Week 12

12: Support Vector Machines (SVMs)

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Support Vector Machine (SVM) - Optimization objective

  • So far, we've seen a range of different algorithms
    • With supervised learning algorithms - performance is pretty similar
      • What matters more often is;
        • The amount of training data
        • Skill of applying algorithms
  • One final supervised learning algorithm that is widely used - support vector machine (SVM)
    • Compared to both logistic regression and neural networks, a SVM sometimes gives a cleaner way of learning non-linear functions
    • Later in the course we'll do a survey of different supervised learning algorithms

An alternative view of logistic regression

  • Start with logistic regression, see how we can modify it to get the SVM
    • As before, the logistic regression hypothesis is as follows
    • And the sigmoid activation function looks like this
    • In order to explain the math, we use z as defined above
  • What do we want logistic regression to do?
    • We have an example where y = 1
      • Then we hope hθ(x) is close to 1
      • With hθ(x) close to 1, (θT x) must be much larger than 0
    • Similarly, when y = 0
      • Then we hope hθ(x) is close to 0
      • With hθ(x) close to 0, (θT x) must be much less than 0
    • This is our classic view of logistic regression
      • Let's consider another way of thinking about the problem
  • Alternative view of logistic regression
    • If you look at cost function, each example contributes a term like the one below to the overall cost function
      • For the overall cost function, we sum over all the training examples using the above function, and have a 1/m term
  • If you then plug in the hypothesis definition (hθ(x)), you get an expanded cost function equation;
    • So each training example contributes that term to the cost function for logistic regression
  • If y = 1 then only the first term in the objective matters
    • If we plot the functions vs. z we get the following graph
      • This plot shows the cost contribution of an example when y = 1 given z
        • So if z is big, the cost is low - this is good!
        • But if z is 0 or negative the cost contribution is high
        • This is why, when logistic regression sees a positive example, it tries to set θT x to be a very large term
  • If y = 0 then only the second term matters
    • We can again plot it and get a similar graph
      • Same deal, if z is small then the cost is low
        • But if s is large then the cost is massive

SVM cost functions from logistic regression cost functions

  • To build a SVM we must redefine our cost functions
    • When y = 1
      • Take the y = 1 function and create a new cost function
      • Instead of a curved line create two straight lines (magenta) which acts as an approximation to the logistic regression y = 1 function
        • Take point (1) on the z axis
          • Flat from 1 onwards
          • Grows when we reach 1 or a lower number
        • This means we have two straight lines
          • Flat when cost is 0
          • Straight growing line after 1
      • So this is the new y=1 cost function
        • Gives the SVM a computational advantage and an easier optimization problem
        • We call this function cost1(z) 
  • Similarly
    • When y = 0
      • Do the equivalent with the y=0 function plot
      • We call this function cost0(z)
  • So here we define the two cost function terms for our SVM graphically
    • How do we implement this?

The complete SVM cost function

  • As a comparison/reminder we have logistic regression below
    • If this looks unfamiliar its because we previously had the - sign outside the expression
  • For the SVM we take our two logistic regression y=1 and y=0 terms described previously and replace with
    • cost1T x)
    • cost0T x)
  • So we get

SVM notation is slightly different

  • In convention with SVM notation we rename a few things here
  • 1) Get rid of the 1/m terms
    • This is just a slightly different convention
    • By removing 1/m we should get the same optimal values for 
      • 1/m is a constant, so should get same optimization
      • e.g. say you have a minimization problem which minimizes to u = 5
        • If your cost function * by a constant, you still generates the minimal value
        • That minimal value is different, but that's irrelevant
  • 2) For logistic regression we had two terms;
    • Training data set term (i.e. that we sum over m) = A
    • Regularization term (i.e. that we sum over n) = B
      • So we could describe it as A + λB
      • Need some way to deal with the trade-off between regularization and data set terms
      • Set different values for λ to parametrize this trade-off
    • Instead of parameterization this as A + λB
      • For SVMs the convention is to use a different parameter called C
      • So do CA + B
      • If C were equal to 1/λ then the two functions (CA + B and A + λB) would give the same value
  • So, our overall equation is
  • Unlike logistic, hθ(x) doesn't give us a probability, but instead we get a direct prediction of 1 or 0
    • So if θT x is equal to or greater than 0 --> hθ(x) = 1
    • Else --> hθ(x) = 0


Large margin intuition

  • Sometimes people refer to SVM as large margin classifiers
    • We'll consider what that means and what an SVM hypothesis looks like
    • The SVM cost function is as above, and we've drawn out the cost terms below
    • Left is cost1 and right is cost0
    • What does it take to make terms small
      • If y =1
        • cost1(z) = 0 only when z >= 1
      • If y = 0
        • cost0(z) = 0 only when z <= -1
    • Interesting property of SVM
      • If you have a positive example, you only really need z to be greater or equal to 0 
        • If this is the case then you predict 1
      • SVM wants a bit more than that - doesn't want to *just* get it right, but have the value be quite a bit bigger than zero
        • Throws in an extra safety margin factor
  • Logistic regression does something similar
  • What are the consequences of this?
    • Consider a case where we set C to be huge
      • C = 100,000
      • So considering we're minimizing CA + B
        • If C is huge we're going to pick an A value so that A is equal to zero
        • What is the optimization problem here - how do we make A = 0?
      • Making A = 0
        • If y = 1
          • Then to make our "A" term 0 need to find a value of θ so (θT x) is greater than or equal to 1
        • Similarly, if y = 0
          • Then we want to make "A" = 0 then we need to find a value of θ so (θT x) is equal to or less than -1
      • So - if we think of our optimization problem a way to ensure that this first "A" term is equal to 0, we re-factor our optimization problem into just minimizing the "B" (regularization) term, because 
        • When A = 0 --> A*C = 0 
      • So we're minimizing B, under the constraints shown below
    • Turns out when you solve this problem you get interesting decision boundaries
    • The green and magenta lines are functional decision boundaries which could be chosen by logistic regression
      • But they probably don't generalize too well
    • The black line, by contrast is the the chosen by the SVM because of this safety net imposed by the optimization graph
      • More robust separator
    • Mathematically, that black line has a larger minimum distance (margin) from any of the training examples
    •  By separating with the largest margin you incorporate robustness into your decision making process
  • We looked at this at when C is very large
    • SVM is more sophisticated than the large margin might look
      • If you were just using large margin then SVM would be very sensitive to outliers 
      • You would risk making a ridiculous hugely impact your classification boundary
        • A single example might not represent a good reason to change an algorithm
        • If C is very large then we do use this quite naive maximize the margin approach
        • So we'd change the black to the magenta
      • But if C is reasonably small, or a not too large, then you stick with the black decision boundary
    • What about non-linearly separable data?
      • Then SVM still does the right thing if you use a normal size C
      • So the idea of SVM being a large margin classifier is only really relevant when you have no outliers and you can easily linearly separable data
    • Means we ignore a few outliers 

Large margin classification mathematics (optional)


Vector inner products 

  • Have two (2D) vectors u and v - what is the inner product (uT v)?
    • Plot u on graph
      • i.e u1 vs. u2
    • One property which is good to have is the norm of a vector
      • Written as ||u||
        • This is the euclidean length of vector u
      • So ||u|| = SQRT(u12 + u22) = real number
        • i.e. length of the arrow above
        • Can show via Pythagoras 
    • For the inner product, take v and orthogonally project down onto u
      • First we can plot v on the same axis in the same way (vvs v1)
      • Measure the length/magnitude of the projection
      • So here, the green line is the projection
        • p = length along u to the intersection
        • p is the magnitude of the projection of vector v onto vector u
    • Possible to show that
      • uT v = p * ||u||
        • So this is one way to compute the inner product
      • uT v = u1v1u2v2
      • So therefore
        • p * ||u|| = u1v1u2v2
        • This is an important rule in linear algebra
      • We can reverse this too
        • So we could do 
          • vT u = v1u1+ v2u2
          • Which would obviously give you the same number
    • p can be negative if the angle between them is 90 degrees or more
      • So here p is negative
  • Use the vector inner product theory to try and understand SVMs a little better


SVM decision boundary


  • For the following explanation - two simplification
    • Set θ0= 0 (i.e. ignore intercept terms)
    • Set n = 2 - (x1, x2)
      • i.e. each example has only 2 features
  • Given we only have two parameters we can simplify our function to
  • And, can be re-written as 
    • Should give same thing 
  • We may notice that
    • The term in red is the norm of θ
      • If we take θ as a 2x1 vector
      • If we assume θ0 = 0 its still true
  • So, finally, this means our optimization function can be re-defined as 
  • So the SVM is minimizing the squared norm
  • Given this, what are the (θT x) parameters doing?
    • Given θ and given example x what is this equal to
      • We can look at this in a comparable manner to how we just looked at u and v
    • Say we have a single positive training example (red cross below)
    • Although we haven't been thinking about examples as vectors it can be described as such
    • Now, say we have our parameter vector θ and we plot that on the same axis
    • The next question is what is the inner product of these two vectors
      • p, is in fact pi, because it's the length of p for example i
        • Given our previous discussion we know 
        • T x) = p* ||θ||
                     = θ
          1xi1 + θ2xi2 
        • So these are both equally valid ways of computing θT xi
  • What does this mean?
    • The constraints we defined earlier 
      • T x) >= 1 if y = 1
      • T x) <= -1 if y = 0
    • Can be replaced/substituted with the constraints
      • p* ||θ|| >= 1 if y = 1
      • p* ||θ|| <= -1 if y = 0
    • Writing that into our optimization objective 
  • So, given we've redefined these functions let us now consider the training example below
    • Given this data, what boundary will the SVM choose? Note that we're still assuming θ0 = 0, which means the boundary has to pass through the origin (0,0)
      • Green line - small margins
        • SVM would not chose this line
          • Decision boundary comes very close to examples
          • Lets discuss why the SVM would not chose this decision boundary
    • Looking at this line 
      • We can show that θ is at 90 degrees to the decision boundary
        • θ is always at 90 degrees to the decision boundary (can show with linear algebra, although we're not going to!)
  • So now lets look at what this implies for the optimization objective
    • Look at first example (x1)
    • Project a line from xon to to the θ vector (so it hits at 90 degrees)
      • The distance between the intersection and the origin is (p1)
    • Similarly, look at second example (x2)
      • Project a line from x2 into to the θ vector
      • This is the magenta line, which will be negative (p2)
    • If we overview these two lines below we see a graphical representation of what's going on;
    • We find that both these p values are going to be pretty small
    • If we look back at our optimization objective 
      • We know we need p1 * ||θ|| to be bigger than or equal to 1 for positive examples
        • If p is small
          • Means that ||θ|| must be pretty large
      • Similarly, for negative examples we need p2 * ||θ|| to be smaller than or equal to -1
        • We saw in this example p2 is a small negative number
          • So ||θ|| must be a large number
    • Why is this a problem?
      • The optimization objective is trying to find a set of parameters where the norm of theta is small
        • So this doesn't seem like a good direction for the parameter vector (because as p values get smaller ||θ|| must get larger to compensate)
          • So we should make p values larger which allows ||θ|| to become smaller
  • So lets chose a different boundary
    • Now if you look at the projection of the examples to θ we find that pbecomes large and ||θ|| can become small
    • So with some values drawn in
    • This means that by choosing this second decision boundary we can make ||θ|| smaller
      • Which is why the SVM choses this hypothesis as better 
      • This is how we generate the large margin effect
      • The magnitude of this margin is a function of the p values
        • So by maximizing these p values we minimize ||θ|| 
  • Finally, we did this derivation assuming θ0 = 0,
    • If this is the case we're entertaining only decision boundaries which pass through (0,0)
    • If you allow θ0 to be other values then this simply means you can have decision boundaries which cross through the x and y values at points other than (0,0)
    • Can show with basically same logic that this works, and even when θ0 is non-zero when you have optimization objective described above (when C is very large) that the SVM is looking for a large margin separator between the classes

Kernels - 1: Adapting SVM to non-linear classifiers

  •  What are kernels and how do we use them
    • We have a training set
    • We want to find a non-linear boundary
    • Come up with a complex set of polynomial features to fit the data
      • Have hθ(x) which 
        • Returns 1 if the combined weighted sum of vectors (weighted by the parameter vector) is less than or equal to 0
        • Else return 0
      • Another way of writing this (new notation) is
        • That a hypothesis computes a decision boundary by taking the sum of the parameter vector multiplied by a new feature vector f, which simply contains the various high order x terms
        • e.g.
          • hθ(x) = θ0+ θ1f1+ θ2f+ θ3f3
          • Where
            • f1= x1
            • f= x1x2
            • f= ...
            • i.e. not specific values, but each of the terms from your complex polynomial function
      • Is there a better choice of feature f than the high order polynomials?
        • As we saw with computer imaging, high order polynomials become computationally expensive
  • New features 
    • Define three features in this example (ignore x0)
    • Have a graph of xvs. x(don't plot the values, just define the space)
    • Pick three points in that space
    • These points l1, l2, and l3, were chosen manually and are called landmarks
      • Given x, define f1 as the similarity between (x, l1)
        • = exp(- (|| x - l||2 ) / 2σ2)
        • || x - l|| is the euclidean distance between the point x and the landmark lsquared
          • Disussed more later
        • If we remember our statistics, we know that 
          • σ is the standard deviation
          • σis commonly called the variance
      • Remember, that as discussed
    • So, f2 is defined as
      • f2 = similarity(x, l1) = exp(- (|| x - l||) / 2σ2)
    • And similarly
      • f3 = similarity(x, l2) = exp(- (|| x - l||) / 2σ2)
    • This similarity function is called a kernel
      • This function is a Gaussian Kernel
    • So, instead of writing similarity between x and l we might write
      • f1 = k(x, l1)

Diving deeper into the kernel

  • So lets see what these kernels do and why the functions defined make sense
    • Say x is close to a landmark
      • Then the squared distance will be ~0
        • So
          • Which is basically e-0
            •  Which is close to 1
      • Say x is far from a landmark
        • Then the squared distance is big
          • Gives e-large number
            • Which is close to zero
      • Each landmark defines a new features
  • If we plot f1 vs the kernel function we get a plot like this
    • Notice that when x = [3,5] then f1 = 1
    • As x moves away from [3,5] then the feature takes on values close to zero
    • So this measures how close x is to this landmark

What does σ do?

  • σis a parameter of the Gaussian kernel
    • Defines the steepness of the rise around the landmark
  • Above example σ2 = 1
  • Below σ2 = 0.5
    • We see here that as you move away from 3,5 the feature f1 falls to zero much more rapidly
  • The inverse can be seen if σ2 = 3


  • Given this definition, what kinds of hypotheses can we learn?
    • With training examples x we predict "1" when
    • θ0+ θ1f1+ θ2f+ θ3f3 >= 0
      • For our example, lets say we've already run an algorithm and got the
        • θ= -0.5
        • θ= 1
        • θ= 1
        • θ= 0
      • Given our placement of three examples, what happens if we evaluate an example at the magenta dot below?
      • Looking at our formula, we know f1 will be close to 1, but f2 and f3 will be close to 0
        • So if we look at the formula we have
          • θ0+ θ1f1+ θ2f+ θ3f3 >= 0
          • -0.5 + 1 + 0 + 0 = 0.5
            • 0.5 is greater than 1
      • If we had another point far away from all three
        • This equates to -0.5
          • So we predict 0
    • Considering our parameter, for points near l1 and l2 you predict 1, but for points near lyou predict 0
    • Which means we create a non-linear decision boundary that goes a lil' something like this;
      • Inside we predict y = 1
      • Outside we predict y = 0
  • So this show how we can create a non-linear boundary with landmarks and the kernel function in the support vector machine
    • But
      • How do we get/chose the landmarks
      • What other kernels can we use (other than the Gaussian kernel)

Kernels II

  • Filling in missing detail and practical implications regarding kernels
  • Spoke about picking landmarks manually, defining the kernel, and building a hypothesis function
    • Where do we get the landmarks from?
    • For complex problems we probably want lots of them

Choosing the landmarks

  • Take the training data
  • For each example place a landmark at exactly the same location
  • So end up with m landmarks
    • One landmark per location per training example
    • Means our features measure how close to a training set example something is
  • Given a new example, compute all the f values
    • Gives you a feature vector f (f0 to fm)
      • f0 = 1 always
  • A more detailed look at generating the f vector
    • If we had a training example - features we compute would be using (xi, yi)
      • So we just cycle through each landmark, calculating how close to that landmark actually xi is
        • f1i, = k(xi, l1)
        • f2i, = k(xi, l2)
        • ...
        • fmi, = k(xi, lm)
      • Somewhere in the list we compare x to itself... (i.e. when we're at fii)
        • So because we're using the Gaussian Kernel this evalues to 1
      • Take these m features (f1, f2 ... fm) group them into an [m +1 x 1] dimensional vector called f
        • fi is the f feature vector for the ith example
        • And add a 0th term = 1
  • Given these kernels, how do we use a support vector machine

SVM hypothesis prediction with kernels

  • Predict y = 1 if (θT f) >= 0
    • Because θ = [m+1 x 1] 
    • And f = [m +1 x 1] 
  • So, this is how you make a prediction assuming you already have θ
    • How do you get θ?

SVM training with kernels

  • Use the SVM learning algorithm
    • Now, we minimize using f as the feature vector instead of x
    • By solving this minimization problem you get the parameters for your SVM
  • In this setup, m = n
    • Because number of features is the number of training data examples we have 
  • One final mathematic detail (not crucial to understand)
    • If we ignore θ0 then the following is true
    • What many implementations do is 
      • Where the matrix M depends on the kernel you use
      • Gives a slightly different minimization - means we determine a rescaled version of θ
      • Allows more efficient computation, and scale to much bigger training sets
      • If you have a training set with 10 000 values, means you get 10 000 features
        • Solving for all these parameters can become expensive
        • So by adding this in we avoid a for loop and use a matrix multiplication algorithm instead 
  • You can apply kernels to other algorithms
    • But they tend to be very computationally expensive
    • But the SVM is far more efficient - so more practical
  • Lots of good off the shelf software to minimize this function
  • SVM parameters (C)
    • Bias and variance trade off
    • Must chose C
      • C plays a role similar to 1/LAMBDA (where LAMBDA is the regularization parameter)
    • Large C gives a hypothesis of low bias high variance --> overfitting
    • Small C gives a hypothesis of high bias low variance --> underfitting
  • SVM parameters (σ2)
    • Parameter for calculating f values
      • Large σ2 - f features vary more smoothly - higher bias, lower variance
      • Small σ2 - f features vary abruptly - low bias, high variance

SVM - implementation and use

  • So far spoken about SVM in a very abstract manner
  • What do you need to do this
    • Use SVM software packages (e.g. liblinear, libsvm) to solve parameters θ
    • Need to specify
      • Choice of parameter C
      • Choice of kernel

Choosing a kernel

  • We've looked at the Gaussian kernel
    • Need to define σ (σ2)
      • Discussed σ2
    • When would you chose a Gaussian?
      • If n is small and/or m is large
        • e.g. 2D training set that's large
    • If you're using a Gaussian kernel then you may need to implement the kernel function
      • e.g. a function
      • fi = kernel(x1,x2)
        • Returns a real number
      • Some SVM packages will expect you to define kernel
      • Although, some SVM implementations include the Gaussian and a few others
        • Gaussian is probably most popular kernel
    • NB - make sure you perform feature scaling before using a Gaussian kernel 
      • If you don't features with a large value will dominate the f value
  • Could use no kernel - linear kernel
    • Predict y = 1 if (θT x) >= 0
      • So no f vector
      • Get a standard linear classifier
    • Why do this?
      • If n is large and m is small then
        • Lots of features, few examples
        • Not enough data - risk overfitting in a high dimensional feature-space
  • Other choice of kernel
    • Linear and Gaussian are most common
    • Not all similarity functions you develop are valid kernels
      • Must satisfy Merecer's Theorem
      • SVM use numerical optimization tricks
        • Mean certain optimizations can be made, but they must follow the theorem
    • Polynomial Kernel
      • We measure the similarity of x and l by doing one of
        • (xT l)2 
        • (xT l)3 
        • (xT l+1)3 
      • General form is
        • (xT l+Con)D 
      • If they're similar then the inner product tends to be large
      • Not used that often
      • Two parameters
        • Degree of polynomial (D)
        • Number you add to l (Con)
      • Usually performs worse than the Gaussian kernel
      • Used when x and l are both non-negative
    • String kernel
      • Used if input is text strings
      • Use for text classification
    • Chi-squared kernel
    • Histogram intersection kernel

Multi-class classification for SVM

  • Many packages have built in multi-class classification packages
  • Otherwise use one-vs all method
  • Not a big issue

Logistic regression vs. SVM

  • When should you use SVM and when is logistic regression more applicable
  • If n (features) is large vs. m (training set)
    • e.g. text classification problem
      • Feature vector dimension is 10 000
      • Training set is 10 - 1000
      • Then use logistic regression or SVM with a linear kernel
  • If n is small and m is intermediate
    • n = 1 - 1000
    • m = 10 - 10 000
    • Gaussian kernel is good
  • If n is small and m is large
    • n = 1 - 1000
    • m = 50 000+
      • SVM will be slow to run with Gaussian kernel
    • In that case
      • Manually create or add more features
      • Use logistic regression of SVM with a linear kernel
  • Logistic regression and SVM with a linear kernel are pretty similar
    • Do similar things
    • Get similar performance
  • A lot of SVM's power is using diferent kernels to learn complex non-linear functions
  • For all these regimes a well designed NN should work
    • But, for some of these problems a NN might be slower - SVM well implemented would be faster
  • SVM has a convex optimization problem - so you get a global minimum
  • It's not always clear how to chose an algorithm
    • Often more important to get enough data
    • Designing new features
    • Debugging the algorithm
  • SVM is widely perceived a very powerful learning algorithm
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