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CMPSC 448: Machine Learning and AI

创建日期：2024-07-02 14:45:06

所属分类：计算机computer science

标签： CMPSC 448

Machine Learning and AI

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CMPSC 448: Machine Learning and AI

This HW includes both theory and implementation problems. Please note,

Your code must work with Python 3.5+ (you may install the Anaconda distribution of Python)
You need to submit a report in PDF including all written deliverable and plots via Gradescope, and all implementation codes via Canvas so one could regenerate your results.
For non-linear classifier problem, you need to submit a Jupyter notebook for each algorithm and all complementary python codes so one could reproduce your results. Submit your code in Problem4*.py and replace * witn the name of classifier.
For clustering problem, you are NOT allowed to use the implementation from scikit-learn or import it and should submit your own implementation. Submit your code in Problem5.py.
The submission for this homework should be a single PDF file via Gradescope and a single ‘zip‘ file containing all of the relevant code, figures, and any text explaining your results via Canvas.

Kernel Methods

Problem 1 (10 points) In this problem you are asked to find the explicit mapping function Φ(·) for the Gaussian kernel k (x, x′) = exp ?− ∥x − x′∥2 /2γ2? by showing that it can be expressed as the inner product of an infinite-dimensional feature space. Assume x, x ∈ Rd and ∥x∥2 = ∥x′∥2 = 1.

Problem 2 (20 points) In the lectures we used the KKT optimally conditions to drive the dual optimization problem for hard SVM. In this problem you are asked to do the same for soft SVM. Consider the constrained optimization problem for soft SVM:

1 2 ?n min ∥w∥ +C ξi

w,b,ξ1,ξ2,…,ξn 2 i=1

subjecttoyi?w⊤xi +b?≥1−ξi,i=1,2,…,n ξi ≥ 0,i = 1,2,…,n

a) Consider two sets of dual variables: α1, . . . , αn for first set of constraints, i.e., yi ?w⊤xi + b? ≥ 1 − ξi and another set of dual variables β1, . . . , βn for second set of constrains ξi ≥ 0 and write down the Lagrangian functions.

b) For the Lagrangian function in part (a), write down the KKT optimally conditions. c) Derive the dual problem.

Boosting

Problem 3 (15 points) Consider the AdaBoost algorithm we discussed in the class 1. Ad- aBoost is an example of ensemble classifiers where the weights in next round are decided based on the training error of the weak classifier learned on the current weighted training set. We wish to run the AdaBoost on the dataset provided in Table 1.

Instance Color Size Shape Edible? D1 Yellow Small Round Yes

D2	Yellow	Small	Round	No
D3	Green	Small	Irregular	Yes
D4	Green	Large	Irregular	No
D5	Yellow	Large	Round	Yes
D6	Yellow	Small	Round	Yes
D7	Yellow	Small	Round	Yes
D8	Yellow	Small	Round	Yes
D9	Green	Small	Round	No
D10	Yellow	Large	Round	No
D11	Yellow	Large	Round	Yes
D12	Yellow	Large	Round	No
D13	Yellow	Large	Round	No
D14	Yellow	Large	Round	No
D15	Yellow	Small	Irregular	Yes
D16	Yellow	Large	Irregular	Yes

Table 1: Mushroom data with 16 instances, three categorical features, and binary labels.

a) Assume we choose the following decision stump f1 (a shallow tree with a single decision node), as the first predictor (i.e., when training instances are weighted uniformly):

  if(Color is Yellow):
        predict  Eadible = Yes

  else:
        predict Eadible = No

What would be the weight of f1 in final ensemble classifier (i.e., α1 in f(x) = ?Ki=1 αifi(x))?

b) After computing f1, we proceed to next round of AdaBoost. We begin by recomputing data weights depending on the error of f1 and whether a point was (mis)classified by f1. What is the weight of each instance in second boosting iteration, i.e., after the points have been re-weighted? Please note that the weights across the training set are to be uniformly initialized.

c) In AdaBoost, would you stop the iteration if the error rate of the current weak classifier on the weighted training data is 0?

1Please read reading assignement from textbook and (optional) the Introduction chapter from Boosting book for detailed explanation https://bit.ly/2HvKObl

Experiment with non-linear classifiers
Problem 4 (40 points) For this problem, you will need to learn to use software libraries for

at least two of the following non-linear classifier types:
• Boosted Decision Trees (i.e., boosting with decision trees as weak learner)

• Random Forests

• Support Vector Machines with Gaussian Kernel

All of these are available in scikit-learn, although you may also use other external libraries (e.g., XGBoost 2 for boosted decision trees and LibSVM for SVMs). You are welcome to implement learning algorithms for these classifiers yourself, but this is neither required nor recommended.

Pick two different types of non-linear classifiers from above for classification of Adult dataset. You can the download the data from a9a in libSVM data repository. The a9a data set comes with two files: the training data file a9a with 32,561 samples each with 123 features, and a9a.t with 16,281 test samples. Note that a9a data is in LibSVM for- mat. In this format, each line takes the form <label> <feature-id>:<feature-value> <feature-id>:<feature-value> ….. This format is especially suitable for sparse datasets. Note that scikit-learn includes utility functions (e.g., load svmlight file in example code below) for loading datasets in the LibSVM format.

For each of learning algorithms, you will need to set various hyperparameters (e.g., the type of kernel and regularization parameter for SVM, tree method, max depth, number of weak classifiers, etc for XGBoost, number of estimators and min impurity decrease for Random Forests). Often there are defaults that make a good starting point, but you may need to adjust at least some of them to get good performance. Use hold-out validation or K-fold cross-validation to do this (scikit-learn has nice features to accomplish this, e.g., you may use train test split to split data into train and test data and sklearn.model selection for K-fold cross validation). Do not make any hyperparameter choices (or any other similar choices) based on the test set! You should only compute the test error rates after you have settled on hyperparameter settings and trained your two final classifiers.

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