For Thursday

• No reading or homework• Exam 2• Take-home due Tuesday• Read chapter 22 for next Tuesday

Program 4

• Any questions?

Hypothesis Space in Decision Tree Induction

• Conducts a search of the space of decision trees which can represent all possible discrete functions.

• Creates a single discrete hypothesis consistent with the data, so there is no way to provide confidences or create useful queries.

Algorithm Characteristics

• Performs hill climbing search so may find a locally optimal solution. Guaranteed to find a tree that fits any noise free training set, but it may not be the smallest.

• Performs batch learning. Bases each decision on a batch of examples and can terminate early to avoid fitting noisy data.

Bias

• Bias is for trees of minimal depth; however, greedy search introduces a complication that it may not find the minimal tree and positions features with high information gain high in the tree.

• Implements a preference bias (search bias) as opposed to a restriction bias (language bias) like candidate elimination.

Simplicity• Occam's razor can be defended on the basis that

there are relatively few simple hypotheses compared to complex ones, therefore, a simple hypothesis that is consistent with the data is less likely to be a statistical coincidence than finding a complex, consistent hypothesis.

• However, – Simplicity is relative to the hypothesis language used. – This is an argument for any small hypothesis space and

holds equally well for a small space of arcane complex hypotheses, e.g. decision trees with exactly 133 nodes where attributes along every branch are ordered alphabetically from root to leaf.

Overfitting• Learning a tree that classifies the training data

perfectly may not lead to the tree with the best generalization performance since – There may be noise in the training data that the tree is

fitting. – The algorithm might be making some decisions toward

the leaves of the tree that are based on very little data and may not reflect reliable trends in the data.

• A hypothesis, h, is said to overfit the training data if there exists another hypothesis, h’, such that h has smaller error than h’ on the training data but h’ has smaller error on the test data than h.

Overfitting and Noise• Category or attribute noise can cause overfitting. • Add noisy instance:

– <<medium, green, circle>, +> (really )

• Noise can also cause directly conflicting examples with same description and different class. Impossible to fit this data and must label leaf with majority category. – <<big, red, circle>, > (really +)

• Conflicting examples can also arise if attributes are incomplete and inadequate to discriminate the categories.

Avoiding Overfitting

• Two basic approaches – Prepruning: Stop growing the tree at some

point during construction when it is determined that there is not enough data to make reliable choices.

– Postpruning: Grow the full tree and then remove nodes that seem to not have sufficient evidence.

Evaluating Subtrees to Prune

• Cross validation: – Reserve some of the training data as a hold out set

(validation set, tuning set) to evaluate utility of subtrees.

• Statistical testing: – Perform some statistical test on the training data to

determine if any observed regularity can be dismissed as likely to to random chance.

• Minimum Description Length (MDL): – Determine if the additional complexity of the

hypothesis is less complex than just explicitly remembering any exceptions.

Learning Theory• Theorems that characterize classes of learning problems or

specific algorithms in terms of computational complexity or sample complexity, i.e. the number of training examples necessary or sufficient to learn hypotheses of a given accuracy.

• Complexity of a learning problem depends on:– Size or expressiveness of the hypothesis space.– Accuracy to which target concept must be approximated.– Probability with which the learner must produce a successful

hypothesis.– Manner in which training examples are presented, e.g. randomly

or by query to an oracle.

Types of Results• Learning in the limit: Is the learner guaranteed to

converge to the correct hypothesis in the limit as the number of training examples increases indefinitely?

• Sample Complexity: How many training examples are needed for a learner to construct (with high probability) a highly accurate concept?

• Computational Complexity: How much computational resources (time and space) are needed for a learner to construct (with high probability) a highly accurate concept?– High sample complexity implies high computational complexity,

since learner at least needs to read the input data.• Mistake Bound: Learning incrementally, how many

training examples will the learner misclassify before constructing a highly accurate concept.

Learning in the Limit• Given a continuous stream of examples where the learner

predicts whether each one is a member of the concept or not and is then is told the correct answer, does the learner eventually converge to a correct concept and never make a mistake again?

• No limit on the number of examples required or computational demands, but must eventually learn the concept exactly.

• By simple enumeration, concepts from any known finite hypothesis space are learnable in the limit, although typically requires an exponential (or doubly exponential) number of examples and time.

• Class of total recursive (Turing computable) functions is not learnable in the limit.

Learning in the Limit vs.PAC Model

• Learning in the limit model is too strong.– Requires learning correct exact concept

• Learning in the limit model is too weak– Allows unlimited data and computational resources.

• PAC Model– Only requires learning a Probably Approximately

Correct Concept: Learn a decent approximation most of the time.

– Requires polynomial sample complexity and computational complexity.

Cannot Learn Exact Conceptsfrom Limited Data, Only Approximations

Negative

Learner Classifier

Positive

Negative

Positive

Cannot Learn Even Approximate Conceptsfrom Pathological Training Sets

Learner ClassifierNegative

Positive

NegativePositive

PAC Learning

• The only reasonable expectation of a learner is that with high probability it learns a close approximation to the target concept.

• In the PAC model, we specify two small parameters, ε and δ, and require that with probability at least (1 δ) a system learn a concept with error at most ε.

Formal Definition of PAC-Learnable• Consider a concept class C defined over an instance space X

containing instances of length n, and a learner, L, using a hypothesis space, H.

• C is said to be PAC-learnable by L using H iff for all cC, distributions D over X, 0<ε<0.5, 0<δ<0.5; learner L by sampling random examples from distribution D, will with probability at least 1 δ output a hypothesis hH such that errorD(h) ε, in time polynomial in 1/ε, 1/δ, n and size(c).

• Example:– X: instances described by n binary features– C: conjunctive descriptions over these features– H: conjunctive descriptions over these features– L: most-specific conjunctive generalization algorithm (Find-S)– size(c): the number of literals in c (i.e. length of the conjunction).

Issues of PAC Learnability• The computational limitation also imposes a

polynomial constraint on the training set size, since a learner can process at most polynomial data in polynomial time.

• How to prove PAC learnability:– First, prove sample complexity of learning C using H is

polynomial.– Second, prove that the learner can train on a

polynomial-sized data set in polynomial time.• To be PAC-learnable, there must be a hypothesis

in H with arbitrarily small error for every concept in C, generally CH.

Version Space

• Bounds on generalizations of a set of examples

Consistent Learners• A learner L using a hypothesis H and training data

D is said to be a consistent learner if it always outputs a hypothesis with zero error on D whenever H contains such a hypothesis.

• By definition, a consistent learner must produce a hypothesis in the version space for H given D.

• Therefore, to bound the number of examples needed by a consistent learner, we just need to bound the number of examples needed to ensure that the version-space contains no hypotheses with unacceptably high error.

ε-Exhausted Version Space• The version space, VSH,D, is said to be ε-exhausted iff every

hypothesis in it has true error less than or equal to ε.• In other words, there are enough training examples to

guarantee than any consistent hypothesis has error at most ε.• One can never be sure that the version-space is ε-exhausted,

but one can bound the probability that it is not.• Theorem 7.1 (Haussler, 1988): If the hypothesis space H is

finite, and D is a sequence of m1 independent random examples for some target concept c, then for any 0 ε 1, the probability that the version space VSH,D is not ε-exhausted is less than or equal to:

|H|e–εm

Sample Complexity Analysis• Let δ be an upper bound on the probability of not

exhausting the version space. So:

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Sample Complexity Result• Therefore, any consistent learner, given at least:

examples will produce a result that is PAC.• Just need to determine the size of a hypothesis space to

instantiate this result for learning specific classes of concepts.

• This gives a sufficient number of examples for PAC learning, but not a necessary number. Several approximations like that used to bound the probability of a disjunction make this a gross over-estimate in practice.

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Sample Complexity of Conjunction Learning• Consider conjunctions over n boolean features. There are 3n of these

since each feature can appear positively, appear negatively, or not appear in a given conjunction. Therefore |H|= 3n, so a sufficient number of examples to learn a PAC concept is:

• Concrete examples:– δ=ε=0.05, n=10 gives 280 examples– δ=0.01, ε=0.05, n=10 gives 312 examples– δ=ε=0.01, n=10 gives 1,560 examples– δ=ε=0.01, n=50 gives 5,954 examples

• Result holds for any consistent learner.

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Sample Complexity of LearningArbitrary Boolean Functions

• Consider any boolean function over n boolean features such as the hypothesis space of DNF or decision trees. There are 22^n of these, so a sufficient number of examples to learn a PAC concept is:

• Concrete examples:– δ=ε=0.05, n=10 gives 14,256 examples– δ=ε=0.05, n=20 gives 14,536,410 examples– δ=ε=0.05, n=50 gives 1.561x1016 examples

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COLT Conclusions

• The PAC framework provides a theoretical framework for analyzing the effectiveness of learning algorithms.

• The sample complexity for any consistent learner using some hypothesis space, H, can be determined from a measure of its expressiveness |H| or VC(H), quantifying bias and relating it to generalization.

• If sample complexity is tractable, then the computational complexity of finding a consistent hypothesis in H governs its PAC learnability.

• Constant factors are more important in sample complexity than in computational complexity, since our ability to gather data is generally not growing exponentially.

• Experimental results suggest that theoretical sample complexity bounds over-estimate the number of training instances needed in practice since they are worst-case upper bounds.

COLT Conclusions (cont)• Additional results produced for analyzing:

– Learning with queries– Learning with noisy data– Average case sample complexity given assumptions about the data

distribution.– Learning finite automata– Learning neural networks

• Analyzing practical algorithms that use a preference bias is difficult.

• Some effective practical algorithms motivated by theoretical results:– Winnow– Boosting– Support Vector Machines (SVM)

Beyond a Single Learner

• Ensembles of learners work better than individual learning algorithms

• Several possible ensemble approaches:– Ensembles created by using different learning

methods and voting– Bagging– Boosting

Bagging

• Random selections of examples to learn the various members of the ensemble.

• Seems to work fairly well, but no real guarantees.

Boosting

• Most used ensemble method• Based on the concept of a weighted training set.• Works especially well with weak learners.• Start with all weights at 1.• Learn a hypothesis from the weights.• Increase the weights of all misclassified examples

and decrease the weights of all correctly classified examples.

• Learn a new hypothesis.• Repeat

Why Neural Networks?

Why Neural Networks?• Analogy to biological systems, the best examples we

have of robust learning systems. • Models of biological systems allowing us to

understand how they learn and adapt. • Massive parallelism that allows for computational

efficiency. • Graceful degradation due to distributed represent-

ations that spread knowledge representation over large numbers of computational units.

• Intelligent behavior is an emergent property from large numbers of simple units rather than resulting from explicit symbolically encoded rules.

Neural Speed Constraints

• Neuron “switching time” is on the order of milliseconds compared to nanoseconds for current transistors.

• A factor of a million difference in speed. • However, biological systems can perform

significant cognitive tasks (vision, language understanding) in seconds or tenths of seconds.

What That Means• Therefore, there is only time for about a

hundred serial steps needed to perform such tasks.

• Even with limited abilties, current AI systems require orders of magnitude more serial steps.

• Human brain has approximately 1011 neurons each connected on average to 104 others, therefore must exploit massive parallelism.

Real Neurons• Cells forming the basis of neural tissue

– Cell body – Dendrites – Axon – Syntaptic terminals

• The electrical potential across the cell membrane exhibits spikes called action potentials.

• Originating in the cell body, this spike travels down the axon and causes chemical neuro-transmitters to be released at syntaptic terminals.

• This chemical difuses across the synapse into dendrites of neighboring cells.

Real Neurons (cont)

• Synapses can be excitory or inhibitory. • Size of synaptic terminal influences

strength of connection. • Cells “add up” the incoming chemical

messages from all neighboring cells and if the net positive influence exceeds a threshold, they “fire” and emit an action potential.

Model Neuron(Linear Threshold Unit)

• Neuron modelled by a unit (j) connected by weights, wji, to other units (i):

• Net input to a unit is defined as:

netj = S wji * oi

• Output of a unit is a threshold function on the net input:– 1 if netj > Tj

– 0 otherwise

Neural Computation

• McCollough and Pitts (1943) show how linear threshold units can be used to compute logical functions.

• Can build basic logic gates – AND: Let all wji be (Tj /n)+e where n = number

of inputs – OR: Let all wji be Tj+e

– NOT: Let one input be a constant 1 with weight Tj+e and the input to be inverted have weight Tj

Neural Computation (cont)

• Can build arbitrary logic circuits, finite state machines, and computers given these basis gates.

• Given negated inputs, two layers of linear threshold units can specify any boolean function using a two layer AND OR network.

Learning

• Hebb (1949) suggested if two units are both active (firing) then the weight between them should increase:

wji = wji + ojoi

– h is a constant called the learning rate – Supported by physiological evidence

Alternate Learning Rule

• Rosenblatt (1959) suggested that if a target output value is provided for a single neuron with fixed inputs, can incrementally change weights to learn to produce these outputs using the perceptron learning rule. – Assumes binary valued input/outputs – Assumes a single linear threshold unit. – Assumes input features are detected by fixed

networks.

Perceptron Learning Rule

• If the target output for output unitj is tj

wji = wji + h(tj - oj)oi

• Equivalent to the intuitive rules: – If output is correct, don't change the weights – If output is low (oj = 0, tj =1), increment weights for all inputs

which are 1. – If output is high (oj = 1, tj =0), decrement weights for all inputs

which are 1.

• Must also adjust threshold:

Tj = Tj + h(tj - oj)

• or equivalently assume there is a weight wj0 = -Tj for an extra input unit 0 that has constant output o0 =1 and that

the threshold is always 0.

Perceptron Learning Algorithm

• Repeatedly iterate through examples adjusting weights according to the perceptron learning rule until all outputs are correct

Initialize the weights to all zero (or randomly)

Until outputs for all training examples are correct

For each training example, e, do

Compute the current output oj

Compare it to the target tj and update the weights

according to the perceptron learning rule.

Algorithm Notes

• Each execution of the outer loop is called an epoch.

• If the output is considered as concept membership and inputs as binary input features, then easily applied to concept learning problems.

• For multiple category problems, learn a separate perceptron for each category and assign to the class whose perceptron most exceeds its threshold.

• When will this algorithm terminate (converge) ??

Representational Limitations

• Perceptrons can only represent linear threshold functions and can therefore only learn data which is linearly separable (positive and negative examples are separable by a hyperplane in n dimensional space)

• Cannot represent exclusive or (xor)

Perceptron Learnability

• System obviously cannot learn what it cannot represent. • Minsky and Papert(1969) demonstrated that many

functions like parity (n input generalization of xor) could not be represented.

• In visual pattern recognition, assumed that input features are local and extract feature within a fixed radius. In which case no input features support learning – Symmetry – Connectivity

• These limitations discouraged subsequent research on

neural networks.

Perceptron Convergence and Cycling Theorems

• Perceptron Convergence Theorem: If there are a set of weights that are consistent with the training data (i.e. the data is linearly separable), the perceptron learning algorithm will converge (Minsky & Papert, 1969).

• Perceptron Cycling Theorem: If the training data is not linearly separable, the Perceptron learning algorithm will eventually repeat the same set of weights and threshold at the end of some epoch and therefore enter an infinite loop.

Perceptron Learning as Hill Climbing

• The search space for Perceptron learning is the space of possible values for the weights (and threshold).

• The evaluation metric is the error these weights produce when used to classify the training examples.

• The perceptron learning algorithm performs a form of hill climbing (gradient descent), at each point altering the weights slightly in a direction to help minimize this error.

• Perceptron convergence theorem guarantees that for the linearly separable case there is only one local minimum and the space is well behaved.

Perceptron Performance

• Can represent and learn conjunctive concepts and M of N concepts (true if any M of a set of N selected binary features are true).

• Although simple and restrictive, this high bias algorithm performs quite well on many realistic problems.

• However, the representational restriction is limiting in many applications.

Multi Layer Neural Networks• Multi layer networks can represent arbitrary functions,

but building an effective learning method for such networks was thought to be difficult.

• Generally networks are composed of an input layer, hidden layer, and output layer and activation feeds forward from input to output.

• Patterns of activation are presented at the inputs and the resulting activation of the outputs is computed.

• The values of the weights determine the function computed.

• A network with one hidden layer with a sufficient number of units can represent any boolean function.

Basic Problem

• General approach to the learning algorithm is to apply gradient descent.

• However, for the general case, we need to be able to differentiate the function computed by a unit and the standard threshold function is not differentiable at the threshold.

Differentiable Threshold Unit• Need some sort of non linear output function to

allow computation of arbitary functions by mulit layer networks (a multi layer network of linear units can still only represent a linear function).

• Solution: Use a nonlinear, differentiable output function such as the sigmoid or logistic function

oj = 1/(1 + e-(netj - Tj) )

• Can also use other functions such as tanh or a Gaussian.

Error Measure

• Since there are mulitple continuous outputs, we can define an overall error measure:

E(W) = 1/2 *( S S (tkd - okd)2) dD kK

where D is the set of training examples, K is the set of output units, tkd is the target output for the kth unit given input d, and okd is network output for the kth unit given input d.

Gradient Descent

• The derivative of the output of a sigmoid unit given the net input is

¶oj/ ¶netj = oj(1 - oj)

• This can be used to derive a learning rule which performs gradient descent in weight space in an attempt to minimize the error function.

wji = -(E / wji)

Backpropogation Learning Rule• Each weight wji is changed by

wji = djoi

dj = oj (1 - oj) (tj - oj) if j is an output unit

dj = oj (1 - oj) Sdk wkj otherwise where h is a constant called the learning rate, tj is the correct output for unit j, dj is an error measure for unit j.

• First determine the error for the output units, then backpropagate this error layer by layer through the network, changing weights appropriately at each layer.

Backpropogation Learning Algorithm• Create a three layer network with N hidden units and fully

connect input units to hidden units and hidden units to output units with small random weights.

Until all examples produce the correct output within e or the mean squared error ceases to decrease (or other termination criteria):

Begin epoch

For each example in training set do:

Compute the network output for this example.

Compute the error between this output and the correct output.

Backpropagate this error and adjust weights to decrease this error.

End epoch

• Since continuous outputs only approach 0 or 1 in the limit, must allow for some e approximation to learn binary functions.

Comments on Training• There is no guarantee of convergence, may

oscillate or reach a local minima. • However, in practice many large networks

can be adequately trained on large amounts of data for realistic problems.

• Many epochs (thousands) may be needed for adequate training, large data sets may require hours or days of CPU time.

• Termination criteria can be: – Fixed number of epochs – Threshold on training set error

Representational PowerMulti layer sigmoidal networks are very expressive. • Boolean functions: Any Boolean function can be represented

by a two layer network by simulating a two layer AND OR network. But number of required hidden units can grow exponentially in the number of inputs.

• Continuous functions: Any bounded continuous function can be approximated with arbitrarily small error by a two layer network. Sigmoid functions provide a set of basis functions from which arbitrary functions can be composed, just as any function can be represented by a sum of sine waves in Fourier analysis.

• Arbitrary functions: Any function can be approximated to arbitarary accuracy by a three layer network.

Sample Learned XOR Network

Hidden unit A represents ¬(X Ù Y)

Hidden unit B represents ¬(X Ú Y)

Output O represents: A Ù ¬B

¬(X Ù Y) Ù (X Ú Y)

X Å Y

A B

X Y

3.11

6.96 -7.38

-5.24 -2.03

-5.57-3.6

-3.58

-5.74

Hidden Unit Representations

• Trained hidden units can be seen as newly constructed features that re represent the examples so that they are linearly separable.

• On many real problems, hidden units can end up representing interesting recognizable features such as vowel detectors, edge detectors, etc.

• However, particularly with many hidden units, they become more “distributed” and are hard to interpret.

Input/Output Coding

• Appropriate coding of inputs and outputs can make learning problem easier and improve generalization.

• Best to encode each binary feature as a separate input unit and for multi valued features include one binary unit per value rather than trying to encode input information in fewer units using binary coding or continuous values.

I/O Coding cont.

• Continuous inputs can be handled by a single input by scaling them between 0 and 1.

• For disjoint categorization problems, best to have one output unit per category rather than encoding n categories into log n bits. Continuous output values then represent certainty in various categories. Assign test cases to the category with the highest output.

• Continuous outputs (regression) can also be handled by scaling between 0 and 1.

Neural Net Conclusions• Learned concepts can be represented by networks

of linear threshold units and trained using gradient descent.

• Analogy to the brain and numerous successful applications have generated significant interest.

• Generally much slower to train than other learning methods, but exploring a rich hypothesis space that seems to work well in many domains.

• Potential to model biological and cognitive phenomenon and increase our understanding of real neural systems. – Backprop itself is not very biologically plausible