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========= Assignment 3: Support Vector Machines ============ | ========= Assignment 3: Support Vector Machines ============ | ||
+ | |||
+ | Due: October 20th at 6pm | ||
===== Part 1: SVM with no bias term ===== | ===== Part 1: SVM with no bias term ===== | ||
- | Formulate a soft-margin SVM without the bias term, i.e. $f(\x) = \w^{\tr} \x$. | + | Formulate a soft-margin SVM without the bias term, i.e. $f(\mathbf{x}) = \mathbf{w}^{T} \mathbf{x}$. |
Derive the saddle point conditions, KKT conditions and the dual. | Derive the saddle point conditions, KKT conditions and the dual. | ||
Compare it to the standard SVM formulation. | Compare it to the standard SVM formulation. | ||
Line 10: | Line 12: | ||
Hint: consider the difference in the constraints. | Hint: consider the difference in the constraints. | ||
- | Discuss the merit of the bias-less formulation as the dimensionality | + | ===== Part 2: Closest Centroid Algorithm ===== |
- | of the data (or the feature space) is varied. | + | |
- | When using this SVM formulation it may be useful to add a constant to the | + | Express the closest centroid algorithm in terms of kernels, i.e. determine how the coefficients $\alpha_i$ will be computed using a given labeled dataset. |
- | kernel matrix. Explain why this can be beneficial. | + | |
+ | ===== Part 3: Soft-margin SVM for separable data ===== | ||
+ | |||
+ | Consider training a soft-margin SVM | ||
+ | with $C$ 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? | ||
+ | |||
+ | ===== Part 4: Using SVMs ===== | ||
+ | |||
+ | 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 {{assignments:scop_motif.data|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. | ||
+ | I chose to represent the proteins in | ||
+ | terms of their motif composition. A sequence motif is a | ||
+ | pattern of nucleotides/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. It is also 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. [[http://bioinformatics.oxfordjournals.org/content/19/suppl_1/i26.abstract | 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 SVM parameter soft-margin parameter. | ||
+ | The use of the SVM class is discussed in the PyML [[http://pyml.sourceforge.net/tutorial.html#svms|tutorial]], and by using help(SVM) in the python interpreter. | ||
+ | |||
+ | By default, a dataset is instantiated with a linear kernel attached to it. | ||
+ | To use a different kernel you need to attach a new kernel to the dataset: | ||
+ | <code python> | ||
+ | >>> from PyML import ker | ||
+ | >>> data.attachKernel(ker.Gaussian(gamma = 0.1)) | ||
+ | </code> | ||
+ | or | ||
+ | <code python> | ||
+ | >>> from PyML import her | ||
+ | >>> data.attachKernel(ker.Polynomial(degree = 3)) | ||
+ | </code> | ||
+ | Alternatively, you can instantiate an SVM with a given kernel: | ||
+ | <code python> | ||
+ | >>> classifier = SVM(ker.Gaussian(gamma = 0.1)) | ||
+ | </code> | ||
+ | This will override the kernel the data is associated with. | ||
+ | |||
+ | In this question we will consider both the Gaussian and polynomial kernels: | ||
+ | $$ | ||
+ | K_{gauss}(\mathbf{x}, \mathbf{x'}) = \exp(-\gamma || \mathbf{x} - \mathbf{x}' ||^2) | ||
+ | $$ | ||
+ | $$ | ||
+ | K_{poly}(\mathbf{x}, \mathbf{x'}) = (1 + \mathbf{x}^T \mathbf{x}') ^{p} | ||
+ | $$ | ||
+ | Plot the accuracy of the SVM, measured using the balanced success rate | ||
+ | as a function of both the soft-margin parameter of the SVM, and the free parameter | ||
+ | of the kernel function. | ||
+ | Show a couple of representative cross sections of this plot for a given value | ||
+ | of the ridge 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). | ||
+ | |||
+ | For this type of sparse dataset it is useful to normalize the input features. | ||
+ | One way to do so is to divide each input example by its norm. This is | ||
+ | accomplished in PyML by: | ||
+ | <code python> | ||
+ | >>> data.normalize() | ||
+ | </code> | ||
+ | Compare the results under this normalization with what you obtain | ||
+ | without normalization. | ||
+ | |||
+ | You can visualize the whole kernel matrix associated with the data using the following commands: | ||
+ | <code python> | ||
+ | >>> from PyML import ker | ||
+ | >>> ker.showKernel(data) | ||
+ | </code> | ||
+ | Explain the structure that you are seeing in the plot (it is more | ||
+ | interesting when the data is normalized). | ||
+ | |||
+ | ===== Submission ===== | ||
+ | |||
+ | Submit your report via RamCT. 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. | ||
+ | Also, please check-in a text file named README that describes what you found most difficult in completing this assignment (or provide that as a comment on ramct). | ||
+ | |||
+ | ===== Grading ===== | ||
+ | |||
+ | Here is what the grade sheet will look like for this assignment. 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. You can use a few lines of python code or pseudo-code. If your code is more than a few lines, you can include it as an appendix to your report. For example, for the first part of the assignment, provide the protocol you use to evaluate classifier accuracy. | ||
+ | * You can provide results in the form of tables, figures or text - whatever form is most appropriate for a given problem. There are no rules about how much space each answer should take. BUT we will take off points if we have to wade through a lot of redundant data. | ||
+ | * 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? | ||
+ | |||
+ | <code> | ||
+ | Grading sheet for assignment 2 | ||
+ | |||
+ | Part 1: 30 points. | ||
+ | (10 points): Lagrangian found correctly | ||
+ | ( 5 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: 15 points. | ||
+ | |||
+ | Part 3: 15 points. | ||
+ | |||
+ | Part 1: 40 points. | ||
+ | (25 points): Accuracy as a function of parameters and discussion of the results | ||
+ | (10 points): Comparison of normalized and non-normalized results | ||
+ | ( 5 points): Visualization of the kernel matrix and observations made about it | ||
+ | |||
+ | Report structure, grammar and spelling: 15 points | ||
+ | ( 5 points): Heading and subheading structure easy to follow and | ||
+ | clearly divides report into logical sections. | ||
+ | ( 5 points): Code, math, figure captions, and all other aspects of | ||
+ | report are well-written and formatted. | ||
+ | ( 5 points): Grammar, spelling, and punctuation. | ||
+ | </code> |