A classification learning example Predicting when Rusell will wait for a table --similar to book...
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Transcript of A classification learning example Predicting when Rusell will wait for a table --similar to book...
A classification learning examplePredicting 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, and a space of hypotheses
– Find the rule that underlies the labeling
• (so you can use it to predict future unlabeled examples)
– Tabularasa, fully supervised
• Idea:– Loop through all hypotheses
• Rank each hypothesis in terms of its match to data
• Pick the best hypothesis
• Main variations:• Bias: the “sort” of rule are you
looking for?– If you are looking for only
conjunctive hypotheses, there are just 3n
– Search:– Greedy search
– Decision tree learner– Systematic search
– Version space learner– Iterative search
– Neural net learner
The main problem is that the space of hypotheses is too large
Given examples described in terms of n boolean variablesThere are 2 different hypothesesFor 6 features, there are 18,446,744,073,709,551,616 hypotheses
2n
It can be shown that sample complexity of PAC learning is proportional to 1/, 1/ AND log |H|
5/5
Bias & Learning AccuracyWhy Simple is Better?
Training error
Test (prediction) error
Fra
ctio
n in
core
ctly
cla
ssifi
ed• Having weak bias (large hypothesis space)– Allows us to capture
more concepts– ..increases learning
cost– May lead to over-
fitting
Also the goal of a compression algorithm is to drive down the training errorBut the goal of a learning algorithm is to drive down the test error
Uses different biases in predicting Russel’s waiting habbits
Russell waits
Wait time? Patrons? Friday?
0.3
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Naïve bayes(bayesnet learning)--Examples are used to --Learn topology --Learn CPTs
Neural Nets--Examples are used to --Learn topology --Learn edge weights
Decision Trees--Examples are used to --Learn topology --Order of questionsIf patrons=full and day=Friday
then wait (0.3/0.7)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
K-nearest neighbors
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?
N+N-
N1+N1-
N2+N2-
Nk+Nk-
Splitting on feature fk
P+ : N+ /(N++N-)P- : N- /(N++N-)
I(P+ ,, P-) = -P+ log(P+) - P- log(P- )
I(P1+ ,, P1-) I(P2+ ,, P2-) I(Pk+ ,, Pk-)
[Ni+ + Ni- ]/[N+ + N-] I(Pi+ ,, Pi-)
i=1
k
The differenceis the informationgain
So, pick the featurewith the largest Info Gain
I.e. smallest residual info
The Information GainComputation
Given k mutually exclusive and exhaustiveevents E1….Ek whose probabilities are p1….pk
The “information” content (entropy) is defined as
i -pi log2 pi
A split is good if it reduces the entropy..
# expected comparisonsneeded to tell whether agiven example is +ve or -ve
Ex Masochistic Anxious Nerdy HATES EXAM
1 F T F Y
2 F F T N
3 T F F N
4 T T T Y
A simple example
V(M) = 2/4 * I(1/2,1/2) + 2/4 * I(1/2,1/2) = 1
V(A) = 2/4 * I(1,0) + 2/4 * I(0,1) = 0
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 onOnce you split on Anxious, the problem is solved
I(1/2,1/2) = -1/2 *log 1/2 -1/2 *log 1/2
= 1/2 + 1/2 =1
I(1,0) = 1*log 1 + 0 * log 0 = 0
Learning curves… Given N examples, partition them into N tr 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)
Evaluating the Decision Trees
Russell Domain“Majority” function(say yes if majority of attributes are yes)
Lesson: Every bias makes some concepts easier to learn and others harder to learn…
m-fold cross-validation Split N examples into m equal sized parts for i=1..m train with all parts except ith
test with the ith part
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 2 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 –ve training examples– Could happen either because
examples are noisy and mis-classified or because you want to stop before reaching pure leafs
• When you reach that node, you return the majority label as the decision.
• (We can associate a confidence with that decision using the P+
and P-)
N+N-
N1+N1-
N2+N2-
Nk+Nk-
Splitting on feature fk
P+= N1+ / N1
++N1-
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– Guess Topology– Estimate CPTs
• Simplest case: Naïve Bayes – 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
Russell waits
Wait time? Patrons? Friday?
0.3
0.5
full
0.3
0.2
some
0.4
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None
F
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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(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 NBCGiven 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)
= 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)
Common factor
USER PROFILE
P(willwait=yes) = 6/12 = .5P(Patrons=“full”|willwait=yes) = 2/6=0.333P(Patrons=“some”|willwait=yes)= 4/6=0.666
P(willwait=yes|Patrons=full) = P(patrons=full|willwait=yes) * P(willwait=yes) ----------------------------------------------------------- P(Patrons=full) = k* .333*.5P(willwait=no|Patrons=full) = k* 0.666*.5
Similarly we can show that P(Patrons=“full”|willwait=no) =0.6666
Example
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
Russell waits
Wait time? Patrons? Friday?
0.3
0.5
full
0.3
0.2
some
0.4
0.3
None
F
T
RW
0.3
0.5
full
0.3
0.2
some
0.4
0.3
None
F
T
RW
Naïve bayes(bayesnet learning)--Examples are used to --Learn topology --Learn CPTs
Neural Nets--Examples are used to --Learn topology --Learn edge weights
Decision Trees--Examples are used to --Learn topology --Order of questionsIf patrons=full and day=Friday
then wait (0.3/0.7)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
K-nearest neighbors
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 ofwith interconnections
Feed ForwardUni-directional connections
Single Layer Multi-Layer
Recurrent
Bi-directional connections
Any linear decision surface can be representedby a single layer neural net
Any “continuous” decision surface (function) can be approximated to any degree of accuracy by some 2-layer neural net
Can act as associative memory
differentiable
threshold units
w1
w2
t=k
I1
I2
w1
w2
t=k
I1
I2
= 1 if w1I1+w2I2 > k= 0 otherwise
A Threshold Unit
…is sort of like a neuron
Threshold Functions
differentiable
The “Brain” Connection
Perceptron Networks
What happened to the“Threshold”? --Can model as an extra weight with static input
w1
w2
t=k
I1
I2
w1
w2
w0= k
I0=-1
t=0
==
Perceptron Learning
• Perceptron learning algorithmLoop 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 *Ij, where Ii is the ith input value and 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 *Ij
– Otherwise, do nothing
Until “convergence”Iterative search!
--node -> network weights
--goodness -> error
Actually a “gradient descent” search
http://neuron.eng.wayne.edu/java/Perceptron/New38.html
A nice applet at:
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Perceptron Learning as Gradient Descent Search in the weight-space
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Often a constant learning rate parameter is used instead
Can Perceptrons Learn All Boolean Functions?--Are all boolean functions linearly separable?
Majority function Russell Domain
Perce
ptro
n
Decision Trees
Decision Trees
Perceptron
Comparing Perceptrons and Decision Treesin Majority Function and 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 into higher dimensional space
• Find a linear surface – Which is guaranteed to
exist
• Transform it back to the original space
• TRICK is to do this without explicitly doing a transformation
• Learn non-linear surfaces directly (as multi-layer neural nets)
• Trick is to do training efficiently– Back Propagation to the
rescue..
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 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
K (A;A0) = ((100A à 1)(100
A 0à 1) à 0:5)6
ïPolynomial Kernel:
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..
Those who ignore easily available domain knowledge are doomed to re-learn it… Santayana’s brother
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
Russell Domain
Decision Trees
Perceptron
Decision Trees
Multi-layernetworks
Multi-Network Learning can learn Russell Domains
…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
K-nearest-neighbor The test example’s class is determined by the class of the majority of its k nearest neighborsNeed to define an appropriate distance measure --sort of easy for real valued vectors --harder for categorical attributes
Other impressive applications: --no-hands across america --learning to speak
Humans make 0.2%Neumans (postmen) make 2%
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 Smarthttp://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
2. Decide on an action, at
3. Perform action
4. Observe new state, st+1
5. Observe reward, rt+1
6. Learn from experience7. Repeat
Goal: Find a control policy that will maximize the observed rewards over the lifetime of the agent
AS R
World
An Example: Gridworld
Canonical RL domain• 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
+1
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: SA → S• Reward function, R: SAS →
Compute the optimal value function for this MDP• And then compute the optimal policy from it
Temporal Difference (TD)
TD-learning estimates the value function directly• Don’t try to learn the underlying MDP
Keep an estimate of V(s) in a table• Update these estimates as we gather more
experience• Estimates depend on exploration policy, • TD is an on-policy method
[Sutton, 88]
TD-Learning Algorithm
Initialize V(s) to 0, sObserve state, sPerform action, (s)Observe new state, s’, and reward, rV(s) ← (1-)V(s) + (r + V(s’))Go to 2
0 ≤ ≤ 1 is the learning rate• How much attention do we pay to new experiences
TD-Learning
V(s) is guaranteed to converge to V*(s)• After an infinite number of experiences• If we decay the learning rate
• will work
In practice, we often don’t need value convergence• Policy convergence generally happens sooner
0tt
0t
2t
tc
ct
Actor-Critic Methods
TD only evaluates a particular policy• Does not learn a better policy
We can change the policy as we learn V• Policy is the actor• Value-function estimate is the critic
Success is generally dependent on the starting policy being “good enough”
ValueFunction
(critic)
World
Policy(actor)
s r
aV
[Barto, Sutton, & Anderson, 83]
Q-Learning
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
[Watkins & Dayan, 92]
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 - )Q(s, a) + (r + maxa’Q(s’, a’))
Go to 2
0 ≤ ≤ 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
-greedy• Pick best (greedy) action with probability • Otherwise, pick a random action
Boltzmann (Soft-Max)• Pick an action based on its Q-value
• , where is the “temperature”
a'
)a' Q(s,
a) Q(s,
e
e s) | P(a
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, aObserve state, sPick an action, a, and do it (just like Q-learning)Observe next state, s’, and reward, rQ(s, a) ← (1-)Q(s, a) + (r + Q(s’, (s’)))Go to 2
0 ≤ ≤ 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
Given enoughexperience
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
These aredense rewards
These aresparse rewards
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
Rule-of-thumb• If you can calculate the reward
function from the state without any additional information, you’re OK
S G
K
Not holding key
Holding key
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
• Estimate the reward gradient, , for each qi
•
iθ
R
iii θ
Rθθ
This is anotherlearning 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
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
[Kohl & Stone, 04]
An Example: Learning to Walk
Basic idea1. Pick an initial = {1, 2, ... , 11}
2. Generate N testing parameter settings by perturbing j = {1 + 1, 2 + 2, ... , 11 + 11}, i {-, 0, }
3. Test each setting, and observe rewardsj → rj
4. For each i Calculate 1
+, 10, 1
- and set
5. Set ← ’, and go to 2Average rewardwhen qn
i = qi - di
largest θ if
largest θ if
largest θ if
θθ'
i
i
i
ii
00
An Example: Learning to Walk
Video: Nate Kohl & Peter Stone, UT Austin
Initial Final
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