Machine Learning

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					          A classification learning example
     Predicting when Rusell will wait for a table




--similar to book preferences, predicting credit card fraud,
  predicting when people are likely to respond to junk mail
                  Inductive Learning
               (Classification Learning)
    •   Given a set of labeled examples,          •   Main variations:
        and a space of hypotheses                 •   Bias: the “sort” of rule are you
         – Find the rule that underlies               looking for?
            the labeling                                    – If you are looking for only
                                                              conjunctive hypotheses,
              • (so you can use it to                         there are just 3n
                 predict future unlabeled         –   Search:
                 examples)                             – Greedy search
         – Tabularasa, fully supervised                     – Decision tree learner
                                                       – Systematic search
    • Idea:
                                                            – Version space learner
         – Loop through all hypotheses                 – Iterative search
              • Rank each hypothesis in                     – Neural net learner
                terms of its match to data
              • Pick the best hypothesis                         It can be shown that
                                                                  sample complexity of PAC
The main problem is that
                                                                  learning is proportional to
 the space of hypotheses is too large
      Given examples described in terms of n boolean variables    1/e, 1/d AND log |H|
                         n
           There are 2 2 different hypotheses
           For 6 features, there are 18,446,744,073,709,551,616 hypotheses
5/5
              Bias & Learning Accuracy
• Having weak bias




                                       Fraction incorectly classified
  (large hypothesis
                                                                        Test (prediction) error
  space)
    – Allows us to capture                                                        Training error

      more concepts
    – ..increases learning
      cost
    – May lead to over-
      fitting
Also the goal of a compression
algorithm is to drive down
the training error
But the goal of a learning algorithm
is to drive down the test error
      Uses different biases in predicting Russel’s waiting habbits
                                                                      Decision Trees
                                                                      --Examples are used to
                                                                        --Learn topology
If patrons=full and day=Friday                     K-nearest            --Order of questions
  then wait (0.3/0.7)                              neighbors
If wait>60 and Reservation=no
  then wait (0.4/0.9)


     Association rules
     --Examples are used to
       --Learn support and
         confidence of association
         rules                                                                                                SVMs
                                                                                               Neural Nets
                                                                                               --Examples are used to
                                                                                                 --Learn topology
                                                                                                 --Learn edge weights
                                                    Naïve bayes
                                                    (bayesnet learning)
                                                    --Examples are used to
                                 Russell waits
                                                      --Learn topology
RW    None   some   full
                                                      --Learn CPTs
T     0.3    0.2    0.5
F     0.4    0.3    0.3


                    Wait time?          Patrons?            Friday?
Learning Decision Trees---How?

                        Basic Idea:
                         --Pick an attribute
                         --Split examples in terms
                               of that attribute
                            --If all examples are +ve
                               label Yes. Terminate
                           --If all examples are –ve
                               label No. Terminate
                           --If some are +ve, some are –ve
                              continue splitting recursively

                        (Special case: Decision Stumps
                         If you don’t feel like splitting
                         any further, return the majority
                         label )




                Which one to pick?
Depending on the order we pick, we can get smaller or bigger trees




      Which tree is better?
       Why do you think so??
Basic Idea:
 --Pick an attribute
 --Split examples in terms
       of that attribute
    --If all examples are +ve
       label Yes. Terminate
   --If all examples are –ve
       label No. Terminate
   --If some are +ve, some are –ve
      continue splitting recursively
        --if no attributes left to split?
          (label with majority element)




           Would you split on
            patrons or Type?
 The Information Gain                                  P+ : N+ /(N++N-)
                                                                                   # expected comparisons
                                                                                   needed to tell whether a
 Computation                                           P- : N- /(N++N-)            given example is +ve or -ve

                                                       I(P+ ,, P-) = -P+ log(P+) - P- log(P- )
                                              N+
                                              N-
                      Splitting on                                                  The difference
                                                                                    is the information
                      feature fk                                                    gain

                            N1+           N2+             Nk+                       So, pick the feature
                            N1-           N2-             Nk-                       with the largest Info Gain
                     I(P1+ ,, P1-)   I(P2+ ,, P2-)   I(Pk+ ,, Pk-)                  I.e. smallest residual info

                                                             k
Given k mutually exclusive and exhaustive
events E1….Ek whose probabilities are                        S       [Ni+ + Ni- ]/[N+ + N-] I(Pi+ ,, Pi-)

p1….pk                                                       i=1
The “information” content (entropy) is defined as

  S i -pi log2 pi
A split is good if it reduces the entropy..
                             A simple example


Ex Masochistic Anxious   Nerdy   HATES
                                 EXAM    V(M) = 2/4 * I(1/2,1/2) + 2/4 * I(1/2,1/2)
1   F          T         F       Y               = 1

2   F          F         T       N

                                         V(A) = 2/4 * I(1,0) + 2/4 * I(0,1)
3   T          F         F       N               = 0
4   T          T         T       Y



                                         V(N) = 2/4 * I(1/2,1/2) + 2/4 * I(1/2,1/2)
                                                 = 1



                   So Anxious is the best attribute to split on
                   Once you split on Anxious, the problem is solved
m-fold cross-validation
 Split N examples into        Evaluating the Decision Trees
    m equal sized parts
for i=1..m train with
  all parts except ith
                                                                        Lesson:
  test with the ith part                                                Every bias makes
                                                                         some concepts easier to
                                                                         learn and others
                                                                         harder to learn…




                                                                           “Majority” function
                         Russell Domain                                    (say yes if majority of
                                                                            attributes are yes)




Learning curves… Given N examples, partition them into Ntr the training set and Ntest the test instances
 Loop for i=1 to |Ntr|
  Loop for Ns in subsets of Ntr of size I
    Train the learner over Ns
    Test the learned pattern over Ntest and compute the accuracy (%correct)
Problems with Info. Gain. Heuristics

 • Feature correlation: We are splitting on one feature at a time
          • The Costanza party problem
     – No obvious easy solution…
 • Overfitting: We may look too hard for patterns where there are none
    – E.g. Coin tosses classified by the day of the week, the shirt I was
      wearing, the time of the day etc.
    – Solution: Don’t consider splitting if the information gain given by
      the best feature is below a minimum threshold
          • Can use the c2 test for statistical significance
   – Will also help when we have noisy samples…
 • We may prefer features with very high branching
     – e.g. Branch on the “universal time string” for Russell restaurant example
     –      Branch on social security number to look for patterns on who will get A
     – Solution: “gain ratio” --ratio of information gain with the attribute A to the
       information content of answering the question “What is the value of A?”
          • The denominator is smaller for attributes with smaller domains.
                     Decision Stumps
• Decision stumps are decision
  trees where the leaf nodes do not
  necessarily have all +ve or all –                          N+
  ve training examples                                       N-
    – Could happen either because           Splitting on
      examples are noisy and mis-           feature fk
      classified or because you want to
      stop before reaching pure leafs            N1+       N2+     Nk+
• When you reach that node, you                  N1-       N2-     Nk-
  return the majority label as the
  decision.
        • (We can associate a confidence       P+= N1+ / N1++N1-
          with that decision using the P+
          and P-)


  Sometimes, the best decision tree for a problem
   could be a decision stump (see coin toss example next)
        Bayes Network Learning
• Bias: The relation between the class label and class
  attributes is specified by a Bayes Network.
• Approach                              RW   None   some   full
                                                                        Russell waits


   – Guess Topology                     T    0.3    0.2    0.5

   – Estimate CPTs                      F    0.4    0.3    0.3



• Simplest case: Naïve Bayes                               Wait time?          Patrons?   Friday?


   – Topology of the network is “class label” causes all the attribute
     values independently
   – So, all we need to do is estimate CPTs P(attrib|Class)
       • In Russell domain, P(Patrons|willwait)
            – P(Patrons=full|willwait=yes)=
              #training examples where patrons=full and will wait=yes
              #training examples where will wait=yes
   – Given a new case, we use bayes rule to compute the class label

           Class label is the disease; attributes are symptoms
  Naïve Bayesian Classification
• Problem: Classify a given example E into one of the
  classes among [C1, C2 ,…, Cn]
   – E has k attributes A1, A2 ,…, Ak and each Ai can take d
     different values
• Bayes Classification: Assign E to class Ci that
  maximizes P(Ci | E)
     P(Ci| E) = P(E| Ci) P(Ci) / P(E)
      • P(Ci) and P(E) are a priori knowledge (or can be easily extracted
        from the set of data)
• Estimating P(E|Ci) is harder
   – Requires P(A1=v1 A2=v2….Ak=vk|Ci)
      • Assuming d values per attribute, we will need ndk probabilities
• Naïve Bayes Assumption: Assume all attributes are
  independent P(E| Ci) = P P(Ai=vj | Ci )
   – The assumption is BOGUS, but it seems to WORK (and
     needs only n*d*k probabilities
NBC in terms of BAYES networks..




 NBC assumption
                   More realistic assumption
Estimating the probabilities for NBC
Given an example E described as A1=v1 A2=v2….Ak=vk we want to
   compute the class of E

    – Calculate P(Ci | A1=v1 A2=v2….Ak=vk) for all classes Ci and say
      that the class of E is the one for which P(.) is maximum
    – P(Ci | A1=v1 A2=v2….Ak=vk)                                 Common factor
            = P P(vj | Ci ) P(Ci) / P(A1=v1 A2=v2….Ak=vk)


Given a set of training N examples that have already been classified into n
   classes Ci
      Let #(Ci) be the number of examples that are labeled as Ci
      Let #(Ci, Ai=vi) be the number of examples labeled as Ci
          that have attribute Ai set to value vj
          P(Ci) = #(Ci)/N
          P(Ai=vj | Ci) = #(Ci, Ai=vi) / #(Ci)



                          USER PROFILE
 Example




P(willwait=yes) = 6/12 = .5                            Similarly we can show that
P(Patrons=“full”|willwait=yes) = 2/6=0.333             P(Patrons=“full”|willwait=no)
P(Patrons=“some”|willwait=yes)= 4/6=0.666                     =0.6666

   P(willwait=yes|Patrons=full) = P(patrons=full|willwait=yes) * P(willwait=yes)
                                   -----------------------------------------------------------
                                               P(Patrons=full)
                            = k* .333*.5
   P(willwait=no|Patrons=full) = k* 0.666*.5
       Using M-estimates to improve
           probablity estimates
  • The simple frequency based estimation of P(Ai=vj|Ck) can be
    inaccurate, especially when the true value is close to zero, and
    the number of training examples is small (so the probability that
    your examples don’t contain rare cases is quite high)
  • Solution: Use M-estimate
          P(Ai=vj | Ci) = [#(Ci, Ai=vi) + mp ] / [#(Ci) + m]
       – p is the prior probability of Ai taking the value vi
           • If we don’t have any background information, assume uniform
             probability (that is 1/d if Ai can take d values)
       – m is a constant—called ―equivalent sample size‖
           • If we believe that our sample set is large enough, we can keep m small.
             Otherwise, keep it large.
           • Essentially we are augmenting the #(Ci) normal samples with m more
             virtual samples drawn according to the prior probability on how Ai takes
             values
                – Popular values p=1/|V| and m=|V| where V is the size of the vocabulary



Also, to avoid overflow errors do addition of logarithms of probabilities
(instead of multiplication of probabilities)
        How Well (and WHY) DOES NBC
                    WORK?
• Naïve bayes classifier is darned easy to implement
    – Good learning speed, classification speed
    – Modest space storage
    – Supports incrementality
• It seems to work very well in many scenarios
    – Lots of recommender systems (e.g. Amazon books recommender) use it
    – Peter Norvig, the director of Machine Learning at GOOGLE said, when
      asked about what sort of technology they use “Naïve bayes”
• But WHY?
    – NBC’s estimate of class probability is quite bad
        •   BUT classification accuracy is different from probability estimate accuracy
    – [Domingoes/Pazzani; 1996] analyze this
      Uses different biases in predicting Russel’s waiting habbits
                                                                      Decision Trees
                                                                      --Examples are used to
                                                                        --Learn topology
If patrons=full and day=Friday                     K-nearest            --Order of questions
  then wait (0.3/0.7)                              neighbors
If wait>60 and Reservation=no
  then wait (0.4/0.9)


     Association rules
     --Examples are used to
       --Learn support and
         confidence of association
         rules                                                                                                SVMs
                                                                                               Neural Nets
                                                                                               --Examples are used to
                                                                                                 --Learn topology
                                                                                                 --Learn edge weights
                                                    Naïve bayes
                                                    (bayesnet learning)
                                                    --Examples are used to
                                 Russell waits
                                                      --Learn topology
RW    None   some   full
                                                      --Learn CPTs
T     0.3    0.2    0.5
F     0.4    0.3    0.3


                    Wait time?          Patrons?            Friday?
       Decision Surface Learning
     (aka Neural Network Learning)
• Idea: Since classification is
  really a question of finding
  a surface to separate the
  +ve examples from the -ve
  examples, why not directly
  search in the space of
  possible surfaces?
• Mathematically, a surface
  is a function
    – Need a way of learning
      functions
    – “Threshold units”
“Neural Net” is a collection of threshold units
with interconnections
     I1
          w1         = 1 if w1I1+w2I2 > k                                     differentiable
                     = 0 otherwise

               t=k
    I2
          w2
                                  Feed Forward                          Recurrent
                              Uni-directional connections                Bi-directional
 Single Layer                                        Multi-Layer         connections




                                              Any “continuous”
  Any linear decision                         decision surface           Can act as
  surface can be represented                  (function) can be          associative
  by a single layer neural net                approximated to any        memory
                                              degree of accuracy by
                                              some 2-layer neural net
           The “Brain” Connection
A Threshold Unit               Threshold Functions



                                             differentiable




                            …is sort of like a neuron
Perceptron Networks


                  What happened to the
                  “Threshold”?
                   --Can model as an extra
                      weight with static input
                     I1
                        w1

                                  t=k
                      I2
                            w2
                                  ==
                                   I0=-1

                           w1      w0= k

                                 t=0

                           w2
Perceptron Learning
• Perceptron learning algorithm
Loop through training examples
   – If the activation level of the output unit is
       1 when it should be 0, reduce the weight
       on the link to the jth input unit by a*Ij,
       where Ii is the ith input value and a a
       learning rate
   – If the activation level of the output unit is
       0 when it should be 1, increase the weight
       on the link to the ith input unit by a*Ij
   – Otherwise, do nothing
Until “convergence”

 A nice applet at:
           http://neuron.eng.wayne.edu/java/Perceptron/New38.html
Perceptron Learning as Gradient
Descent Search in the weight-space

          1
     E      (T  O) 2
          2 i
                                               2
                1                      
     E (W )          T  g  W j I j  
                                       
                2 i           j                              1
                                                    g ( x)                   ( sigmoid fn )
      E                                                    1  ex
            I j (T  O) g   W j I j 
                                        
     W j                      
                               j                    g ' ( x)  g ( x)(1  g ( x))
                                 


                                                        Often a constant
                                           
     W j  W j  I j (T  O) g   W j I j 
                                                         learning rate parameter
                                                        is used instead
                             
                                  j
                                         



                                                   Ij
                                                                                               I
Can Perceptrons Learn All Boolean Functions?
       --Are all boolean functions linearly separable?
                   Comparing Perceptrons and Decision Trees
                    in Majority Function and Russell Domain




              Majority function                                          Russell Domain




Majority function is linearly seperable..         Russell domain is apparently not....

                                            Encoding: one input unit per attribute. The unit takes as many
                                            distinct real values as the size of attribute domain
        Max-Margin Classification &
         Support Vector Machines
•   Any line that separates the +ve &
    –ve examples is a solution
•   And perceptron learning finds one
    of them
     – But could we have a preference
       among these?
     – may want to get the line that
       provides maximum margin
       (equidistant from the nearest +ve/-
       ve)
          • The nereast +ve and –ve holding
            up the line are called support
            vectors
•   This changes the problem into an
    optimization one
     – Quadratic Programming can be
       used to directly find such a line



                        Learning is Optimization after all!
Lagrangian Dual
      Two ways to learn non-linear
           decision surfaces
• First transform the data     • Learn non-linear surfaces
  into higher dimensional        directly (as multi-layer
  space                          neural nets)
• Find a linear surface        • Trick is to do training
   – Which is guaranteed to      efficiently
     exist                        – Back Propagation to the
• Transform it back to the          rescue..
  original space
• TRICK is to do this
  without explicitly doing a
  transformation
Linear Separability in High Dimensions




 “Kernels” allow us to consider separating surfaces in high-D
   without first converting all points to high-D
Kernelized Support Vector Machines
•   Turns out that it is not always
    necessary to first map the data into
    high-D, and then do linear separation
•   The quadratic programming
    formulation for SVM winds up using
    only the pair-wise dot product of
    training vectors
•   Dot product is a form of similarity
    metric between points
•   If you replace that dot product by
    any non-linear function, you will, in
    essence, be transforming data into
    some high-dimensional space and
    then finding the max-margin linear
    classifier in that space
     –   Which will correspond to some
         wiggly surface in the original
         dimension
•   The trick is to find the RIGHT
    similarity function
     –   Which is a form of prior knowledge
Kernelized Support Vector Machines
•   Turns out that it is not always
    necessary to first map the data into
    high-D, and then do linear separation
•   The quadratic programming
    formulation for SVM winds up using
    only the pair-wise dot product of
    training vectors
•   Dot product is a form of similarity
    metric between points
•   If you replace that dot product by
    any non-linear function, you will, in
    essence, be tranforming data into                                                        0
    some high-dimensional space and           Polynomial Kernel: K (A ; A ) = (( 100 à 1)( 100 à 1) à 0:5) ï
                                                                         0        A         A              6


    then finding the max-margin linear
    classifier in that space
     –   Which will correspond to some
         wiggly surface in the original
         dimension
•   The trick is to find the RIGHT
    similarity function
     –   Which is a form of prior knowledge
                    Those who ignore easily available domain knowledge are
                    doomed to re-learn it…
                        Santayana’s brother
 Domain-knowledge & Learning
• Classification learning is a problem addressed by both people
  from AI (machine learning) and Statistics
• Statistics folks tend to ―distrust‖ domain-specific bias.
   – Let the data speak for itself…
   – ..but this is often futile. The very act of ―describing‖ the data points
     introduces bias (in terms of the features you decided to use to
     describe them..)
• …but much human learning occurs because of strong domain-
  specific bias..
• Machine learning is torn by these competing influences..
   – In most current state of the art algorithms, domain knowledge is
     allowed to influence learning only through relatively narrow
     avenues/formats (E.g. through ―kernels‖)
       • Okay in domains where there is very little (if any) prior knowledge (e.g.
         what part of proteins are doing what cellular function)
       • ..restrictive in domains where there already exists human expertise..
Multi-layer Neural Nets




                          How come back-prop
                          doesn’t get stuck in
                           local minima?

                          One answer: It is actually
                           hard for local minimas to
                           form in high-D, as the “trough”
                           has to be closed in all dimensions
Multi-Network Learning can learn Russell Domains




                 Russell Domain



       …but does it slowly…
     Practical Issues in Multi-layer
            network learning
• For multi-layer networks, we need to learn both
  the weights and the network topology
   – Topology is fixed for perceptrons
• If we go with too many layers and connections, we
  can get over-fitting as well as sloooow
  convergence
   – Optimal brain damage
      • Start with more than needed hidden layers as well as
        connections; after a network is learned, remove the nodes and
        connections that have very low weights; retrain
                                           Humans make 0.2%
                                           Neumans (postmen) make 2%


Other impressive applications:
 --no-hands across K-nearest-neighbor
    america             The test example’s class is determined
                        by the class of the majority of its k nearest
 --learning to speak
                             neighbors
                          Need to define an appropriate distance measure
                           --sort of easy for real valued vectors
                           --harder for categorical attributes
True hypothesis eventually dominates…
 probability of indefinitely producing uncharacteristic data 0
Bayesian prediction is optimal
 (Given the hypothesis prior,
   all other predictions are less likely)
Also, remember the Economist article that shows
 that humans have strong priors..
..note that the Economist article says humans are
 able to learn from few examples only because of priors..
So, BN learning is just probability estimation!
  (as long as data is complete!)
        How Well (and WHY) DOES NBC
                    WORK?
• Naïve bayes classifier is darned easy to implement
    – Good learning speed, classification speed
    – Modest space storage
    – Supports incrementality
• It seems to work very well in many scenarios
    – Lots of recommender systems (e.g. Amazon books recommender) use it
    – Peter Norvig, the director of Machine Learning at GOOGLE said, when
      asked about what sort of technology they use “Naïve bayes”
• But WHY?
    – NBC’s estimate of class probability is quite bad
        •   BUT classification accuracy is different from probability estimate accuracy
    – [Domingoes/Pazzani; 1996] analyze this
Reinforcement Learning




Based on slides from Bill Smart
    http://www.cse.wustl.edu/~wds/
             What is RL?

 “a way of programming agents by reward and
punishment without needing to specify how the
             task is to be achieved”

                         [Kaelbling, Littman, & Moore, 96]
                  Basic RL Model

    1.   Observe state, st                 World
    2.   Decide on an action, at
    3.   Perform action
    4.   Observe new state, st+1   S   R                A
    5.   Observe reward, rt+1
    6.   Learn from experience
    7.   Repeat



Goal: Find a control policy that will maximize the observed
rewards over the lifetime of the agent
         An Example: Gridworld
Canonical RL domain                         +1

  •   States are grid cells
  •   4 actions: N, S, E, W
  •   Reward for entering top right cell
  •   -0.01 for every other move


Minimizing sum of rewards  Shortest path
  • In this instance
The Promise of Learning
            The Promise of RL
Specify what to do, but not how to do it
   • Through the reward function
   • Learning “fills in the details”


Better final solutions
   • Based of actual experiences, not programmer
     assumptions


Less (human) time needed for a good solution
       Learning Value Functions
We still want to learn a value function
   • We’re forced to approximate it iteratively
   • Based on direct experience of the world


Four main algorithms
   •   Certainty equivalence
   •   Temporal Difference (TD) learning
   •   Q-learning
   •   SARSA
         Certainty Equivalence
Collect experience by moving through the world
  • s0, a0, r1, s1, a1, r2, s2, a2, r3, s3, a3, r4, s4, a4, r5, s5, ...


Use these to estimate the underlying MDP
  • Transition function, T: SA → S
  • Reward function, R: SAS → 


Compute the optimal value function for this MDP
  • And then compute the optimal policy from it
     Temporal Difference (TD)
                                                [Sutton, 88]


TD-learning estimates the value function directly
  • Don’t try to learn the underlying MDP


Keep an estimate of Vp(s) in a table
  • Update these estimates as we gather more
    experience
  • Estimates depend on exploration policy, p
  • TD is an on-policy method
         TD-Learning Algorithm
Initialize Vp(s) to 0, s
Observe state, s
Perform action, p(s)
Observe new state, s’, and reward, r
Vp(s) ← (1-a)Vp(s) + a(r + gVp(s’))
Go to 2

0 ≤ a ≤ 1 is the learning rate
   •   How much attention do we pay to new experiences
                       TD-Learning
Vp(s) is guaranteed to converge to V*(s)
  • After an infinite number of experiences
  • If we decay the learning rate
                        

       a              a             
                                    2
              t                 t
       t 0              t 0


          c
  • at      will work
         ct

In practice, we often don’t need value convergence
  • Policy convergence generally happens sooner
          Actor-Critic Methods        [Barto, Sutton, & Anderson, 83]


TD only evaluates a particular policy                        Policy
  • Does not learn a better policy                           (actor)
                                                                        a
                                                                  V
                                                             Value
We can change the policy as we learn V                      Function
  • Policy is the actor                                      (critic)

  • Value-function estimate is the critic            s            r
                                                             World

Success is generally dependent on the starting
policy being “good enough”
                 Q-Learning             [Watkins & Dayan, 92]


Q-learning iteratively approximates the state-action
value function, Q
  • Again, we’re not going to estimate the MDP directly
  • Learns the value function and policy simultaneously


Keep an estimate of Q(s, a) in a table
  • Update these estimates as we gather more
    experience
  • Estimates do not depend on exploration policy
  • Q-learning is an off-policy method
          Q-Learning Algorithm
Initialize Q(s, a) to small random values, s, a
Observe state, s
Pick an action, a, and do it
Observe next state, s’, and reward, r
Q(s, a) ← (1 - a)Q(s, a) + a(r + gmaxa’Q(s’, a’))
Go to 2

0 ≤ a ≤ 1 is the learning rate
   •   We need to decay this, just like TD
              Picking Actions
We want to pick good actions most of the time, but
also do some exploration
  • Exploring means that we can learn better policies
  • But, we want to balance known good actions with
    exploratory ones
  • This is called the exploration/exploitation problem
                 Picking Actions
e-greedy
  • Pick best (greedy) action with probability e
  • Otherwise, pick a random action


Boltzmann (Soft-Max)
  • Pick an action based on its Q-value
                       Q(s, a) 
                               
                       t       
                  e
  • P(a | s)            Q(s, a' ) 
                                        , where t is the “temperature”
                                   
                 e
                 a'
                         t         
                       SARSA
SARSA iteratively approximates the state-action
value function, Q
  • Like Q-learning, SARSA learns the policy and the
    value function simultaneously


Keep an estimate of Q(s, a) in a table
  •   Update these estimates based on experiences
  •   Estimates depend on the exploration policy
  •   SARSA is an on-policy method
  •   Policy is derived from current value estimates
             SARSA Algorithm
Initialize Q(s, a) to small random values, s, a
Observe state, s
Pick an action, a, and do it (just like Q-learning)
Observe next state, s’, and reward, r
Q(s, a) ← (1-a)Q(s, a) + a(r + gQ(s’, p(s’)))
Go to 2

0 ≤ a ≤ 1 is the learning rate
   •   We need to decay this, just like TD
       On-Policy vs. Off Policy
On-policy algorithms
  • Final policy is influenced by the exploration policy
  • Generally, the exploration policy needs to be “close”
    to the final policy
  • Can get stuck in local maxima


Off-policy algorithms
  • Final policy is independent of exploration policy
  • Can use arbitrary exploration policies
  • Will not get stuck in local maxima
     Convergence Guarantees
The convergence guarantees for RL are “in the
limit”
   • The word “infinite” crops up several times


Don’t let this put you off
   • Value convergence is different than policy
     convergence
   • We’re more interested in policy convergence
   • If one action is really better than the others, policy
     convergence will happen relatively quickly
                     Rewards
Rewards measure how well the policy is doing
  • Often correspond to events in the world
     • Current load on a machine
     • Reaching the coffee machine
     • Program crashing
  • Everything else gets a 0 reward


Things work better if the rewards are incremental
  • For example, distance to goal at each step
  • These reward functions are often hard to design
         The Markov Property
RL needs a set of states that are Markov
  • Everything you need to know to make a decision is
    included in the state
  • Not allowed to consult the past    Not holding key
                                           Holding key

Rule-of-thumb                          K
  • If you can calculate the reward
    function from the state without
    any additional information,
    you’re OK
                                       S                 G
          But, What’s the Catch?
RL will solve all of your problems, but
   •   We need lots of experience to train from
   •   Taking random actions can be dangerous
   •   It can take a long time to learn
   •   Not all problems fit into the MDP framework
     Learning Policies Directly
An alternative approach to RL is to reward whole
policies, rather than individual actions
  • Run whole policy, then receive a single reward
  • Reward measures success of the whole policy


If there are a small number of policies, we can
exhaustively try them all
  • However, this is not possible in most interesting
    problems
       Policy Gradient Methods
Assume that our policy, p, has a set of n real-
valued parameters, q = {q1, q2, q3, ... , qn }
   • Running the policy with a particular q results in a
     reward, rq
                                     R
   • Estimate the reward gradient,        , for each qi
                                     θ i
                 R
   • θi  θi  a
                 θi

         This is another
          learning rate
      Policy Gradient Methods
This results in hill-climbing in policy space
   • So, it’s subject to all the problems of hill-climbing
   • But, we can also use tricks from search, like random
     restarts and momentum terms


This is a good approach if you have a
parameterized policy
   • Typically faster than value-based methods
   • “Safe” exploration, if you have a good policy
   • Learns locally-best parameters for that policy
An Example: Learning to Walk
                                             [Kohl & Stone, 04]

RoboCup legged league
  • Walking quickly is a big advantage


Robots have a parameterized gait controller
  • 11 parameters
  • Controls step length, height, etc.


Robots walk across soccer pitch and are timed
  • Reward is a function of the time taken
 An Example: Learning to Walk
Basic idea
  1. Pick an initial q = {q1, q2, ... , q11}
  2. Generate N testing parameter settings by perturbing q
       qj = {q1 + d1, q2 + d2, ... , q11 + d11},   di  {-e, 0, e}
  3. Test each setting, and observe rewards
       qj → rj
  4. For each qi  q                              d             
                                                             if θi largest     
                                                                              
       Calculate q1+, q10, q1- and set θ'i  θi  0              0
                                                             if θi largest     
                                                   d                        
  5.   Set q ← q’, and go to 2                              if θi largest     

                                                          Average reward
                                                          when qni = qi - di
An Example: Learning to Walk




     Initial                    Final


               Video: Nate Kohl & Peter Stone, UT Austin
Value Function or Policy Gradient?
When should I use policy gradient?
  • When there’s a parameterized policy
  • When there’s a high-dimensional state space
  • When we expect the gradient to be smooth


When should I use a value-based method?
  • When there is no parameterized policy
  • When we have no idea how to solve the problem
            Summary for Part I
Background
  • MDPs, and how to solve them
  • Solving MDPs with dynamic programming
  • How RL is different from DP
Algorithms
  •   Certainty equivalence
  •   TD
  •   Q-learning
  •   SARSA
  •   Policy gradient