learning, testing, and approximating halfspaces rocco servedio columbia university dimacs-rutcor jan...
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Learning, testing, and approximatinghalfspaces
Rocco Servedio
Columbia University
DIMACS-RUTCOR
Jan 2009
Approximation
Given a function goal is to obtain a “simpler”
function such that
• Measure distance between functions under uniform distribution.
Approximating classes of functions
Interested in statements of the form:
“Every function in class has a simple approximator.”
Every -size decision tree can be -approximated by a decision tree of depth
Example statement:
0 1 0 1 01
0 11
0 11
1
TestingGoal: infer “global” property of function via few “local” inspections
Tester makes black-box queries to arbitrary
Tester must output
• “yes” whp if • “no” whp if is -far from
every
• Any answer OK if is -close to some
distance
Usual focus: information-theoretic
# queries required
oracle for
Some known property testing results
parity functions [BLR93]
deg- polynomials [AKK+03]
literals [PRS02]
conjunctions [PRS02]
-juntas [FKRSS04]
-term monotone DNF [PRS02]
-term DNF [DLM+07]
size- decision trees [DLM+07]
-sparse polynomials [DLM+07]
Class of functions over # of queries
Halfspaces
A function
is a halfspace if
such that
for all
• Also called linear threshold functions (LTFs), threshold gates, etc.
• Fundamental to learning theory– Halfspaces are at the heart of many learning algorithms: Perceptron,
Winnow, boosting, Support Vector Machines,…
• Well studied in complexity theory
Some examples of halfspaces
Weights can be all the same…
(decision list)
•
•
•
…but don’t have to be…
•
•
What’s a “simple” halfspace?
Every halfspace has a representation with integer weights:
– finite domain, so can “nudge” weights to rational #s, scale to integers
Some halfspaces over require integer weights
[MTT61, H94]
Low-weight halfspaces are nice for complexity, learning.
is equivalent to
Approximating halfspaces using small weights?
Let be an arbitrary halfspace.
If is a halfspace which -approximates how large do the weights of need to be?
Consider (view as n-bit binary numbers)
This is a halfspace:
but it’s easy to -approximate with weight
Any halfspace for requires weight …
Let’s warm up with a concrete example.
Approximating all halfspaces using small weights?
So there are halfspaces that require weight
but can be -approximated with weight
Let be an arbitrary halfspace.
If is a halfspace which -approximates how large do the weights of need to be?
Can every halfspace be approximated by a small-weight halfspace?
Yes
Every halfspace has a low-weight approximator
• Can’t do better in terms of ; may need some
• Dependence on must be [H94]
Theorem: [S06]
Let be any halfspace. For any
there is an -approximator with
integer weights that has
How good is this bound?
Idea behind the approximation
Let
• If weights decrease rapidly, then
is well approximated by a junta
WOLOG have
Key idea: look at how these weights decrease.
• If weights decrease slowly, then
is “nice” – can get a handle on
distribution of
A few more details
Def: Critical index of is the first index such that
is “small relative to the remaining weights”:
Let
How do these weights decrease?
critical index
Sketch of approximation: case 1
• “First weights all decrease rapidly” – factor of
• Remaining weight after very small
• Can show
is -close to , so can approximate just by truncating
• has relevant variables so can be expressed with integer weights each at most
Critical index is first
index such that
First case:
Why does truncating work?
Have
only if either
or
each of these weights
small, so unlikely by
Hoeffding bound
unlikely by more complicated argument (split up
into blocks; symmetry argument on each
block bounds prob by ½; use independence)
Let’s write for
Sketch of approximation: case 2
Second case:
Critical index is first
index such that
• “weights are smooth”
• Intuition: behaves like Gaussian
• Can show it’s OK to round weights to small integers
(at most )
Why does rounding work?
Let
so
Have only if either
or
each small, so
unlikely by
Hoeffding bound
unlikely since
Gaussian is “anticoncentrated”
}
Sketch of approximation: case 2
Second case:
Critical index is first
index such that
• “weights are smooth”
• Intuition: behaves like Gaussian
• Can show it’s OK to round weights to small integers
(at most )
• Need to deal with first weights, but at most
many – they cost at most
END OF
SKETCH
Extensions
Let be any halfspace. For any
there is an -approximator with
integer weights that has
We saw:
Recent improvement [DS09]: replace with
For
with bit flipped
Standard fact: Every halfspace has (but can be much less)
Proof uses structural properties of
halfspaces from testing & learning.
Can be viewed as (exponential)
sharpening of Friedgut’s theorem:
Every Boolean is -close to a
function on variables.
We show:
Every halfspace is -close to a
function on
variables.approximation
Combines
• Littlewood-Offord type theorems on
“anticoncentration” of
• delicate linear programming
arguments
Gives new proof of original
bound that does not use the
“critical index”
So halfspaces have low-weight approximators.What about testing?
Use approximation viewpoint: two possibilities depending on critical index.
First case: critical index large
• close to junta halfspace over variables
• Implicitly identify the junta variables (high influence)
• Do Occam-type “implicit learning” similar to [DLMORSW07]
(building on [FKRSS02]): check every possible halfspace over the junta variables
– If is a halfspace, it’ll be close to some function you check
– If far from every halfspace, it’ll be close to no function you check
So halfspaces have low-weight approximators.What about testing?
Second case: critical index small
• every restriction of high-influence vars makes “regular”
– all weights & influences are small
• Low-influence halfspaces have nice Fourier properties
• Can use Fourier analysis to check that each restriction
is close to a low-influence halfspace
• Also need to check:
– cross-consistency of different restrictions (close to low-influence halfspaces with same weights)?
– global consistency with a single set of high-influence weights most
s
A taste of FourierA helpful Fourier result about low-influence halfspaces:
“Theorem”: [MORS07] Let be any Boolean function such that:
• all the degree-1 Fourier coefficients of are small
• the degree-0 Fourier coefficient synchs up with the degree-1 coeffs
Then is close to a halfspace
A taste of FourierA helpful Fourier result about low-influence halfspaces:
“Theorem”: [MORS07] Let be any Boolean function such that:
• all the degree-1 Fourier coefficients of are small
• the degree-0 Fourier coefficient synchs up with the degree-1 coeffs
Then is close to a halfspace – in fact, close to the halfspace
• Useful for soundness portion of test
Testing halfspaces
When all the dust settles:
Theorem: [MORS07]
The class of halfspaces over is testable with queries.
approximation
testing
What about learning?
Learning halfspaces from random
labeled examples is easy using
poly-time linear programming.
1. The RFA model
2. Agnostic learning under uniform distribution
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?!There are other harder learning models…
The RFA learning model
• Introduced by [BDD92]: “restricted focus of attention”
• For each labeled example the learner gets to choose
one bit of the example that he can see
(plus the label of course).
• Examples are drawn from uniform distribution over
• Goal is to construct -accurate hypothesis
Question: [BDD92, ADJKS98, G01]
Are halfspaces learnable in RFA model?
The RFA learning model in action
learner oracle
May I have a random example, please?
Sure, which bit would you like to see?
Oh, man…uh, x7.
Thanks, I guess
Watch your manners
Here’s your example:
Very brief Fourier interlude
Every has a unique Fourier representation
The coefficients
are sometimes called the Chow parameters of
Another view of the RFA learning model
Every has a unique Fourier representation
The coefficients
are sometimes called the Chow parameters of
RFA model: learner gets
Not hard to see:
In the RFA model, all the learner can do is estimate
the Chow parameters
• With examples, can estimate any given Chow parameter
to additive accuracy
(Approximately) reconstructing halfspaces from their (approximate) Chow parameters
Theorem [C61]: If is a halfspace &
has for all then
Perfect information about Chow parameters suffices for halfspaces:
To solve 1-RFA learning problem, need a version of Chow’s theorem which is both robust and effective
• robust: only get approximate Chow parameters (and only hope for approximation to )
• effective: want an actual poly(n) time algorithm!
Previous results
Theorem: Let be a weight- halfspace. Let
be any Boolean function satisfying
for all Then is an -approximator for
[Goldberg01] proved:
[ADJKS98] proved:
Theorem: Let be any halfspace. Let be any function
satisfying
for all Then is an -approximator for
• Good for low-weight halfspaces, but could be
• Better bound for high-weight halfspaces, but superpolynomial in n.
Neither of these results is algorithmic.
Robust, effective version of Chow’s theorem
Theorem: [OS08] For any constant and any halfspace given accurate enough approximations of the Chow
parameters of
algorithm runs in time and w.h.p. outputs a halfspace
that is -close to
Fastest runtime dependence on of any algorithm for learning halfspaces, even in usual random-examples model
– Previous best runtime: time for learning to constant accuracy
– Any algorithm needs examples, i.e. bits of input
Corollary: [OS08] Halfspaces are learnable to any constant accuracy in
time in the RFA model.
A tool from testing halfspaces
If itself is a low-influence halfspace, means we can plug in
degree-1 Fourier coefficients as weights and get a good approximator.
Also need to deal with high-influence case…a hassle, but doable.
Recall helpful Fourier result about low-influence halfspaces:
“Theorem”: Let be any function which is such that:
• all the degree-1 Fourier coefficients of are small
• the degree-0 Fourier coefficient synchs up with the degree-1 coeffs
Then is close to
We know (approximations to)
these in the RFA setting!polynomial time!
Recap of whole talk++ ++
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approximation
testing
learning
1. Every halfspace can be approximated to any constant accuracy with small integer weights.
2. Halfspaces can be tested with queries.
3. Halfspaces can be efficiently learned from
(approximations of) their degree-0 and degree-1 Fourier coefficients.
Halfspaces over
Future directionsBetter quantitative results (dependence on ?)
– Testing:
– Approximating:
– Learning (from Chow parameters):
What about {approximating, testing, learning} w.r.t. other distributions?– Rich theory of distribution-independent PAC learning
– Less fully developed theory of distribution-independent testing [HK03,HK04,HK05,AC06]
– Things are harder; what is doable?
– [GS07] Any distribution-independent algorithm for testing whether is a halfspace requires queries.
II. Learning a concept class
Setup: Learner is given a sample of labeled examples
• Target function is unknown to learner
• Each example in sample is independent, uniform over
Goal: For every , with probability learner should output a hypothesis such that
“PAC learning concept class under the uniform distribution”