ML 11: Support Vector Machines (SVMs)

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 h_θ(x) is close to 1
    • With h_θ(x) close to 1, (θ^T x) must be much larger than 0
      
  • Similarly, when y = 0
    • Then we hope h_θ(x) is close to 0
    • With h_θ(x) close to 0, (θ^T 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 (h_θ(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 θ^T 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
  • cost₁(θ^T x)
  • cost₀(θ^T 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, h_θ(x) doesn’t give us a probability, but instead we get a direct prediction of 1 or 0
  • So if θ^T x is equal to or greater than 0 –> h_θ(x) = 1
  • Else –> h_θ(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 (θ^T 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 (θ^T 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 (u^T 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 (v₁ vs 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
    • u^T v = p * ||u||
      • So this is one way to compute the inner product
    • u^T v = u₁v₁+ u₂v₂
    • So therefore
      • p * ||u|| = u₁v₁+ u₂v₂
      • This is an important rule in linear algebra
    • We can reverse this too
      • So we could do 
        • v^T 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 2×1 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 (θ^T 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 
        (θ^T xⁱ ) = pⁱ * ||θ||
                   = θ₁xⁱ₁ + θ₂xⁱ₂ 
      • So these are both equally valid ways of computing θ^T xⁱ
• What does this mean?
  • The constraints we defined earlier 
    • (θ^T x) >= 1 if y = 1
    • (θ^T x) <= -1 if y = 0
  • Can be replaced/substituted with the constraints
    • pⁱ * ||θ|| >= 1 if y = 1
    • pⁱ * ||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 x¹ on 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 p¹ becomes 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 h_θ(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.
        • h_θ(x) = θ₀+ θ₁f₁+ θ₂f₂ + θ₃f₃
        • Where
          • f₁= x₁
          • f₂ = x₁x₂
          • f₃ = …
          • 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 x₁ vs. x₂ (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 – l¹ ||² ) / 2σ²)
        = 
      • || x – l¹ || is the euclidean distance between the point x and the landmark l¹ squared
        • Disussed more later
      • If we remember our statistics, we know that 
        • σ is the standard deviation
        • σ² is commonly called the variance
    • Remember, that as discussed
      
  • So, f2 is defined as
    
    • f2 = similarity(x, l¹) = exp(- (|| x – l² ||² ) / 2σ²)
  • And similarly
    • f3 = similarity(x, l²) = exp(- (|| x – l¹ ||² ) / 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?

• σ² is 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+ θ₂f₂ + θ₃f₃ >= 0
    • For our example, lets say we’ve already run an algorithm and got the
      • θ₀ = -0.5
      • θ₁ = 1
      • θ₂ = 1
      • θ₃ = 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+ θ₂f₂ + 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 l³ you 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 f_m)
    • 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)
      • …
      • f_mⁱ, = k(xⁱ, l^m)
    • Somewhere in the list we compare x to itself… (i.e. when we’re at f_iⁱ)
      • So because we’re using the Gaussian Kernel this evalues to 1
    • Take these m features (f₁, f₂ … f_m) 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 (θ^T 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 (θ^T 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
      • (x^T l)² 
      • (x^T l)³ 
      • (x^T l+1)³ 
    • General form is
      • (x^T 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