15-859(B) Machine Learning Theory 02/20/12
* Better MB->Batch conversion
* SVMs
* L_2 Margin bounds
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We've seen several algorithms for learning linear separators (Winnow,
Perceptron). Today will talk about SVMs and more about sample-size
bounds for learning over iid data.
Recap of facts so far
=====================
1. VC-dim of linear separators in n-dim space is n+1. So this says
that O(1/epsilon[n*log(1/epsilon)+log(1/delta)] examples are
sufficient for learning.
2. But, we've seen that if there's a large margin, can get away with less.
Let's assume target is w^* . x > 0, and has zero error. |w^*| = 1,
and all |x| <=1. Given a sample S, define the L_2 margin to be:
gamma = min_{x in S} |w^* . x|
In this case, the Perceptron algorithm makes at most M = 1/gamma^2
mistakes. Can then convert this into a sample-size bound for iid
data.
Topic 1: Better MB=>Batch conversion
====================================
Let's do the improved online->batch conversion formally, since we
didn't quite do it earlier in class.
Theorem: If have conservative alg with mistake-bound M, can use to get
PAC sample-complexity O((1/epsilon)[M + log(1/delta)])
Proof: To do this, we will split data into a ``training set'' S_1 of size
max[(4M/epsilon), (16/epsilon)*ln(1/delta)]
and a ``test set'' S_2 of size
(32/epsilon)*ln(M/delta)
We will run the algorithm on S_1 and test all hypotheses produced on S_2.
Claim 1: w.h.p., at least one hyp produced on S_1 has error < epsilon/2.
Proof (formally, use martingales):
- If all are >= epsilon/2 then the expected number of mistakes >= 2M.
- By Chernoff, Pr[# mistakes <= M] <= e^{(-expect)/8} <= delta.
- View as game: after M mistakes, alg forced to reveal target. If
alg keeps giving bad hyps, then whp will be forced to do it.
Claim 2: w.h.p., best one on S_2 has error < epsilon.
Proof: Suffices to show that good one is likely to look better
than 3*epsilon/4, and all with true error > epsilon are likely to look
worse than 3*epsilon/4. Just apply Chernoff again to the set of M
hypotheses as in your homework.
Note that this could be better or worse than the dimension bound.
In fact, can show that whp *any* large-margin separator you can
find will generalize well from roughly this much data. So, this
motivates why SVMs find large margin separators.
Support-Vector Machines
=======================
Support vector machines do convex optimization to find the maximum
margin separator, and more generally to optimize a given tradeoff
between margin and hinge-loss. Let's first do the easier case, where
we assume data is linearly separable and we want the separator of
maximum margin. Then we could write that as: minimize |w|^2 subject
to the constraint that l(x)*(w.x) >= 1 for all examples x in our
training set (l(x) is the label of x). This is a convex optimization
problem, so we can do it. Equivalently we could fix |w|^2 <= 1 and
maximize gamma s.t. l(x)(w.x) >= gamma, but people like to do it this
way, for reasons that will make more sense in a minute. Note that if
we set the RHS to 1, then 1/|w| is the margin and so |w|^2 = 1/gamma^2.
More generally, we want a tradeoff betwen margin and hinge-loss, so
what SVMs really do is (C is a given constant):
minimize: |w|^2 + C sum_{x in S} epsilon_x
subject to: l(x)(w.x) >= 1 - epsilon_x, for all x in S.
and epsilon_x >= 0 for all x in S.
These "epsilon_x" variables are called slack variables.
[also write down objective function divided by |S| since will be
easier conceptually for motivation below]
Here is the motivation. What we *really* want is to minimize the true
error err(h), but what we observe is our empirical error err_S(h).
So, let's split err(h) into two parts: (1) err(h) - err_S(h), which
is the amount we're overfitting, and (2) err_S(h). One bound on part
(1) is approximately (1/gamma^2)/|S|, if err_S(h) is small (since our
sample complexity bound is approximately |S| = (1/gamma^2)(1/epsilon)).
So, this is 1/|S| times the first part of the objective function. The
second part of the optimization (for C=1) is our total hinge-loss,
summed over all the examples. Dividing by |S| we get our average
hinge-loss, which is an upper bound on err_S(h). So, an upper bound
on (2) is 1/|S| times the second part of the objective function.
So, the two parts of the objective function are upper bounds on the
two quantities we care about: overfitting and empirical error. And
if you feel your upper bound on (1) is too loose, you might want to
set C to be larger.
Kernel functions
================
One nice thing about Perceptron and SVM is they can be kernelized.
Remember, a kernel is a function K(x,y) such that for some implicit
phi, K(x,y) = .
Intuitively, you want to choose a kernel that you think would be a
reasonable way to measure similarity for the problem at hand, though
the theoretical bounds we have been giving are in terms of the margin.
Next lecture we will look at ways to connect these two views. (Note:
ideal kernel: K(x,y)=1 if x,y have the same label, K(x,y)=-1 if they
have different label).
On hwk, we looked at how you can build legal kernels out of other
kernels. Can use extensions to show that e^{-||x-y||^2} is a legal
kernel too.
More about margins
==================
- We've seen that having a large margin is a good thing because then
the perceptron algorithm will behave well. It turns out another
thing we can say is that whp, *any* separator with a large margin
over the data will have low error. This then motivates finding the
maximum margin separator, which is what is done in Support Vector
Machines. (Can also approximate it using Perceptron by doing an
update on examples that it just gets "barely correct".)
Sample complexity analysis
==========================
The sample complexity analysis is done in two steps.
First thing to show: what is the maximum number of points that can be
split in all possible ways by a separator of margin at least gamma?
a.k.a., "fat-shattering dimension". Ans: O(1/gamma^2).
Can anyone see a simple proof?
Proof: simple proof is just to consider perceptron algorithm. Suppose
gamma-fat-shattering dimension is d. Then can force perceptron alg to
make d mistakes, and yet still have a separator w^* of margin gamma.
But we know the number of mistakes is at most 1/gamma^2. So, that's it.
Second part: now want to apply this to get a sample-complexity bound.
Sauer's lemma still applies [to bound the number of ways to split a
set of m points by margin gamma, as a function of the fat-shattering
dimension] so seems like analysis we used for VC-dimension should just
go right through, but it's actually quite not so easy. Plus there's
one technical fact we'll need. Let's do the analysis and will just
give a citation for the technical fact we need.
Analysis: Draw 2m points from D. Want to show it is unlikely there
exists a separator that gets first half correct by margin gamma, but
has more than epsilon*m mistakes on the 2nd half. This then implies the
conclusion we want, by same reasoning as when we argued the VC bounds.
As in VC proof, will show stronger statement: for *any* sets S1, S2 of size
m, each, whp this is true over randomization of pairs {x_i, x_i'} into
T1, T2. (Let S = S1 \union S2). In VC argument, we said: fix some h
that makes at least epsilon*m mistakes. Said that Prob(all mistakes
are in T2) is at most 2^{-epsilon*m}. Then applied union bound over all
labelings of data using h in C. For us, it's tempting to say: let's
count the number of separators of S with margin gamma over
all of S. But this might be undercounting since what about
separators where h only has margin gamma on T1? Instead, we'll do the
following more complicated thing. First, let's assume all ||x||=1. Now
let's group the separators together. Define h(x) = but truncated
at +/- gamma. Let dist_S(h1,h2) to be max_{x in S}|h1(x) - h2(x)|. We want
a "gamma/2-cover": a set H of separators such that every other
separator is within gamma/2 of some separator in H. Claim is: there
exists an H that is not too large, as a function of fat-shattering
dimension [Alon et al]. Can view this as a souped-up version of
Sauer's lemma. Roughly you get |H| ~ (m/gamma^2)^(log(m)/gamma^2).
Now, for these functions, define "correct" as "correct by margin at least
gamma/2" and define "mistake" as "mistake OR correct by less than
gamma/2". Our standard VC argument shows that so long as m is large
compared to (1/epsilon)*log(|H|/delta), whp, none of these will get T1
all correct, and yet make > epsilon*m "mistakes" on T2. This then
implies (by defn of H) that whp *no* separator gets T1 correct by
margin >= gamma and has > epsilon*m real mistakes on T2.
log(|H|) is approximately log^2(m)/gamma^2, so in the end you get a
bound of m = O(1/epsilon [1/gamma^2 log^2(1/(gamma*epsilon)) + log(1/delta)]).
Notice this is almost as good as the Perceptron bound.
Luckiness functions
===================
Basic idea of margins was in essense to view some separators as
"simpler" than others, using margin as the notion of "simple". What
makes this different from our Occam bounds, is that the notion of
"simple" depends on the data. Basically, we have a data-dependent
ordering of functions such that if we're lucky and the the target has
low complexity in this ordering, then we don't need much training
data. More generally, things like this are called "luckiness
functions". If a function is a "legal notion of luckiness"
(basically, the ordering depends only on the data points and not their
labels, and not too many splits of data with small complexity) then
you can apply sample complexity bounds.