Reinfrocement Learning

Learning Through Interaction
Sutton:
“When an infant plays, waves its arms, or looks about, it has no explicit
teacher, but it does have a direct sensorimotor connection to its
environment. Exercising this connection produces a wealth of
information about cause and effect, about the consequences of
actions, and about what to do in order to achieve goals”
• Reinforcement learning is a computational approach for this type of
learning. It adopts AI perspective to model learning through
interaction.

• As a single (agent) approaches the system, it takes an
action. Upon this action he gets a reward and jumps to
the next state. Online learning becomes plausible
3
Reinforcement Learning

Reinforcement Objective
• Learning the relation between the current situation (state) and the
action to be taken in order to optimize a “payment”
Predicting the expected future reward given the current state (s) :
1. Which actions should we take in order to maximize our gain
2. Which actions should we take in order to maximize the click rate
• The action that is taken influences on the next step “closed loop”
• The learner has to discover which action to take (in ML terminology
we can write the feature vector as some features are function of
others)

RL- Elements
• State (s) - The place where the user agent is right now.
Examples:
1. A position on a chess board
2. A potential customer in a sales web
• Action (a)- An action that a user can take while he is in a state.
Examples:
1. Knight pawn captures bishop
2. The user buys a ticket
• Reward (r) - The reward that is obtained due to the action
Examples:
1. A better worse position
2. More money or more clicks

Basic Elements (Cont)
• Policy (π)- The “strategy” in which the agent decides which action to take.
Abstractly speaking the policy is simply a probability function that is defined for
each state
• Episode – A sequence of states and their actions
• 𝑉π
(𝑠) - The value function of a state 𝑠 when using policy π. Mostly it is the
expected reward (e.g. in a chess the expected final outcome of the game if we
follow a given strategy)
• V(s) - Similar to 𝑉π (𝑠) without a fixed policy (The expected reward over all
possible trajectories starting from 𝑠 )
• Q(s,a) - The analog for V(s) : the planar value function for state s and action a

• We wish to find the best slot machine (best = max reward).
Strategy
Play ! .. and find the machine with the biggest reward (on average)
• At the beginning we pick each slot randomly
• After several steps we gain some knowledge
How do we choose which machine to play?
1. Should we always use the best machine ?
2. Should we pick it randomly?
3. Any other mechanism?
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Slot Machines n-armed bandit

• The common trade-off
1. Play always with best machine -Exploitation
We may miss better machines due to statistical “noise”
2. Choose machine randomly - Exploration
We don’t take the optimal machine,
Epsilon Greedy
We exploit in probability (1- ε) and explore with probability ε
Typically ε=0.1
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Exploration ,Exploitation & Epsilon Greedy

• Some problems (like n-bandit) are -Next Best Action.
1. A single given state
2. A set of options that are associated with this state
3. A reward for each action
• Sometimes we wish to learn journeys
Examples:
1. Teach a robot to go from point A to point B
2. Find the fastest way to drive home
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Episodes

• Episode
1. A “time series” of states {S1, S2, S3.. SK}
2. For all state Si There are set of options {O1, O2,..Oki }
3. Learning formula (the “gradient”) depends not only on the immediate
rewards but on the next state as well
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Episode (Cont.)

• The observed sequence:
st ,at , Rt+1, st+1 ,at+1 , Rt+2 ,………….., sT ,aT , RT+1 , s-state , a-action, r-reward
• We optimize our goal function (commonly maximizing the average):
Gt = Rt+1 +γRt+2 +γ2 Rt+3 + …… + γ𝑙Rt+l+1 0< γ ≤ 1 –aging factor
Classical Example
The Pole Balancing
Find the exact force to implement
in order to keep the pole up
The reward is 1 for every time step that
The pole didn’t fall
Reinforcement Learning – Foundation

Markov Property
Pr{ St+1 = s’, Rt+1 = r | S0, A0, R1, . . . , St-1, At-1, Rt , St , At }= Pr{ St+1 = s’, Rt+1 = r | St , At }
i.e. : The current state captures the entire history
• Markov processes are fully determined by the transition matrix P
Markov Process (or Markov Chain)
A tuple <S,P> where
S - set of states (mostly finite),
P a state transition probability matrix. Namely: Pss’= P [St+1 = s’ | St = s]
Markov Decision Process -MDP

A Markov Reward Process -MRP (Markov Chain with Values)
A tuple < S,P, R, γ>
S ,P as in Markov process,
R a reward function Rs = E [Rt+1 | St = s]
γ is a discount factor, γ ∈ [0, 1] (as in Gt )
State Value Function for MRP:
v(s) = E [Gt | St = s]
MDP-cont

Bellman Eq.
• v(s) = E [Gt | St = s] = E [Rt+1 + γRt+2 +γ2 Rt+3 +... | St = s]=
E [Rt+1 + γ (Rt+2 + γRt+3+ ...) | St = s] = E [R t+1 + γG t+1 | S t = s ]
We get a recursion rule:
v(s) = E[Rt+1 + γ v(s t+1) | St = s]
Similalry we can define on a value on state-action space:
Q(s,a)= E [Gt | St = s, At =a]
MDP - MRP with a finite set of actions A
MDP-cont

• Recall - policy π is the strategy – it maps between states and actions.
π(a|s) = P [At = a | St = s]
We assume that for each time t ,and state S π( | St) is fixed (π is stationary )
Clearly for a MDP, a given policy π modifies the MDP:
R -> Rπ P->Pπ
We modify V & Q
Vπ(s) = Eπ [G t | S t = s]
Qπ(s,a) = Eπ [G t | S t = s, At =a]
Policy

• For V (or Q) the optimal value function v* ,for each state s :
v*(s) = max
π
vπ(s) π -policy
Solving MDP ≡ Finding the optimal value function!!!!
Optimal Policy
π ≥ π’ if vπ(s) ≥ v π’(s) ∀s
Theorem
For every MDP there exists optimal policy
Optimal Value Function

• If we know 𝑞∗ (s,a) we can find the optimal policy:
Optimal Value (Cont)

• Dynamic programming
• Monte Carlo
• TD methods
Objectives
Prediction - Find the optimal function
Control – Find the optimal policy
Solution Methods

• A class of algorithms used in many applications such as graph theory
(shortest path) and bio informatics. It has two essential properties:
1. Can be decomposed to sub solutions
2. Solutions can be cashed and reused
RL-MDP satisfies these both
• We assume a full knowledge of MDP !!!
Prediction
Input: MDP and policy
Output: Optimal Value function vπ
Control
Input: MDP
Output: Optimal Value function v* Optimal policy π *
Dynamic Programming

• Assume policy π and MDP we wish to find the optima V π(s)
V π(s) = Eπ [Rt+1 + γvπ(St+1) | St =s]
• Since policy and MDP are known it is a linear eq. in vi
but…. Extremely tedious !!!! Let’s do something iterative (Sutton &Barto)
Prediction – Optimal Value Function

• Following the previous algorithm one can use an algorithm (often a greedy algorithm) to improve the
policy which will lead to an optimal function
Policy Improvement (policy iteration)

• Policy iteration requires policy updating which can be heavy.
• We can study 𝑉∗ and obtain the policy through
• The idea is that
• Hence we can find 𝑉∗ iteratively (and derive the optimal policy)
Value Iteration

• The formula supports online update
• Bootstrapping
• Mostly we don’t have MDP
DP -Remarks

• A model free (we don’t need MDP)
1. It learns from generating episodes.
2. It must complete an episode for having the required average.
3. It is unbiased
• For a policy π
S0,A0,R1….. St ~ π
We use empirical mean return rather expected return.
V(St) =V(St) +
1
𝑁(𝑡)
[ Gt –V(St ) ] N(t) – Amount of visits at time t
For non-stationary cases we update differently:
V(St) =V(St) +α [ Gt –V(St ) ]
In MC one must terminate the episode to get the value (we calculate the mean
explicitly ) Hence in grid problems it may work bad
Monte Carlo Methods

• Learn the optimal policy (using Q function):
Monte Carlo Control

Temporal Difference –TD
• Motivation –Combining DP & MC
As MC -Learning from experience , no explicit MDP
As DP- Bootstrapping, no need to complete the episodes
Prediction
Recall that for MC we have
Where Gt is known only at the end of the episode.

TD –Methods (Cont.)
• TD method needs to wait until the next step (TD(0))
We can see that it leads to different targets:
MC- Gt
TD - Rt+1 + γ V(S t+1)
• Hence it is a Bootstrapping method
The estimaion of V given a policy is straightforwad since the policy
chooses S t+1.

TD Vs. MC -Summary
MC
• High variance unbiased
• Good convergence
• Easy to understand
• Low sensitivity for i.c
TD
• Efficiency
• Convergence to V π
• More sensitive to i.c.

SARSA
• On Policy method for Qlearning (update after every step):
The next step is using SARSA to develop also a control algorithm, we
learn on policy the Q function and update the policy toward
greedyness

Qlearning –Off Policy
• Rather learning from an action that has been offered we simply take
the best action for the state
The control algorithm is straightforward

Value Function Approx.
• Sometimes we have a large scale RL
1. TD backgammon (Appendix)
2. GO – (Deep Mind)
3. Helicopter (continuous)
• Our objectives are still :control & predictions but we have huge
amount of states.
• The tabular solutions that we presented are not scalable.
• Value Function approx. will allow us to use models!!!

Value Function (Cont)
• Consider a large (continuous ) MDP
Vπ (s)= 𝑉′
π (s,w)
Qπ (s,a) =𝑄′
π (s,a,w) w –set of function parameters
• We can train them by both TD & MC .
• We can expand values to unseen states

Type of Approximations
1. Linear Combinations
2. Neural networks (lead to DQN)
3. Wavelet solutions

Function Approximation on the technics
• Define features vectors (X(S)) for the state S. e.g.
Distance from target
Trend in stock
Chess board configuration
• Training methods for W
• SGD
Linear Function get the form: 𝑉′
π =<X(S),W>

RL -Based problems
• No supervisor, only rewards solutions become:

Deep -RL
Why using Deep RL?
• It allows us to find an optimal model (value/policy)
• It allows us to optimize a model
• Commonly we will use SGD
Examples
• Automatic cars
• Atari
• Deep Mind
• TD- Gammon

Q – network
• We follow the value function approx. approach
Q(s,a,w)≈𝑄∗(s,a)

Q-Learning
• We will simply follow TD target function with supervised manners:
Target
r+ γmax
𝑎′
𝑄(𝑠′, 𝑎′, 𝑤)
Loss -MSE
(r+ γ max
𝑎′
𝑄(𝑠′, 𝑎′, 𝑤) −Q(s,a,w) )2
• We solve it with SGD

Q Network –Stability Issues
Divergences
• Correlation between successive samples ,non-iid
• Policy is not necessarily stationary (influences on Q value)
• Scale of rewards and Q value is unknown

Deep –Q network
Experience Replay
Replay the data from the past with the current W
It allows to remove correlation in data:
• Pick at upon a greedy algorithm
• Store in memory the tuple(st, at, rt+1, st+1 ) - Replay
• Now calculate the MSE

DQN (Cont)
Fixed Target Q- Network
In order to handle oscillations
We calculate targets with respect to old parameters 𝑤−
r+ γ max
𝑎′
𝑄(𝑠′, 𝑎′, 𝑤− )
The loss becomes
(r+ γ max
𝑎′
𝑄(𝑠′
, 𝑎′
, 𝑤−
) −Q(s,a,w) )2
𝑤−
<- w

DQN –Summary
Many further methods:
• RewardValue
• Double DQN
• Parallel Updates
Requires another lecture

Gradient Policy
• We have discussed:
1. Function approximations
2. Algorithms in which policy is learned through the value functions
We can parametrize policy using parameters θ :
πθ (s, a) =P[a| s, θ]
Remark: we focus on model free!!

Policy Based Good & Bad
Good
Better in High dimensions
Convergence faster
Bad
Less efficient for high variance
Local minima
Example: Rock-Paper-Scissors

How to optimize a policy?
• We assume it is differentiable and calculate the log-likelihood
• We assume further Gibbs distribution i.e.
policy exponent in value function
πθ (s, a) α 𝑒−θΦ(𝑠,𝑎)
Deriving by θ implies:
We can also use Gaussian policy

Optimize policy (Cont.)
Actor-Critic
Critic – Update the action-state function by w
Actor –Update the policy θ upon the critic suggestion

• Rather Learning value functions we learn probabilities. Let At the action
that is taken at time t
Pr(At =a) = πt (a) =
𝑒Ht (a)
𝑏=1
𝑘
𝑒Ht (b)
H – Numerical Preference
We assume Gibbs Boltzmann Distribution
R¯t - The average until time t
Rt - The reward at time t
Ht+1(At) = Ht(At) + α (Rt − R¯t )(1 − πt(At) )
Ht+1(a) = Ht(a) − α (Rt − R¯t ) πt(a) ∀a ≠ At
Gradient Bandit algorithm

Further Reading
• Sutton & Barto
http://ufal.mff.cuni.cz/~straka/courses/npfl114/2016/sutton-
bookdraft2016sep.pdf
Pole balancing - https://www.youtube.com/watch?v=Lt-KLtkDlh8
• DeepMind papers
• David Silver –Youtube and ucl.ac.uk
• TD-Backgammon

Reinfrocement Learning

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