15-859(B) Machine Learning Theory 03/05/14
* Characterizing what's learnable in SQ model via Fourier analysis
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Statistical Query model recap
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No noise. Algorithm asks ``what is the probability a labeled example will
have property chi? Please tell me up to additive error tau.''
Formally, chi: X x {0,1} --> {0,1} must be poly-time computable. tau > 1/poly.
Asking for Pr_D(chi(x,c(x))=1).
Examples:
- Can ask for error rate of current hypothesis h (what is chi(x,y)?)
- Can ask for Pr(x_1=x_2 and label = +).
(can also handle chi with range [0,1])
May repeat this polynomially many times, and in the end the algorithm
must output a hypothesis with error < epsilon (no delta in this
model).
THEOREM: If can learn C in SQ model, then can learn in PAC with
random noise model.
Characterization of SQs using Fourier analysis
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SQ-dimension
============
- We will say that functions f and g are uncorrelated if
Pr_D [f(x)=g(x)] = 1/2.
We will define the "SQ-dimension" of a class C under distribution D to
be the size of the largest set S of "nearly uncorrelated" functions in
C. Specifically, the largest S such that for all f,g in S,
| Pr_D [f(x)=g(x)] - 1/2 | < 1/|S|.
Theorem 1: If SQdim_D(C) = poly(n) then you *can* weak-learn C over D by SQ
algorithms.
Theorem 2: if SQdim_D(C) > poly(n) then you *can't* weak-learn C over D by
SQ algorithms.
Example: parity
- Let D be the uniform distribution over {0,1}^n.
- Fact: any two parity functions are uncorrelated over the uniform distrib.
- So, C = {all 2^n parity functions over {0,1}^n} is not SQ-learnable
over the uniform distribution.
Example: parity_log
- Let C_log = { parity functions of size log(n) }.
- |C_log| = n^(log n) > poly(n). So, this is also not SQ-learnable.
- This means that the class of poly-size DNF and the class of
poly-size decision trees are not SQ-learnable. Why? Because these
contain C_log.
Can anyone think of a non-SQ alg to learn parity?
Learnability of parity with noise is big open crypto and coding-theory
problem.
Theorem #1 is the simpler of the two.
Proof of Theorem 1:
Let d = SQdim_D(C). The algorithm will be specific for the
distribution D. In particular, we store in advance a maximal set H
of functions such that for all f,g in H, |Pr[f(x) != g(x)] - 1/2| <
1/(d+1). So, |H| <= d.
To learn, ask for Pr[c(x) = h(x)] for each h in H, with
tau=0.25/(d+1). We know there must be some h such that the correct
answer differs from 1/2 by at least 1/(d+1) = 4*tau, else H wouldn't be
maximal. So, just keep asking until we find one whose response
differs from 1/2 by at least 3*tau, which means the *true*
answer for that h differs from 1/2 by at least 2*tau. So,
output either h or not(h) as appropriate.
(Note: for this to imply PAC learning for all D, need small SQ-dim
for all D and also need to be able to construct hyps in poly time.)
Now, let's go to Theorem 2, which is the more interesting one. To
make things simpler, let's change "nearly uncorrelated to
"uncorrelated". That is, we will show that if C contains a super-poly
size set of uncorrelated functions over D, then C is not even weakly
SQ-learnable over D.
Fourier analysis of finite functions
-------------------------------------
Not inherently complicated. But it *is* subtle conceptually. I think it's
neat, so *please* stop me when it gets confusing....
First of all, let's think of positive as +1 and negative as -1, so a
boolean concept is a function from {0,1}^n to {-1, +1}.
Think of a function f as a vector of 2^n elements, one for each
input. We'll associate with f the vector:
(sqrt(Pr(000)) * f(000), sqrt(Pr(001)) * f(001), ..., sqrt(Pr(111) * f(111))
In other words, this is the truth table of f, but weighted by sqrt(Pr(x)).
What is the (L^2) length of f as a vector? = 1.
What is ?
= sum_x Pr(x)f(x)g(x) = E_D[f(x)g(x)] = Pr(f(x)=g(x)) - Pr(f(x)!=g(x)).
Let's call this the correlation of f and g with respect to distrib D.
= 0 means that f and g are uncorrelated. They are orthogonal as
vectors.
So, this is a nice way of looking at a set of functions. Each
function is a vector in this (2^n)-dimensional space, and the dot
product between two vectors tells us how correlated they are.
"Fourier analysis" is just a way of saying that we want to talk about
what happens when you make a change of basis.
An "orthonormal basis" is a set of orthogonal unit vectors that span
the space. For instance, in 2-d, let x' be the unit vector in the
x direction and y' be the unit vector in the y direction. Then a
vector v = (3,2) is 3*x' + 2*y'. Say we have two other orthogonal
unit vectors a and b. Then could also write v in terms of a and b as:
v = a + b
This is called a change of basis.
In our 2^n dimensional space, an orthonormal basis is a set of 2^n
orthogonal unit vectors. For instance, if D is the uniform
distribution, then the parity functions are one possible orthonormal basis.
Lets fix one: phi_1, phi_2, ..., phi_{2^n}.
Given a vector f, let f_i be the ith entry in the standard basis.
I.e., f_i = f(i)*sqrt(Pr(i)).
then hat{f}_i = is the ith entry in the phi basis.
For instance, we could write the vector f as: f = sum_i hat{f}_i * phi_i
The hat{f}_i are called the "fourier coefficients of f" in the phi basis.
Since, as a vector, we have f = sum_i hat{f}_i * phi_i, this also
means that as functions,
f(x) = sum_i hat{f}_i phi_i(x)
The reason is that the left-hand-side is the xth coordinate of f,
divided by sqrt(Pr(x)). So, all we're saying is that if one vector is
the sum of a bunch of other vectors, then the xth coordinate of the
vector is the sum of the xth coordinates of the other vectors.
So, nothing fancy so far, but here's one neat thing that comes out of
the point of view we're using: Since f is a unit vector, the sum of
the squares of its coefficients is equal to 1. So,
\sum_i (\hat{f}_i)^2 = 1.
(This is called "Parseval's identity")
One thing this implies is: at most t^2 of the \hat{f}_i can be greater
than 1/t.
In other words, any boolean function f can be weakly correlated with
at most a polynomial number of the phi's. More generally if we have
some set S of pairwise uncorrelated functions, then f can be weakly
correlated with at most a polynomial number of functions in S (since
we can extend this set arbitrarily into an orthonormal basis).
Here is the point: Suppose we ask a query "what is the correlation of
the target with my hypothesis h?" with some tau = 1/poly. Then if the
target function is chosen randomly from some superpolynomial-sized set
of uncorrelated functions (like parity functions) then w.h.p. the
correct answer is infinitesimally small, so a legal response is 0.
(each question throws out only a poly number of possibilities).
Or, to think about what this means going back to the PAC-style
setting: if you have a fixed set of data and you're doing
hill-climbing, then each time you do a check on your current error,
whp the correct answer is infinitesimally close to 1/2, and all the
variation is due to the variation in the data sample. It will take
more than poly number of steps to stumble on a good query.
It turns out that ANY Stat query can be basically viewed in this way.
Proof:
Step 1: Say the set S of orthogonal functions is phi_1, ..., phi_m.
Let's extend this to an orthonormal basis arbitrarily
phi_1, ..., phi_m,..., phi_{2^n}
(only the first m of these are functions in our concept class)
Step 2: chi(x,l) is a function from {0,1}^n x {-1,1} to [-1,1].
(Might as well extend range to [-1,1]). So we can think of chi as a
vector but in 2^{n+1} dimensions. To use fourier analysis, we need to
extend our basis to this higher-dimensional space. Can do it like this:
* define distrib D' = D x (uniform on {-1, +1})
* Define phi_i(x,l) = phi_i(x).
still orthogonal:
Pr_D'[phi_i(x,l) = phi_j(x,l)] = Pr_D[phi_i(x)=phi_j(x)] = 1/2
* Need 2^n more basis functions. Let's define
h_i(x,l) = phi_i(x) * l. Need to verify this is OK:
- check that h_i and h_j are orthogonal (for i != j)
- check that h_i and phi_j are orthogonal
Pr[h_i(x,l) = phi_j(x)] = Pr[l*phi_i(x) = phi_j(x)]
even if i=j we get 1/2.
Step 3: Do a fourier decomposition of chi.
chi = sum_i alpha_i*phi_i + sum_i beta_i*h_i
where sum_i (alpha_i)^2 + sum_i (beta_i)^2 = 1.
So, we can write the quantity we're asking about
E_D[chi(x, c(x))]
= E_D[ sum_i alpha_i*phi_i(x) + sum_i beta_i*h_i(x,c(x))]
= sum_i alpha_i*E_D[phi_i(x)] + sum_i beta_i*E_D[c(x)phi_i(x)]
The first term is some constant offset that depends on the query but
is independent of the target function.
What about the second term? Well, since c = phi_target is one of the
phi's, and since they are orthogonal, that expectation is 0 except for
phi_target. So, the second term is just beta_target.
And, since at most 1/tau^2 of the betas can be larger than tau, since
the target was chosen randomly from superpolynomial-sized set, this is
whp less than tau. So, can whp respond with just the first part
that is independent of the target function.
That's it.