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-========= Assignment 3: Support Vector Machines ============
+~~NOTOC~~
-Due:  October 16th at 11pm
+====== Assignment 3 ======
-===== Part 1:  SVM with no bias term =====
+**Due:** 10/3 at 11:59pm.
-Formulate a soft-margin SVM without the bias term, i.e. one where the discriminant function is equal to $\mathbf{w}^{T} \mathbf{x}$.
+===== Preliminaries =====
-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.
-===== Part 2:  Soft-margin SVM for separable data =====
+In this assignment you will explore ridge regression applied to the task of predicting wine quality.
+You will use the [[http://archive.ics.uci.edu/ml/datasets/Wine+Quality | wine quality]] dataset from the UCI machine learning repository, and compare accuracy obtained using ridge regression to the results from a [[http://www.sciencedirect.com/science/article/pii/S0167923609001377# | recent publication]] (if you have trouble accessing that version of the paper, here's a link to a [[http://www3.dsi.uminho.pt/pcortez/wine5.pdf| preprint]].
+The wine data is composed of two datasets - one for white wines, and one for reds.  In this assignment perform all your analyses on just the red wine data.
-Consider training a soft-margin SVM
+The features for the wine dataset are not standardized, so make sure you do this, especially since we are going to consider the magnitude of the weight vector (recall that standardization entails subtracting the mean and then dividing by the standard deviation for each feature; you can use the [[http://docs.scipy.org/doc/numpy/reference/routines.statistics.html | Numpy statistics module]] to perform the required calculations).
-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!
-===== Part 3:  Using SVMs =====
+==== Part 1 ====
-The data for this question comes from a database called SCOP (structural
+Implement ridge regression in a class called RidgeRegression that implements the classifier API, i.e. ''fit'' and ''predict'' methods with the same signature as the classifiers you implemented in the previous assignment.  Also implement functions for computing the following measures of error:
-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
-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:
-<code>
+  * The Root Mean Square Error (RMSE).
-d1scta_,a.1.1.2 31417:1.0 32645:1.0 39208:1.0 42164:1.0 ....
+  * The Maximum Absolute Deviation (MAD).
-</code>
-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:value''which 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 [[http://scikit-learn.org/stable/datasets/#datasets-in-svmlight-libsvm-format | 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. [[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.
+For a hypothesis $h$, they are defined as follows:
-In this part of the assignment we will explore the dependence of classifier accuracy on
+$$RMSE(h) = \sqrt{\frac{1}{N}\sum_{i=1}^N (y_i - h(\mathbf{x}_i))^2}$$
-the kernel, kernel parameters, kernel normalization, and the SVM soft-margin parameter.
-In your implementation you can use the scikit-learn [[http://scikit-learn.org/stable/modules/generated/sklearn.svm.SVC.html | 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
+$$MAD(h) = \frac{1}{N}\sum_{i=1}^N |y_i - h(\mathbf{x}_i)|.$$
-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).
+With the code you just implemented, your next task is to explore the dependence of error on the value of the regularization parameter, $\lambda$.
-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.
+In what follows set aside 30% of the data as a test-set, and compute the in-sample error, and the test-set error as a function of the parameter $\lambda$ on the red wine data.  Choose the values of $\lambda$ on a logarithmic scale with values 0.01, 0.1, 1, 10, 100, 1000 and plot the RMSE only.
-Perform your comparison by comparing the accuracy measured by the area under the ROC curve in five-fold cross validation.
+Repeat the same experiment where instead of using all the training data, choose 20 random examples out of the training set, and train your model using those 20 examples.
-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 [[http://scikit-learn.org/stable/tutorial/statistical_inference/model_selection.html
- | grid-search]] class for model selection.
+Now answer the following:
-Finally, visualize the kernel matrix associated with the dataset.
+  * What is the optimal value of $\lambda$?
-Explain the structure that you are seeing in the plot (it is more
+  * What observations can you make on the basis of these plots?  (The concepts of overfitting/underfitting should be addressed in your answer).
-interesting when the data is normalized).
+  * Finally, compare the results that you are getting with the published results in the paper linked above.  In particular, is the performance you have obtained is comparable to that observed in the paper?
+==== Part 2 ====
+Regression Error Characteristic (REC) curves are an interesting way of visualizing regression error as described
+in the following [[http://machinelearning.wustl.edu/mlpapers/paper_files/icml2003_BiB03.pdf|paper]].
+Write a function that plots the REC curve of a regression method, and plot the REC curve of the best regressor you found in Part 1 of the assignment (i.e. the one that gave the lowest error on the validation set).  Plot the REC curve for both the validation set and the training set.
+What can you learn from this curve that you cannot learn from an error measure such as RMSE or MAD?
+==== Part 3 ====
+As we discussed in class, the magnitude of the weight vector can be interpreted as a measure of feature importance.
+Train a ridge regression classifier on a subset of the dataset that you reserved for training.
+We will explore the relationship between the magnitude of weight vector components and their relevance to the classification task in several ways.
+Each feature is associated with a component of the weight vector.  It can also be associated with the correlation of that feature with the vector of labels.
+As we discussed in class, the magnitude of the weight vector can give an indication of feature relevance; another measure of relevance of a feature is its correlation with the labels.  To compare the two,
+create a scatter plot of weight vector components against the [[https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient | Pearson correlation coefficient]] of the corresponding feature with the labels (again, you can use the [[http://docs.scipy.org/doc/numpy/reference/routines.statistics.html | Numpy statistics module]] to compute it).
+What can you conclude from this plot?
+The paper ranks features according to their importance using a different approach.  Compare your results with what they obtain.
+Next, perform the following experiment:
+Incrementally remove the feature with the lowest absolute value of the weight vector and retrain the ridge regression classifier.
+Plot RMSE as a function of the number of features that remain on the test set which you have set aside and comment on the results.
 ===== Submission =====
-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).
+Submit your report via Canvas.  Python code can be displayed in your report if it is short, and helps understand what you have done. The sample LaTex document provided in assignment 1 shows how to display Python code.  Submit the Python code that was used to generate the results as a file called ''assignment3.py'' (you can split the code into several .py files; Canvas allows you to submit multiple files).  Typing
-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.
+<code>
+$ python assignment3.py
+</code>
+should generate all the tables/plots used in your report.
 ===== Grading =====
@@ Line 103: / Line 78: @@
 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!
+  * Your answers should be concise and to the point.
-  * You can provide results in the form of tables, figures or text - whatever form is most appropriate for a given problem.
+  * You need to use LaTex to write the report.
+  * The report is well structured, the writing is clear, with good grammar and correct spelling; good formatting of math, code, figures and captions (every figure and table needs to have a caption that explains what is being shown).
+  * Whenever you use information from the web or published papers, a reference should be provided.  Failure to do so is considered plagiarism.
+We will take off points if these guidelines are not followed.
+  * 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.
+  * 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?
-  * 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.
 <code>
 Grading sheet for assignment 3
-Part 1:  40 points.
+Part 1:  50 points.
-(10 points):  Primal SVM formulation is correct
+(15 points):  Ridge regression is correctly implemented.
-( 7 points):  Lagrangian found correctly
+(15 points):  Plots of RMSE as a function of lambda are generated correctly.
-( 8 points):  Derivation of saddle point equations
+(20 points):  Discussion of the results
-(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 2:  25 points.
+(15 points):  REC curves are generated correctly
+(10 points):  discussion of REC curves
-Part 3:  40 points.
+Part 3:  25 points.
-(20 points):  Accuracy as a function of parameters and discussion of the results
+(20 points):  Weight vector analysis
-(15 points):  Comparison of normalized and non-normalized kernels and correct model selection
+( 5 points):  Comparison to the published results
-( 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.
 </code>

CS545 fall 2016

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