Title: New Theoretical Frameworks for Machine Learning
1New Theoretical Frameworks for Machine Learning
MariaFlorina Balcan
2Thanks to My Committee
Avrim Blum
Tom Mitchell
Manuel Blum
Yishay Mansour
Santosh Vempala
3The Goals of the Thesis
New Frameworks for Important Learning Problems
Models, algorithms, generalization bounds for
Prominent methods in ML today.
 Learning with General Similarity Fns
Machine Learning Algorithmic Game
Theory
 ML both for designing and analyzing auctions
(Revenue Maximization)
4New Frameworks for Machine Learning
(Not captured by standard learning models)
Important Learning Paradigms
Incorporating Unlabeled Data in the Learning
Process
Kernel based Learning
Qualitative gap between theory and practice
Semisupervised Learning
Unified theoretical treatment lacking
Active Learning
Our Contributions
Our Contributions
A theory of learning with general similarity
functions
Semisupervised learning
 a unified discriminative framework
Active Learning
New discriminative model for Clustering
 new positive theoretical results
5New Frameworks for Machine Learning
(Not captured by standard learning models)
Important Learning Paradigms
Incorporating Unlabeled Data in the Learning
Process
Kernel, Similarity based Learning and Clustering
Qualitative gap between theory and practice
Unified theoretical treatment lacking
Our Contributions
Our Contributions
A theory of learning with general similarity
functions
Semisupervised learning
 a unified discriminative framework
Active Learning
New discriminative model for Clustering
 new positive theoretical results
6Machine Learning and Algorithmic Game Theory
Oneslide Overview of Our Results
Machine Learning for Auction Design and Pricing
generic
Incentive compatible auction design
Standard algorithm design
reduction
BalcanBlumHartlineMansour, FOCS 2005
BalcanBlumHartlineMansour, JCSS 2008
Approximation
and Online Algorithms Pricing revenue
maximization in combinatorial auctions
Other related work
Single minded customers
Customers with general valuations
BB, EC 2006
BB, TCS 2007
BBM, EC 2008
BBCH, WINE 2007
7The Goals of the Thesis
New Frameworks for Important Learning Problems
Machine Learning Algorithmic Game
Theory
8New Frameworks for Important Learning Problems
Prominent methods in Machine Learning today.
Lots of Books, Workshops.
Significant percentage of ICML, NIPS, COLT.
ICML 2007, Business meeting
9Structure of the Talk
New Frameworks for Important Learning Paradigms
Incorporating Unlabeled Data in the Learning
Process
Kernels, Similarity based learning and Clustering
Semisupervised learning (SSL)
 Kernels, margins feature selection
 An Augmented PAC model for SSL
BalcanBlumVempala, ALT 2004 MLJ 2006
BalcanBlum, COLT 2005 book chapter,
SemiSupervised Learning, 2006
 General theory of learning with similarity
functions
Active Learning (AL)
BalcanBlum, ICML 2006
 Generic agnostic AL procedure
BalcanBlumSrebro, MLJ 2008
BalcanBlumSrebro, COLT 2008
BalcanBeygelzimerLangford, ICML 2006 JCSS
2008
 Discriminative model for Clustering
 Margin based AL of linear separators
BalcanBlumVempala, STOC 2008
BalcanBroderZhang, COLT 2007
BalcanBlumGupta, Manuscript 2008
BalcanHannekeWortman, COLT 2008 MLJ 2008
(best student paper)
10Part I, Incorporating Unlabeled Data in the
Learning Process
SemiSupervised Learning
A general discriminative framework
BalcanBlum, COLT 2005 book chapter,
SemiSupervised Learning, 2006
11 Standard Supervised Learning
 S(x, l)  set of labeled examples
 labeled examples  drawn i.i.d. from distr. D
over X and labeled by some target concept c
 labels 2 1,1  binary classification
 Do optimization over S, find hypothesis h 2 C.
 Goal h has small error over D.
c in C, realizable case
err(h)Prx 2 D(h(x) ? c(x))
c not in C, agnostic case
 Classic models for learning from labeled data.
 Statistical Learning Theory (Vapnik)
12 Standard Supervised Learning
Sample Complexity
 E.g., Finite Hypothesis Spaces, Realizable Case
 In PAC, can also talk about efficient algorithms.
13SemiSupervised Learning
Suxi  unlabeled examples i.i.d. from D
Sl(xi, yi) labeled examples i.i.d. from D,
labeled by target c.
Data Source
Learning Algorithm
Expert / Oracle
Unlabeled examples
Unlabeled examples
Labeled Examples
Algorithm outputs a classifier
14SemiSupervised Learning
 Variety of methods and experimental results
 Transductive SVM Joachims 98
 Cotraining Blum Mitchell 98,
BalcanBlumYang04  Graphbased methods Blum Chawla01,
ZhuLaffertyGhahramani03  Etc
 Scattered and very specific theoretical results
We provide a general discriminative (PAC, SLT
style) framework for SSL.
Challenge capture many of the assumptions
typically used.
Different SSL algorithms based on different
assumptions.
15Example of typical assumption Margins
Belief target goes through low density regions
(large margin).
16Another Example Selfconsistency
Agreement between two parts cotraining
BlumMitchell98.
 examples contain two sufficient sets of
features, x h x1, x2 i
 belief the parts are consistent, i.e. 9 c1,
c2 s.t. c1(x1)c2(x2)c(x)
For example, if we want to classify web pages
x h x1, x2 i
17New discriminative model for SSL BB05
Problems with thinking about SSL in standard
models
 PAC or SLT learn a class C under (known or
unknown) distribution D.
 Unlabeled data doesnt give any info about which
c 2 C is the target.
Key Insight
Unlabeled data useful if we have beliefs not only
about the form of the target, but also about its
relationship with the underlying distribution.
18Proposed Model, Main Ideas
Augment the notion of a concept class C with a
notion of compatibility ? between a concept and
the data distribution.
learn C becomes learn (C,?) (learn class C
under ?)
Express relationships that one hopes the target
function and underlying distribution possess.
Idea I use unlabeled data belief that the
target is compatible to reduce C down to just
the highly compatible functions in C.
abstract prior ?
Class of fns C
unlabeled data
Compatible fns in C
e.g., linear separators
e.g., large margin linear separators
finite sample
Idea II require that the degree of compatibility
can be estimated from a finite sample.
19Types of Results in the BB05 Model
Fundamental Sample Complexity issues
 How much unlabeled data we need
 both complexity of C and of the compatibility
notion.
 Ability of unlabeled data to reduce of
labeled examples
 compatibility of the target
 (various) measures of the helpfulness of the
distribution
?Cover bounds much better than Uniform
Convergence bounds.
Main PolyTime Algorithmic Result improved alg
cotraining of linear separators (improves over
BM98 substantially)
Subsequent Work used our framework
P. Bartlett, D. Rosenberg, AISTATS 2007
Kakade et al, COLT 2008
J. ShaweTaylor et al., Neurocomputing 2007
20Part II, Incorporating Unlabeled Data in the
Learning Process
Active Learning
Brief Overview of the results
21Active Learning (AL)
Data Source
Expert / Oracle
Unlabeled examples
Learning Algorithm
Request for the Label of an Example
A Label for that Example
Request for the Label of an Example
A Label for that Example
. . .
Algorithm outputs a classifier
 Classic example where AL helps thresholds on the
real line



22First Agnostic Active Learning Procedure
We provide A2 the first algorithm which is
robust to noise.
Balcan, Beygelzimer, Langford, ICML06
Balcan, Beygelzimer, Langford, JCSS08
Region of disagreement style Pick a few
points at random from the current region of
uncertainty, query their labels, throw out
hypothesis if you are statistically confident
they are suboptimal.
(similar to CAL92 realizable case)
Guarantees for A2
 Fallback exponential improvements.
 C thresholds, low noise, exponential
improvement.
 C  homogeneous linear separators in Rd,
 D  uniform over unit sphere, low noise, only
d2 log (1/?) labels to find h with error ?.
A lot of subsequent work.
Hanneke07, DHM07, BBZ07, BHW08
23First Agnostic Active Learning Procedure
We provide A2 the first algorithm which is
robust to noise.
Balcan, Beygelzimer, Langford, ICML06
Balcan, Beygelzimer, Langford, JCSS08
Region of disagreement style Pick a few
points at random from the current region of
uncertainty, query their labels, throw out
hypothesis if you are statistically confident
they are suboptimal.
(similar to CAL92 realizable case)
Guarantees for A2
 Fallback exponential improvements.
 C thresholds, low noise, exponential
improvement.
 C  homogeneous linear separators in Rd,
 D  uniform over unit sphere, low noise, only
d2 log (1/?) labels to find h with error ?.
 Realizable d3/2 log (1/?) labels
 Improved in subsequent work d log2 (1/?)
BalcanBroderZhang, COLT 07
24Margin Based ActiveLearning Algorithm
Realizable case, can get d log2 (1/?) labels
BalcanBroderZhang, COLT 07
Use O(d) examples to find w1 of error 1/8.
 iterate k2, , log(1/?)
 rejection sample mk samples x from D
 satisfying wk1T x ?k
 label them
 find wk 2 B(wk1, 1/2k ) consistent with all
these examples.  end iterate
Other Work BHW08  new perspective on AL.
25 Part III, Learning with Kernels and
More General Similarity Functions
BalcanBlum, ICML 2006
BalcanBlumSrebro, MLJ 2008
BalcanBlumSrebro, COLT 2008
26Kernel Methods
Prominent method for supervised classification
today.
The learning alg. interacts with the data via a
similarity fns
What is a Kernel?
A kernel K is a legal def of
dotproduct i.e. there exists an implicit
mapping ? such that K( , )? ( )? (
).
E.g., K(x,y) (x y 1)d
? (ndimensional space) ! nddimensional space
Why Kernels matter?
Many algorithms interact with data only via
dotproducts.
So, if replace x y with K(x,y), they act
implicitly as if data was in the
higherdimensional ?space.
27Example
K(x,y) (xy)d corresponds to
 E.g., for n2, d2, the kernel
original space
?space
z2
28Generalize Well if Good Margin
 If data is linearly separable by margin in
?space, then good sample complexity.
If margin ? in ?space, then need sample size
of only Õ(1/?2) to get confidence in
generalization.
?(x) 1
 (another example of a generalization bound)
29Limitations of the Current Theory
In practice kernels are constructed by viewing
them as measures of similarity.
Existing Theory in terms of margins in implicit
spaces.
Difficult to think about, not great for intuition.
Kernel requirement rules out many natural
similarity functions.
Better theoretical explanation?
30Better Theoretical Framework
Yes! We provide a more general and intuitive
theory that formalizes the intuition that a good
kernel is a good measure of similarity.
In practice kernels are constructed by viewing
them as measures of similarity.
Existing Theory in terms of margins in implicit
spaces.
Difficult to think about, not great for intuition.
Kernel requirement rules out natural similarity
functions.
BalcanBlum, ICML 2006
BalcanBlumSrebro, MLJ 2008
BalcanBlumSrebro, COLT 2008
Better theoretical explanation?
31More General Similarity Functions
We provide a notion of a good similarity function
 Simpler, in terms of natural direct quantities.
Main notion
 no implicit highdimensional spaces
 no requirement that K(x,y)?(x) ? (y)
Good kernels
K can be used to learn well.
First attempt
2) Is broad includes usual notion of good
kernel.
has a large margin sep. in ?space
3) Allows one to learn classes that have no good
kernels.
32A First Attempt
P distribution over labeled examples (x, l(x))
Goal output classification rule good for P
K is good if most x are on average more
similar to points y of their own type than to
points y of the other type.
K is (?,?)good for P if a 1? prob. mass of x
satisfy
EyPK(x,y)l(y)l(x) EyPK(x,y)l(y)?l(x)?
gap
Average similarity to points of opposite label
Average similarity to points of the same label
33A First Attempt
K is (?,?)good for P if a 1? prob. mass of x
satisfy
EyPK(x,y)l(y)l(x) EyPK(x,y)l(y)?l(x)?
0.4
Example
0.3
0.5
E.g., K(x,y) 0.2, l(x) l(y)
1
1
K(x,y) random in 1,1, l(x) ? l(y)
1
34A First Attempt
K is (?,?)good for P if a 1? prob. mass of x
satisfy
EyPK(x,y)l(y)l(x) EyPK(x,y)l(y)?l(x)?
Algorithm
 Draw sets S, S of positive and negative
examples.
 Classify x based on average similarity to S
versus to S.
S
S
x
x
35A First Attempt
K is (?,?)good for P if a 1? prob. mass of x
satisfy
EyPK(x,y)l(y)l(x) EyPK(x,y)l(y)?l(x)?
Algorithm
 Draw sets S, S of positive and negative
examples.
 Classify x based on average similarity to S
versus to S.
Theorem
If S and S are ?((1/?2)
ln(1/??)), then with probability 1?, error
??.
 For a fixed good x prob. of error w.r.t. x (over
draw of S, S) is ². Hoeffding
 At most ? chance that the error rate over GOOD is
?.
36A First Attempt Not Broad Enough
EyPK(x,y)l(y)l(x) EyPK(x,y)l(y)?l(x)?
30o
30o
½ versus ¼




½ versus ½ 1 ½ ( ½)


Similarity function K(x,y)x y
 has a large margin separator
does not satisfy our definition.
37A First Attempt Not Broad Enough
EyPK(x,y)l(y)l(x) EyPK(x,y)l(y)?l(x)?
R
30o
30o
Broaden 9 nonnegligible R s.t. most x are
on average more similar to y 2 R of same label
than to y 2 R of other label.
even if do not know R in advance
38Broader Definition
 K is (?, ?, ?) if 9 a set R of reasonable y
(allow probabilistic) s.t. 1? fraction of x
satisfy
EyPK(x,y)l(y)l(x), R(y) EyPK(x,y)l(y)?l(
x), R(y)?
At least ? prob. mass of reasonable positives
negatives.
Property
 Draw Sy1, ?, yd set of landmarks.
F(x) K(x,y1), ,K(x,yd).
x !
Rerepresent data.
P
 If enough landmarks (d?(1/?2 ? )), then with
high prob. there exists a good L1 large margin
linear separator.
w0,0,1/n,1/n,0,0,0,1/n,0,0
39Broader Definition
 K is (?, ?, ?) if 9 a set R of reasonable y
(allow probabilistic) s.t. 1? fraction of x
satisfy
EyPK(x,y)l(y)l(x), R(y) EyPK(x,y)l(y)?l(
x), R(y)?
At least ? prob. mass of reasonable positives
negatives.
Algorithm
duÕ(1/(?2? ))
dlO(1/(?2²acc ln (du) ))
 Draw Sy1, ?, yd set of landmarks.
F(x) K(x,y1), ,K(x,yd)
x !
Rerepresent data.
P
X
X
X
X
X
O
X
X
O
O
O
O
X
O
O
X
O
O
X
O
 Take a new set of labeled examples, project to
this space, and run a good L1 linear separator
alg.
40Kernels versus Similarity Functions
Main Technical Contributions
Theorem
K is also a good similarity function.
K is a good kernel
(but ? gets squared).
If K has margin ? in implicit space, then for any
?, K is (?,?2,?)good in our sense.
41Kernels versus Similarity Functions
Main Technical Contributions
Strictly more general
Theorem
K is also a good similarity function.
K is a good kernel
(but ? gets squared).
Can also show a Strict Separation.
Theorem
For any class C of n pairwise uncorrelated
functions, 9 a similarity function good for all f
in C, but no such good kernel function exists.
42Kernels versus Similarity Functions
Can also show a Strict Separation.
Theorem
For any class C of n pairwise uncorrelated
functions, 9 a similarity function good for all f
in C, but no such good kernel function exists.
 In principle, should be able to learn from
O(?1log(C/?)) labeled examples.
 Claim 1 can define generic (0,1,1/C)good
similarity function achieving this bound. (Assume
D not too concentrated)
 Claim 2 There is no (?,?) good kernel in hinge
loss, even if ?1/2 and ?1/ C1/2. So, margin
based SC is d?(1/C).
43Generic Similarity Function
 Partition X into regions R1,,RC with P(Ri) gt
1/poly(C).  Ri will be R for target fi.
 For y in Ri, define K(x,y)fi(x)fi(y).
 So, for any target fi in C, any x, we get
 Eyl(x)l(y)K(x,y) y in Ri El(x)2l(y)2 1.
 So, K is (0,1,1/poly(C))good.
Gives bound O(?1log(C))
44Similarity Functions for Classification
Conceptual Contributions
Before
After, Our Work
Much more intuitive theory
Difficult theory
Formalizes a common intuition.
Not helpful for intuition.
Provably more general.
Limiting.
Algorithmic Implications
 Can use nonPSD similarities, no need to
transform them into PSD functions and plug into
SVM.
E.g., Liao and Noble, Journal of Computational
Biology
45Similarity Functions for Classification
Algorithmic Implications
 Can use nonPSD similarities, no need to
transform them into PSD functions and plug into
SVM.
E.g., Liao and Noble, Journal of Computational
Biology
 Give justification to the following rule
 Also show that anything learnable with SVM is
learnable this way!
46Part IV, A Novel View on Clustering
BalcanBlumVempala, STOC 2008
A General Framework for analyzing clustering
accuracy without strong probabilistic assumptions
47What if only Unlabeled Examples Available?
sports
fashion
S set of n objects.
documents
9 ground truth clustering.
x, l(x) in 1,,t.
topic
Goal h of low error where
err(h) min?PrxS?(h(x)) ? l(x)
Problem unlabeled data only!
But have a Similarity Function!
48What if only Unlabeled Examples Available?
sports
fashion
Protocol
9 ground truth clustering for S
The similarity function K has to be related to
the groundtruth.
i.e., each x in S has l(x) in 1,,t.
S, a similarity function K.
Input
Clustering of small error.
Output
(err(h) min?PrxS?(h(x)) ? l(x))
49What if only Unlabeled Examples Available?
sports
fashion
Fundamental Question
What natural properties on a similarity function
would be sufficient to allow one to cluster well?
50 Contrast with Standard Approaches
Approximation algorithms
Mixture models
Input embedding into Rd
Input graph or embedding into Rd
 score algs based on apx ratios
 score algs based on error rate
 analyze algs to optimize various criteria over
edges
 strong probabilistic assumptions
Clustering Theoretical Frameworks
Discriminative, not generative.
Our Approach
Much better when input graph/ similarity is
based on heuristics.
BalcanBlumVempala, STOC 2008
Input graph or similarity info
E.g., clustering documents by topic, web
search results by category
 score algs based on error rate
 no strong probabilistic assumptions
51Condition that trivially works.
What natural properties on a similarity function
would be sufficient to allow one to cluster well?
sports
fashion
C
C
K(x,y) gt 0 for all x,y, l(x) l(y).K(x,y) lt 0
for all x,y, l(x) ? l(y).
A
A
52What natural properties on a similarity function
would be sufficient to allow one to cluster well?
All x more similar to all y in own cluster than
any z in any other cluster
Problem same K can satisfy it for two very
different, equally natural clusterings of the
same data!
K(x,x)1
K(x,x)0.5
K(x,x)0
53Relax Our Goals
1. Produce a hierarchical clustering s.t.
correct answer is approximately some pruning of
it.
54Relax Our Goals
1. Produce a hierarchical clustering s.t.
correct answer is approximately some pruning of
it.
All topics
sports
fashion
tennis
Lacoste
soccer
Gucci
2. List of clusterings s.t. at least one has
low error.
Tradeoff strength of assumption with size of list.
Obtain a rich, general model.
55Examples of Properties and Algorithms
Strict Separation Property
All x are more similar to all y in own cluster
than any z in any other cluster
Sufficient for hierarchical clustering
(single linkage algorithm)
Stability Property
C
C
For all clusters C, C, for all Aµ C, A µ C,
neither A nor A more attracted to the other
one than to the rest of its own cluster.
A
A
(K(A,A)  average attraction between A and A)
Sufficient for hierarchical clustering
(average linkage algorithm)
56Examples of Properties and Algorithms
Average Attraction Property
Ex 2 C(x)K(x,x) gt Ex 2 C K(x,x)? (8
C?C(x))
Not sufficient for hierarchical clustering
Can produce a small list of clusterings.
(sampling based algorithm)
Stability of Large Subsets Property
C
C
For all clusters C, C, for all Aµ C, A µ C,
AA sn, neither A nor A more attracted to
the other one than to the rest of its own cluster.
A
A
Sufficient for hierarchical clustering
Find hierarchy using a multistage
learningbased algorithm.
57Stability of Large Subsets Property
C
C
For all C, C, all A ½ C, A µ C,
K(A,CA) gt K(A,A),
AA sn
A
A
Algorithm
 Generate list L of candidate clusters (average
attraction alg.)
Ensure that any groundtruth cluster is fclose
to one in L.
 For every (C, C0) in L s.t. all three parts are
large
If K(C Å C0, C \ C0) K(C Å C0, C0 \ C),
then throw out C0
Else throw out C.
3) Clean and hook up the surviving clusters
into a tree.
58Stability of Large Subsets
C
C
For all C, C, all A½C, AµC, AA sn
K(A,CA) gt K(A,A)?
A
A
If sO(?2/k2), fO(?2 ?/k2), then produce
a tree s.t. the groundtruth is ?close to a
pruning.
Theorem
59Similarity Functions for Clustering, Summary
 Minimal conditions on K to be useful for
clustering.
 For robust theory, relax objective hierarchy,
list.
 A general model that parallels PAC, SLT, Learning
with Kernels and Similarity Functions in
Supervised Classification.
60Similarity Functions, Overall Summary
Supervised Classification
Unsupervised Learning
First Clustering model for analyzing accuracy
without strong probabilistic assumptions.
Generalize and simplify the existing theory
of Kernels.
BalcanBlum, ICML 2006
BalcanBlumSrebro, COLT 2008
BalcanBlumVempala, STOC 2008
BalcanBlumSrebro, MLJ 2008
61Future Directions
Connections between Computer Science and Economics
Active learning and online learning techniques
for better pricing algorithms and auctions.
New Frameworks and Algorithms for Machine Learning
 Similarity Functions for Learning and Clustering
Learn a good similarity based on data from
related problems.
Other notions of useful, other types of
feedback.
Other navigational structures e.g., a small DAG.
62Thank you !