Due: October 16th at 11pm

Formulate a soft-margin SVM without the bias term, i.e. one where the discriminant function is equal to $\mathbf{w}^{T} \mathbf{x}$. Derive the saddle point conditions, KKT conditions and the dual. Compare it to the standard SVM formulation that was derived in class. In class we discussed SMO-type algorithms for optimizing the dual SVM. At each step SMO optimizes two variables at a time, which is the smallest number possible. Is this still the case for the formulation you have derived? In other words, is two the smallest number of variables that can be optimized at a time? Hint: consider the difference in the constraints.

Consider training a soft-margin SVM with the soft margin constant $C$ set to some positive constant. Suppose the training data is linearly separable. Since increasing the $\xi_i$ can only increase the objective of the primal problem (which we are trying to minimize), at the optimal solution to the primal problem, all the training examples will have $\xi_i$ equal to zero. True or false? Explain! Given a linearly separable dataset, is it necessarily better to use a a hard margin SVM over a soft-margin SVM? Explain!

The data for this question comes from a database called SCOP (structural classification of proteins), which classifies proteins into classes according to their structure (download it from here). The data is a two-class classification problem of distinguishing a particular class of proteins from a selection of examples sampled from the rest of the SCOP database using features derived from their sequence (a protein is a chain of amino acids, so as computer scientists, we can consider it as a sequence over the alphabet of the 20 amino acids). I chose to represent the proteins in terms of their motif composition. A sequence motif is a pattern of amino acids that is conserved in evolution. Motifs are usually associated with regions of the protein that are important for its function, and are therefore useful in differentiating between classes of proteins. A given protein will typically contain only a handful of motifs, and so the data is very sparse. Therefore, only the non-zero elements of the data are represented. Each line in the file describes a single example. Here's an example from the file:

d1scta_,a.1.1.2 31417:1.0 32645:1.0 39208:1.0 42164:1.0 ....

The first column is the ID of the protein, the second is the class it belongs to (the values for the class variable are a.1.1.2, which is the given class of proteins, and rest which is the negative class representing the rest of the database); the remainder consists of elements of the form feature_id:valuewhich provide an id of a feature and the value associated with it. This is an extension of the format used by LibSVM, that scikit-learn can read. See a discussion of this format and how to read it here.

We note that the data is very high dimensional since the number of conserved patterns in the space of all proteins is large. The data was constructed as part of the following analysis of detecting distant relationships between proteins:

• A. Ben-Hur and D. Brutlag. Remote homology detection: a motif based approach. In: Proceedings, eleventh international conference on intelligent systems for molecular biology. Bioinformatics 19(Suppl. 1): i26-i33, 2003.

In this part of the assignment we will explore the dependence of classifier accuracy on the kernel, kernel parameters, kernel normalization, and the SVM soft-margin parameter. In your implementation you can use the scikit-learn svm class.

In this question we will consider both the Gaussian and polynomial kernels: $$ K_{gauss}(\mathbf{x}, \mathbf{x'}) = \exp(-\gamma || \mathbf{x} - \mathbf{x}' ||^2) $$ and $$ K_{poly}(\mathbf{x}, \mathbf{x'}) = (\mathbf{x}^T \mathbf{x}' + 1) ^{p}. $$

Plot the accuracy of the SVM, measured using the area under the ROC curve as a function of both the soft-margin parameter of the SVM, and the free parameter of the kernel function. Accuracy should be measured in five-fold cross-validation. Show a couple of representative cross sections of this plot for a given value of the soft margin parameter, and for a given value of the kernel parameter. Comment on the results. When exploring the values of a continuous classifier/kernel parameter it is useful to use values that are distributed on an exponential grid, i.e. something like 0.01, 0.1, 1, 10, 100 (note that the degree of the polynomial kernel is not such a parameter).

Next, we will compare the accuracy of an SVM with a Gaussian kernel on the raw data with accuracy obtained when the data is normalized to be unit vectors (the values of the features of each example are divided by its norm). This is different than standardization which operates at the level of individual features. Normalizing to unit vectors is more appropriate for this dataset as it is sparse, i.e. most of the features are zero. Perform your comparison by comparing the accuracy measured by the area under the ROC curve in five-fold cross validation. The optimal values of kernel parameters should be measured by cross-validation, where the optimal SVM/kernel parameters are chosen using grid search on the training set of each fold. Use the scikit-learn grid-search class for model selection.

Finally, visualize the kernel matrix associated with the dataset. Explain the structure that you are seeing in the plot (it is more interesting when the data is normalized).

Submit the pdf of your report via Canvas. Python code can be displayed in your report if it is succinct (not more than a page or two at the most) or submitted separately. The latex sample document shows how to display Python code in a latex document. Code needs to be there so we can make sure that you implemented the algorithms and data analysis methodology correctly. Canvas allows you to submit multiple files for an assignment, so DO NOT submit an archive file (tar, zip, etc). Canvas will only allow you to submit pdfs (.pdf extension) or python code (.py extension). For this assignment there is a strict 8 page limit (not including references and code that is provided as an appendix). We will take off points for reports that go over the page limit. In addition to the code snippets that you include in your report, make sure you provide complete code from which we can see exactly how your results were generated.

A few general guidelines for this and future assignments in the course:

• Always provide a description of the method you used to produce a given result in sufficient detail such that the reader can reproduce your results on the basis of the description (UNLESS the method has been provided in class or is there in the book). Your code needs to be provided in sufficient detail so we can make sure that your implementation is correct. The saying that “the devil is in the details” holds true for machine learning, and is sometimes the makes the difference between correct and incorrect results. If your code is more than a few lines, you can include it as an appendix to your report, or submit it as a separate file. Make sure your code is readable!
• You can provide results in the form of tables, figures or text - whatever form is most appropriate for a given problem.
• In any machine learning paper there is a discussion of the results. There is a similar expectation from your assignments that you reason about your results. For example, for the learning curve problem, what can you say on the basis of the observed learning curve?
• Write succinct answers. We will take off points for rambling answers that are not to the point, and and similarly, if we have to wade through a lot of data/results that are not to the point.

Grading sheet for assignment 3

Part 1:  40 points.
(10 points):  Primal SVM formulation is correct
( 7 points):  Lagrangian found correctly
( 8 points):  Derivation of saddle point equations
(10 points):  Derivation of the dual
( 5 points):  Discussion of the implication of the form of the dual for SMO-like algorithms

Part 2:  10 points.

Part 3:  40 points.
(20 points):  Accuracy as a function of parameters and discussion of the results
(15 points):  Comparison of normalized and non-normalized kernels and correct model selection
( 5 points):  Visualization of the kernel matrix and observations made about it

Report structure, grammar and spelling:  10 points
(10 points):  Heading and subheading structure easy to follow and clearly divides report into logical sections.  
              Code, math, figure captions, and all other aspects of the report are well-written and formatted.
              Grammar, spelling, and punctuation.  Answers are clear and to the point.