory
- Value functions
- Markov decision processes
- Temporal-Difference Learning
- Policy Iteration
- The Space of RL Methods
- Function Approximation
- Traces
- Kinds of Backups
- Lessons for Artificial Intelligence
- Concluding Remarks

Holland

Sutton

Sarsa BackupOn transition:

Update:

TD Error

The value of this pair

Backup!

Is updated from the value of the next pair

Dimensions of RL algorithms

1. Action Selection and Exploration

2. Function Approximation

3. Eligibility Trace

4. On-line Experience vs Simulated Experience

5. Kind of Backups

.

.

.

1. Action Selection (Exploration)

Ok, you've learned a value function,

How do you pick Actions?

-Greedy Action Selection:

Be greedy most of the time

Occasionally take a random action

Surprisingly, this is the state of the art.

Greedy Action Selection:

Always select the action that looks best:

Exploitation

vs

Exploration!

2. Function Approximation

Could be:

- table
- Backprop Neural Network
- Radial-Basis-Function Network
- Tile Coding (CMAC)
- Nearest Neighbor, Memory Based
- Decision Tree

Function

Approximator

targets or errors

gradient-

descent

methods

Adjusting Network Weights

weight vector

standard

backprop

gradient

e.g., gradient-descent Sarsa:

estimated value

target value

Sparse Coarse Coding

.

.

.

.

Linear

last

layer

.

fixed expansive

Re-representation

.

.

.

.

.

.

features

Coarse: Large receptive fields

Sparse: Few features present at one time

The Mountain Car Problem

Moore, 1990

SITUATIONS: car's position and velocity

ACTIONS: three thrusts: forward, reverse, none

REWARDS: always –1 until car reaches the goal

No Discounting

Goal

Gravity wins

Minimum-Time-to-Goal Problem

RL Applied to the Mountain Car

1. TD Method = Sarsa Backup

2. Action Selection = Greedy

with initial optimistic values

3. Function approximator = CMAC

4. Eligibility traces = Replacing

Video Demo

(Sutton, 1995)

The Acrobot Problem

e.g., Dejong & Spong, 1994

Sutton, 1995

Minimum–Time–to–Goal:

4 state variables:

2 joint angles

2 angular velocities

CMAC of 48 layers

RL same as Mountain Car

3. Eligibility Traces (*)

- The in TD(), Q(), Sarsa() . . .
- A more primitive way of dealing with delayed reward
- The link to Monte Carlo methods
- Passing credit back farther than one action
- The general form is:

specific to

global

~

change in

at time t

(TD error) • (eligibility of at time t )

at time t

e.g., was taken at t ?

or shortly before t ?

Complex TD Backups (*)

primitive TD backups

trial

e.g. TD ( )

Incrementally computes a weighted mixture of these backups as states are visited

1–

(1–)

= 0 Simple TD

= 1 Simple Monte Carlo

2

(1–)

= 1

3

Dynamic

search

programming

full

backups

sample

Temporal-

backups

difference

learning

l

bootstrapping,

shallow

deep

backups

backups

Kinds of BackupsMonte Carlo

Outline

- Core RL Theory
- Value functions
- Markov decision processes
- Temporal-Difference Learning
- Policy Iteration
- The Space of RL Methods
- Function Approximation
- Traces
- Kinds of Backups
- Lessons for Artificial Intelligence
- Concluding Remarks

World-Class Applications of RL

• TD-Gammon and Jellyfish Tesauro, Dahl

World's best backgammon player

• Elevator Control Crites & Barto

World's best down-peak elevator controller

• Inventory Management Van Roy, Bertsekas, Lee & Tsitsiklis

10-15% improvement over industry standard methods

• Dynamic Channel Assignment Singh & Bertsekas, Nie & Haykin

World's best assigner of radio channels to mobile telephone calls

Lesson #1:Approximate the Solution, Not the Problem

- All these applications are large, stochastic, optimal control problems
- Too hard for conventional methods to solve exactly
- Require problem to be simplified
- RL just finds an approximate solution
- . . . which can be much better!

Backgammon

Tesauro, 1992,1995

SITUATIONS: configurations of the playing board (about 10 )

ACTIONS: moves

REWARDS: win: +1

lose: –1

else: 0

20

Pure delayed reward

TD-Gammon

Start with a random network

Play millions of games against self

Learn a value function from this simulated experience

Action selection

by 2–3 ply search

Value

TD error

This produces arguably the best player in the world

Lesson #2:The Power of Learning from Experience

TD-Gammon

self-play

Expert examples are expensive and scarce

Experience is cheap and plentiful!

And teaches the real solution

70%

Tesauro, 1992

performance

against

gammontool

Neurogammon

same network, but

trained from 15,000

expert-labeled examples

50%

10

20

40

80

0

# hidden units

Lesson #3:The Central Role of Value Functions

...of modifiable moment-by-moment estimates of how well things are going

All RL methods learn value functions

All state-space planners compute value functions

Both are based on "backing up" value

Recognizing and reacting to the ups and downs of life is an important part of intelligence

interaction

environment

learning

experience

value/policy

planning

simulation

environment

model

Lesson #4:Learning and Planningcan be radically similarHistorically, planning and trial-and-error learning have been seen as opposites

But RL treats both as processing of experience

1-Step Tabular Q-Planning

1. Generate a state, s, and action, a

2. Consult model for next state, s', and reward, r

3. Learn from the experience, s,a,r,s' :

4. go to 1

With function approximation and cleverness in search control (Step 1), this is a viable, perhaps even superior, approach to state-space planning

Lesson #5: Accretive Computation"Solves" Planning Dilemmas

Reactivity/Deliberation dilemma

"solved" simply by not opposing search and memory

Intractability of planning

"solved" by anytime improvement of solution

value/policy

Processes only loosely coupled

Can proceed in parallel, asynchronously

Quality of solution accumulates in value & model memories

acting

planning

direct

RL

model

experience

model

learning

Dyna Algorithm

1. s current state

2. Choose an action, a, and take it

3. Receive next state, s’, and reward, r

4. Apply RL backup to s, a, s’, r

e.g., Q-learning update

5. Update Model(s, a) with s’, r

6. Repeat k times:

- select a previously seen state-action pair s,a

- s’, r Model(s, a)

- Apply RL backup to s, a, s’, r

7. Go to 1

Actions –> Behaviors

MDPs seems too too flat, low-level

Need to learn/plan/design agents at a higher level

at the level of behaviors

rather than just low-level actions

- e.g., open-the-door rather than twitch-muscle-17
- walk-to-work rather than 1-step-forward

- Behaviors are based on entire policies,
- directed toward subgoals
- enable abstraction, hierarchy, modularity, transfer...

Rooms Example

4 rooms

4 hallways

4 unreliable

primitive actions

ROOM

HALLWAYS

up

Fail 33%

left

right

of the time

O

1

down

G?

8 multi-step options

O

G?

(to each room's 2 hallways)

2

Given goal location,

quickly plan shortest route

Goal states are given

All rewards zero

a terminal

value of 1

= .9

Behaviors aremuch like Primitive Actions

• Define a higher level Semi-MDP,

overlaid on the original MDP

• Can be used with same planning methods

- faster planning

- guaranteed correct results

•Interchangeable with primitive actions

• Models and policies can be learned by TD methods

Provide a clear semantics for multi-level, re-usable

knowledge for the general MDP context

Lesson #6: Generality is no impediment to

working with higher-level knowledge

Summary of Lessons

1. Approximate the Solution, Not the Problem

2. The Power of Learning from Experience

3. The Central Role of Value Functions

in finding optimal sequential behavior

4. Learning and Planning can be Radically Similar

5. Accretive Computation "Solves Dilemmas"

6. A General Approach need not be Flat, Low-level

All stem from trying to solve MDPs

- Core RL Theory
- Value functions
- Markov decision processes
- Temporal-Difference Learning
- Policy Iteration
- The Space of RL Methods
- Function Approximation
- Traces
- Kinds of Backups
- Lessons for Artificial Intelligence
- Concluding Remarks

Strands of History of RL

Trial-and-error

learning

Temporal-difference

learning

Optimal control,

value functions

Hamilton (Physics)

1800s

Thorndike ()

1911

Secondary

reinforcement ()

Shannon

Samuel

Minsky

Bellman/Howard (OR)

Holland

Klopf

Witten

Werbos

Barto et al

Sutton

Watkins

Radical Generalityof the RL/MDP problem

Determinism

Goals of achievement

Complete knowledge

Closed world

Unlimited deliberation

Classical

RL/MDPs

General stochastic dynamics

General goals

Uncertainty

Reactive decision-making

Getting the degree of abstraction right

• Actions can be low level (e.g. voltages to motors), high level (e.g., accept a job offer), "mental" (e.g., shift in focus of attention)

• Situations: direct sensor readings, or symbolic descriptions, or "mental" situations (e.g., the situation of being lost)

• An RL agent is not like a whole animal or robot, which consist of many RL agents as well as other components

• The environment is not necessarily unknown to the RL agent, only incompletely controllable

• Reward computation is in the RL agent's environment because the RL agent cannot change it arbitrarily

Some of the Open Problems

• Incomplete state information

• Exploration

• Structured states and actions

• Incorporating prior knowledge

• Using teachers

• Theory of RL with function approximators

• Modular and hierarchical architectures

• Integration with other problem–solving and planning methods

Dimensions of RL Algorithms

• TD --- Monte Carlo ()

• Model --- Model free

• Kind of Function Approximator

• Exploration method

• Discounted --- Undiscounted

• Computed policy --- Stored policy

• On-policy --- Off-policy

• Action values --- State values

• Sample model --- Distribution model

• Discrete actions --- Continuous actions

A Unified View

• There is a space of methods, with tradeoffs and mixtures

- rather than discrete choices

• Value function are key

- all the methods are about learning, computing, or estimating values

- all the methods store a value function

• All methods are based on backups

- different methods just have backups of different

shapes and sizes,

sources and mixtures

Psychology and RL

PAVLOVIAN CONDITIONING

The single most common and important kind of learning

Ubiquitous - from people to paramecia

learning to predict and anticipate, learning affect

TD methods are a compelling match to the data

Most modern, real-time methods are based on Temporal-Difference Learning

INSTRUMENTAL LEARNING (OPERANT CONDITIONING)

"Actions followed by good outcomes are repeated"

Goal-directed learning, The Law of Effect

Biological version of Reinforcement Learning

But careful comparisons to data have not yet been made

Hawkins & Kandel

Hopfield & Tank

Klopf

Moore et al.

Byrne et al.

Sutton & Barto

Malaka

Neuroscience and RL

RL interpretations of brain structures are being explored in several areas:

Basal Ganglia

Dopamine Neurons

Value Systems

Final Summary

• RL is about a problem:

- goal-directed, interactive, real-time decision making

- all hail the problem!

• Value functions seem to be the key to solving it,

and TD methods the key new algorithmic idea

• There are a host of immediate applications

- large-scale stochastic optimal control and scheduling

• There are also a host of scientific problems

• An exciting time

Bibliography

Sutton, R.S., Barto, A.G., Reinforcement Learning: An Introduction. MIT Press 1998.

Kaelbling, L.P., Littman, M.L., and Moore, A.W. "Reinforcement learning: A survey". Journal of Artificial Intelligence Research, 4, 1996.

Special issues on Reinforcement Learning: Machine Learning, 8(3/4),1992, and 22(1/2/3), 1996.

Neuroscience: Schultz, W., Dayan, P., and Montague, P.R. "A neural substrate of prediction and reward". Science, 275:1593--1598, 1997.

Animal Learning: Sutton, R.S. and Barto, A.G. "Time-derivative models of Pavlovian reinforcement". In Gabriel, M. and Moore, J., Eds., Learning and Computational Neuroscience, pp. 497--537. MIT Press, 1990.

See also web pages, e.g., starting from http://www.cs.umass.edu/~rich.

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