Maria-Florina Balcan

Learning with Similarity Functions

Maria-Florina Balcan & Avrim BlumCMU, CSD

Maria-Florina Balcan

Kernels and Similarity Functions

• Useful in practice for dealing with many different kinds of data.

• Elegant theory about what makes a given kernel good for a given learning problem.

Our Goal: analyze more general similarity functions.• In the process we describe ways of constructing good data dependent kernels.

Kernels have become a powerful tool in ML.

Maria-Florina Balcan

Kernels• A kernel K is a pairwise similarity function s.t. 9 an implicit mapping s.t. K(x,y)=(x) ¢ (y).

• Point is: many learning algorithms can be written so only interact with data via dot-products.

• If replace x¢y with K(x,y), it acts implicitly as if data was in higher-dimensional -space.

• If data is linearly separable by large margin in -space, don’t have to pay in terms of data or comp time.

If margin in -space, only need 1/2 examples to learn well.

w

(x)

1

Maria-Florina Balcan

General Similarity Functions

Goal: definition of good similarity function for a learning problem that:

1) Talks in terms of natural direct properties:

• no implicit high-dimensional spaces• no requirement of positive-

semidefiniteness2) If K satisfies these properties for our given

problem, then has implications to learning.

3) Is broad: includes usual notion of “good kernel”.(induces a large margin separator in -space)

Maria-Florina Balcan

A First Attempt: Definition satisfying properties (1) and (2)

• K:(x,y) ! [-1,1] is an (,)-good similarity for P if at least a 1- probability mass of x satisfy:

Ey~P[K(x,y)|l(y)=l(x)] ¸ Ey~P[K(x,y)|l(y)l(x)]+

Note: this might not be a legal kernel.

• Suppose that positives have K(x,y) ¸ 0.2, negatives have K(x,y) ¸ 0.2, but for a positive and a negative K(x,y) are uniform random in [-1,1].

Let P be a distribution over labeled examples (x, l(x))

A

BC

+

--

Maria-Florina Balcan

A First Attempt: Definition satisfying properties (1) and (2). How to use it?

• K:(x,y) ! [-1,1] is an (,)-good similarity for P if at least a 1- probability mass of x satisfy:

Ey~P[K(x,y)|l(y)=l(x)] ¸ Ey~P[K(x,y)|l(y)l(x)]+

Algorithm

• Draw S+ of O((1/2) ln(1/2)) positive examples.• Draw S- of O((1/2) ln(1/2)) negative examples.• Classify x based on which gives better score.

Maria-Florina Balcan

A First Attempt: How to use it?• K:(x,y) ! [-1,1] is an (,)-good similarity for P if at least a 1- probability mass of x satisfy:

Algorithm

• Draw S+ of O((1/2) ln(1/2)) positive examples.• Draw S- of O((1/2) ln(1/2)) negative examples.• Classify x based on which gives better score.

• Hoeffding: for any given “good x”, probability of error w.r.t. x (over draw of S+, S-) at most 2.

• By Markov, at most chance that the error rate over GOOD is more than . So overall error rate · + .

Guarantee: with probability ¸ 1-, error · + Proof

Ey~P[K(x,y)|l(y)=l(x)] ¸ Ey~P[K(x,y)|l(y)l(x)]+

Maria-Florina Balcan

A First Attempt: Not Broad Enough• K:(x,y) ! [-1,1] is an (,)-good similarity for P if at least a 1- probability mass of x satisfy:

Ey~P[K(x,y)|l(y)=l(x)] ¸ Ey~P[K(x,y)|l(y)l(x)]+

• K(x,y)=x ¢ y has good (large margin) separator but doesn’t satisfy our definition.

+ +++++

-- -- --

more similar to negs than to typical pos

Maria-Florina Balcan

A First Attempt: Not Broad Enough• K:(x,y) ! [-1,1] is an (,)-good similarity for P if at least a 1- probability mass of x satisfy:

Idea: would work if we didn’t pick y’s rom top-left.

Broaden to say: OK if 9 large region R s.t. most x are on average more similar to y2R of same label than to y2 R of other label.

Ey~P[K(x,y)|l(y)=l(x)] ¸ Ey~P[K(x,y)|l(y)l(x)]+

R+ ++++

+

-- -- --

Maria-Florina Balcan

Broader/Main Definition

• K:(x,y) ! [-1,1] is an (,)-good similarity for P if exists a weighting function w(y) 2 [0,1] at least a 1- probability mass of x satisfy:

Ey~P[w(y)K(x,y)|l(y)=l(x)] ¸ Ey~P[w(y)K(x,y)|l(y)l(x)]+

Maria-Florina Balcan

Main Definition, How to Use It• K:(x,y) ! [-1,1] is an (,)-good similarity for P if exists a weighting function w(y) 2 [0,1] at least a 1- probability mass of x satisfy:

Ey~P[w(y)K(x,y)|l(y)=l(x)] ¸ Ey~P[w(y)K(x,y)|l(y)l(x)]+

Algorithm

• Draw S+={y1, , yd}, S-={z1, , zd}, d=O((1/2) ln(1/2)).

• Use to “triangulate” data:

F(x) = [K(x,y1), …,K(x,yd), K(x,zd),…,K(x,zd)].

• Take a new set of labeled examples, project to this space, and run your favorite alg for learning lin. separators.

Point is: with probability ¸ 1-, exists linear separator of error · + at margin /4.

(w = [w(y1), …,w(yd),-w(zd),…,-w(zd)])

Maria-Florina Balcan

Main Definition, Implications

Algorithm

• Draw S+={y1, , yd}, S-={z1, , zd}, d=O((1/2) ln(1/2)).• Use to “triangulate” data:F(x) = [K(x,y1), …,K(x,yd), K(x,zd),…,K(x,zd)].

Guarantee: with prob. ¸ 1-, exists linear separator of error · + at margin /4.

Implications legal kernel

K arbitrary sim. function

(,)-good sim. function

(+,/4)-good kernel function

Maria-Florina Balcan

Good Kernels are Good Similarity Functions

Main Definition: K:(x,y) ! [-1,1] is an (,)-good similarity for P if exists a weighting function w(y) 2 [0,1] at least a 1- probability mass of x satisfy:

Ey~P[w(y)K(x,y)|l(y)=l(x)] ¸ Ey~P[w(y)K(x,y)|l(y)l(x)]+

• An (,)-good kernel is an (’,’)-good similarity function under main definition.

Theorem

Our current proofs incur some penalty: ’ = + extra, ’ = 3extra.

Maria-Florina Balcan

Good Kernels are Good Similarity Functions

• An (,)-good kernel is an (’,’)-good similarity function under main definition, where

Theorem

’ = + extra, ’ = 3extra.

Proof Sketch

• Suppose K is a good kernel in usual sense.• Then, standard margin bounds imply:

– if S is a random sample of size Õ(1/(2)), then whp we can give weights wS(y) to all examples y 2 S so that the weighted sum of these examples defines a good LTF.

• But, we want sample-independent weights [and bounded].

– Boundedness not too hard (imagine a margin-perceptron run over just the good y).

– Get sample-independence using an averaging argument.

Maria-Florina Balcan

Learning with Multiple Similarity Functions

• Let K1, …, Kr be similarity functions s. t. some (unknown) convex combination of them is (,)-good.

• Draw S+={y1, , yd}, S-={z1, , zd}, d=O((1/2) ln(1/2)).

• Use to “triangulate” data:

F(x) = [K1(x,y1), …,Kr(x,yd), K1(x,zd),…,Kr(x,zd)].

Guarantee: The induced distribution F(P) in R2dr has a separator of error · + at margin at least

Algorithm

Sample complexity is roughly

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Implications & Conclusions

• Develop theory that provides a formal way of understanding kernels as similarity function.

• Our algorithms work for similarity fns that aren’t necessarily PSD (or even symmetric).

Open ProblemsOpen Problems

• Better results for learning with multiple similarity functions. Extending [SB’06].

• Improve existing bounds.