15-859(A) Machine Learning Theory 04/13/04 * Margins, Kernels, luckiness functions. ===================================================================== We've seen a lot of results on learning linear separators. Today, will talk about more, plus something called the "kernel trick" for using linear-separator algorithms when your data doesn't actually have a good linear separator. Learning linear separators ========================== Given data in R^n, labeled +,-. Want to find a linear separator. Can do with a number of algorithms including linear programming. Notice that this includes decision lists, so pretty much everything we've seen can be transformed into a linear separator problem, with suitable definition of features. (e.g., for k-decision lists) Some things we've seen: - 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. - 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. Given a sample S, define the margin to be: gamma = min_{x in S} |w^* . x|/|x| = |cos(w^*,x)| Perceptron algorithm makes at most 1/gamma^2 mistakes. This means, using our "online->batch" conversions, we just need a training set of size: O(1/epsilon[M + log(1/delta)]) M = mistake bound = 1/gamma^2. - Old view: perceptron alg is not so good, since gamma can be exponentially small in n. (like in case of decision lists) New view: on the other hand, gamma can often be fairly large "the large margin case", in which case this is a better bound: depends only on the margin, not on the dimension!! More about margins ================== - We've seen one reason why having a large margin is a good thing: it allows the perceptron algorithm to work from a small sample. It turns out another thing we can say is that whp, *any* separator with a large margin over the data will have low error. Will get back to this in a minute. This suggests an alternative algorithm: first scale all examples to have length 1, and then solve for w of |w|<=1 such that positives have w.x >= gamma, and negatives have w.x <= -gamma, and maximize gamma. (this is a convex optimization problem). Called a "maximal margin hyperplane". Also "support vector machine". - Often people fix gamma=1 and minimize w.w. - [Aside: what if not possible to separate? One thing you can do is add slack variables: replace gamma with "gamma - epsilon_i" on ith constraint. Require epsilon_i >= 0. Then maximize gamma - alpha * sum_i epsilon_i, where "alpha" trades off the quantities. This is what we talked about last time.] Sample complexity analysis ========================== 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). 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. Seems like analysis we used for VC-dimension should just go right through, but it's actually not so clean. Let's do the analysis and will just point out the one part that's messy, and give a citation for the thing 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/2 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* set S of size 2m, whp this is true over randomization of split into two pieces S1,S2 of size m. In VC argument, we said: fix some h that makes at least epsilon*m/2 mistakes. Showed that Prob(all mistakes are on 2nd half) is at most 2^{epsilon*m/2). 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 S1? Instead, we'll do the following complicated thing. First, let's assume all |x|=1. Now let's group the separators together. Define h(x) = 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]. Now, for these guys, 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 whp, none of these will get S1 all correct, and yet make > epsilon*m/2 "mistakes" on S2. This then implies (by defn of H) that whp *no* separator gets S1 correct by margin >= gamma and has > epsilon*m/2 real mistakes on S2. More about margins ================== Here is yet another way to see why having a large margin is a good thing: if you take two vectors of angle alpha, and do a *random* linear projection down to a space of dimension O(k^2 * log(1/delta)), then with prob 1-delta, the angle of the projected vectors will be alpha +/- 1/k. This is a Johnson-Lindenstrauss lemma argument. Implication: if you have a set S of points that are separable with margin gamma, and randomly project down to a space of dimension O((1/gamma^2) log(|S|/delta)), then whp they will still be separable (in fact, separable by margin gamma/2 if we put an extra "4" into the O(..)). So it really wasn't that high-dimensional of a problem after all. Kernels ======= What if your data doesn't have a good linear separator? Here's a neat idea, called the "kernel trick". One thing we might like to do is map our data to a higher dimensional space, e.g., look at all products of pairs of features, in the hope that data will be linearly separable there. If we're lucky, will be separable by a large margin so we don't have to pay a lot in terms of data. But this is going to a pain computationally. However, one thing we notice is that most learning algorithms only access data through performing dot-products (will get back to how to interpret algs like perceptron in this way in a minute). So, maybe we can do our mapping in such a way that we have an efficient way of computing dot-products. This leads to idea of a kernel. A Kernel is a function K(x,y) such that for some mapping phi, K(x,y) = phi(x) . phi(y). Some examples: K(x,y) = (1 + )^d. K(x,y) = (1 + x_1*y_1)(1 + x_2*y_2)...(1 + x_n*y_n) [corresponds to mapping x,y to list of all products of subsets] Also string kernels [count how many substrings of length p two strings have in common] More generally, nice for the case where examples aren't so easy to map directly into R^n, but we have a reasonable notion of similarity we can encode in a kernel K. Neat fact: most learning algorithms can be run using kernels. E.g., perceptron algorithm: Initialize w = first positive example. If w.x < 0 on positive x, let w = w+x if w.x > 0 on negative x, let w = w-x. How to kernelize? notice that w is just a weighted sum of examples. E.g., w = x1 - x2 + x3, where x1,x2,x3 are examples we've seen so far. So to compute phi(w).phi(x), just do: K(x1,x)-K(x2,x)+K(x3,x). The examples that the hypothesis is written in terms of are called "support vectors". If you find the maximum margin separator for a given dataset, that is also something that can be written in terms of support vectors (not hard to see). That's the reason for the name "support vector machines". 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. PAC-MDL bounds ============== A nice way to view this is in terms of a game between two players A and B. A and B both have same set of unlabeled data. A is then given labels for a random subset of the data, and has to transmit a string that will uncompress to a labeling of the entire data set. Claim is: if can do this with not too many bits, then can be confident in labels on the test set. Point is that communication language is allowed to refer to the unlabeled data. E.g., for VC-bounds: since only C[2m] ways of splitting whole bunch, A can just give index of the split she is thinking of. Can also interpret things like "PAC-Bayes" bounds in this way. A super-simple algorithm for weak-learning a large-margin separator =================================================================== While we're at it, here is a super-simple algorithm for weak-learning a large-margin separator. Algorithm: pick h at random. if error(h) < 1/2 - gamma/4 then done. if error(h) > 1/2 + gamma/4 then done (use -h). Else repeat. How to analyze this? Let w be the vector defining target function. Since we are looking at both h and -h, we can assume wlog that w.h>=0. Now, let x be one of the examples. Let's look at: Pr_h(h.x > 0 | w.h >= 0). To analyze this, we only need to consider the 2-dim plane defined by w and x. h's projection down to this plane (ignoring length) looks like a random vector subject to having w.h >= 0. What fraction of these label x incorrectly? The answer is: angle(w,x)/pi. Approximating theta=cos(theta), this is at most (pi/2 - gamma)/pi = 1/2 - gamma/pi. So, Pr(h is wrong) <= 1/2 - gamma/pi. This means that for random h, E[min(err(h),err(-h))] <= 1/2 - gamma/pi. Now, it could be that with small probability we get a really good h and with large probability we get a not so good h. But since the quantity is bounded between 0 and 1/2, the probability of success can't be that small. Note: can then apply boosting (since that just reweights the data, and doesn't affect margins). Margins and boosting ==================== For boosting, can also get good results based on the margin of the vote (good generalization on the examples where the vote is far from 50/50). One key difference between the notion of "margin" we were discussing and the notion of margin here is that we have been discussing an "L_2 margin": we scaled examples to have length=1 in the L_2 sense. Here, you have an "L_1 margin", since you are effectively scaling the vote so the weights have length=1 in L_1 sense. Will discuss this topic later.