04/19/23 1

Reinforcement Learning

Presented by:

Bibhas Chakraborty and Lacey Gunter

04/19/23 2

What is Machine Learning?

A method to learn about some phenomenon from data, when there is little scientific theory (e.g., physical or biological laws) relative to the size of the feature space.

The goal is to make an “intelligent” machine, so that it can make decisions (or, predictions) in an unknown situation.

The science of learning plays a key role in areas like statistics, data mining and artificial intelligence. It also arises in engineering, medicine, psychology and finance.

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Types of Learning

Supervised Learning

- Training data: (X,Y). (features, label)

- Predict Y, minimizing some loss.

- Regression, Classification. Unsupervised Learning

- Training data: X. (features only)

- Find “similar” points in high-dim X-space.

- Clustering.

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Types of Learning (Cont’d)

Reinforcement Learning

- Training data: (S, A, R). (State-Action-Reward)

- Develop an optimal policy (sequence of

decision rules) for the learner so as to

maximize its long-term reward.

- Robotics, Board game playing programs.

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Example of Supervised Learning

Predict the price of a stock in 6 months from now, based on economic data. (Regression)

Predict whether a patient, hospitalized due to a heart attack, will have a second heart attack. The prediction is to be based on demographic, diet and clinical measurements for that patient. (Logistic Regression)

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Example of Supervised Learning

Identify the numbers in a handwritten ZIP code, from a digitized image (pixels). (Classification)

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Example of Unsupervised Learning

From the DNA micro-array data, determine which genes are most “similar” in terms of their expression profiles. (Clustering)

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Examples of Reinforcement Learning

How should a robot behave so as to optimize its “performance”? (Robotics)

How to automate the motion of a helicopter? (Control Theory)

How to make a good chess-playing program?

(Artificial Intelligence)

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History of Reinforcement Learning

Roots in the psychology of animal learning (Thorndike,1911).

Another independent thread was the problem of optimal control, and its solution using dynamic programming (Bellman, 1957).

Idea of temporal difference learning (on-line method), e.g., playing board games (Samuel, 1959).

A major breakthrough was the discovery of Q-learning (Watkins, 1989).

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What is special about RL?

RL is learning how to map states to actions, so as to maximize a numerical reward over time.

Unlike other forms of learning, it is a multistage decision-making process (often Markovian).

An RL agent must learn by trial-and-error. (Not entirely supervised, but interactive)

Actions may affect not only the immediate reward but also subsequent rewards (Delayed effect).

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Elements of RL

A policy - A map from state space to action space. - May be stochastic. A reward function - It maps each state (or, state-action pair) to a real number, called reward. A value function - Value of a state (or, state-action pair) is the total expected reward, starting from that state (or, state-action pair).

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The Precise Goal

To find a policy that maximizes the Value function.

There are different approaches to achieve this goal in various situations.

Q-learning and A-learning are just two different approaches to this problem. But essentially both are temporal-difference methods.

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The Basic Setting

Training data: n finite horizon trajectories, of the form

Deterministic policy: A sequence of decision rules

Each π maps from the observable history (states and actions) to the action space at that time point.

}.,,,,...,,,{ 1000 TTTT srasras

}.,...,,{ 10 T

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Value and Advantage

Time t state value function, for history is

Time t state-action value function, Q-function, is

Time t advantage, A-function, is

.,|),,(),( 1111

ttt

T

tjtjjjjttt SrEV aAsSASas

),( 1tt as

.,|),(),,(),( 111 tttttttttttttt VSrEQ aAsSASASas

).,(),(),( 1 ttttttttt VQ asasas

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Optimal Value and Advantage

Optimal time t value function for history is

Optimal time t Q-function is

Optimal time t A-function is

.,|),,(max),( 1111*

ttt

T

tjtjjjjttt SrEV aAsSASas

),( 1tt as

.,|),(),,(),( 1*

11*

tttttttttttttt VSrEQ aAsSASASas

).,(),(),( 1***

ttttttttt VQ asasas

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Return (sum of the rewards)

The conditional expectation of the return is

where the advantages μ are

and the , called temporal difference errors, are

)(),(),(, 00

1

011

00

SVArET

ttttt

T

ttt

T

tTTt

SASAS

),(),(),( 1 ttttttttt VQ ASASAS

),(),(),(),( 1111111 tttttttttttt QVr ASASASAS

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Return (continued)

Conditional expectation of the return is a telescoping sum

Temporal difference errors have conditional mean zero

)(][][

)(,

00

1

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0

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),(],|),(),([ 11111111 ttttttttttt QVrE ASASASAS

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Q-LearningWatkins,1989

Estimate the Q-function using some approximator (for example, linear regression or neural networks or decision trees etc.).

Derive the estimated policy as an argument of the maximum of the estimated Q-function.

Allow different parameter vectors at different time points.

Let us illustrate the algorithm with linear regression as the approximator, and of course, squared error as the appropriate loss function.

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Q-Learning Algorithm

Set For

The estimated policy satisfies

.01 TQ

.);,(1

minargˆ

).ˆ;,,(max

,0,...,1,

2

1,,,

11111

n

iitittitt

ttttta

tt

QYn

aQrY

TTt

t

AS

AS

.),ˆ;,(maxarg),(ˆ 1, tQ tttta

tttQt

asas

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What is the intuition?

Bellman equation gives

If and the training set were infinite, then Q-learning minimizes

which is equivalent to minimizing

.,|),,(max),( 11*

1*

1

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atttt aQrEQ

t

ASASAS

*11 tt QQ

.)];,([ 2* tttt QQE AS

.);,(),,(max2

11*

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t QaQrEt

ASAS

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A Success Story

TD Gammon (Tesauro, G., 1992)

- A Backgammon playing program.

- Application of temporal difference learning.

- The basic learner is a neural network.

- It trained itself to the world class level by playing against itself and learning from the outcome. So smart!!

- More information: http://www.research.ibm.com/massive/tdl.html

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A-Learning Murphy, 2003 and Robins, 2004

Estimate the A-function (advantages) using some approximator, as in Q-learning.

Derive the estimated policy as an argument of the maximum of the estimated A-function.

Allow different parameter vectors at different time points.

Let us illustrate the algorithm with linear regression as the approximator, and of course, squared error as the appropriate loss function.

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A-Learning Algorithm (Inefficient Version)

For

The estimated policy satisfies

)ˆ;,,(max)ˆ;,()ˆ;,(

.),|();,(1

minargˆ

).ˆ;,(

,0,...,1,

1,

2

1,1,,,,

1

ttttta

tttttttt

n

iitittitittitt

T

tjjjjj

T

tjjt

a

EYn

rY

TTt

t

ASASAS

ASAS

AS

.),ˆ;,(maxarg),(ˆ 1, ttttta

tttAt

asas

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Differences between Q and A-learning

Q-learning At time t we model the main effects of the history,

(St,,At-1) and the action At and their interaction

Our Yt-1 is affected by how we modeled the main

effect of the history in time t, (St,,At-1)

A-learning At time t we only model the effects of At and its

interaction with (St,,At-1)

Our Yt-1 does not depend on a model of the main

effect of the history in time t, (St,,At-1)

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Q-Learning Vs. A-Learning

Relative merits and demerits are not completely known till now.

Q-learning has low variance but high bias.

A-learning has high variance but low bias. Comparison of Q-learning with A-learning

involves a bias-variance trade-off.

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References

Sutton, R.S. and Barto, A.G. (1998). Reinforcement Learning- An Introduction.

Hastie, T., Tibshirani, R. and Friedman, J. (2001). The Elements of Statistical Learning-Data Mining, Inference and Prediction.

Murphy, S.A. (2003). Optimal Dynamic Treatment Regimes. JRSS-B.

Blatt, D., Murphy, S.A. and Zhu, J. (2004).

A-Learning for Approximate Planning. Murphy, S.A. (2004). A Generalization Error for Q-Learning.