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COMP9417 - Machine Learning

创建日期：2024-04-03 16:38:42

所属分类：计算机computer science

标签： COMP9417

Machine Learning

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COMP9417 - Machine Learning
Homework 1: Gradient Descent & Friends
Introduction In this homework, you will be required to manually implement (Stochastic) Gradient Descent
in Python to learn the parameters of a linear regression model. You will make use of the publicly avail-
able ‘real estate.csv’ dataset, which you can download directly from the course Moodle page. The dataset
contains 414 real estate records, each of which contains the following features:
• transactiondate: date of transaction
• age: age of property
• nearestMRT: distance of property to nearest supermarket
• nConvenience: number of convenience stores in nearby locations
• latitude
• longitude
The target variable is the property price. The goal is to learn to predict property prices as a function of a
subset of the above features.
Question 1. (Pre-processing)
A number of pre-processing steps are required before we attempt to fit any models to the data.
(a) Remove any rows of the data that contain a missing (‘NA’) value. List the indices of the removed
data points. Then, delete all features from the dataset apart from: age, nearestMRT and nConve-
nience.
(b) An important pre-processing step: feature normalisation. Feature normalisation involves rescaling
the features such that thay all have similar scales. This is also important for algorithms like gradient
descent to ensure the convergence of the algorithm. One common technique is called min-max
normalisation, in which each feature is scaled to the range [0, 1]. To do this, for each feature we
must find the minimum and maximum values in the available sample, and then use the following
formula to make the transformation:
xnew =
x−min(x)
max(x)−min(x) .
After applying this normalisation, the minimum value of your feature will be 0, and the maximum
will be 1. For each of the features, provide the mean value over your dataset.
Question 2. (Train and Test sets)
Now that the data is pre-processed, we will create train and test sets to build and then evaluate our
1
models on. Use the first half of observations (in the same order as in the original csv file excluding
those you removed in the previous question) to create the training set, and the remaining half for the
test set. Print out the first and last rows of both your training and test sets.
Question 3. (Loss Function)
Consider the loss function
Lc(x, y) =
√
1
c2
(x− y)2 + 1− 1,
where c ∈ R is a hyper-parameter. Consider the (simple) linear model
yˆ(i) = w0 + w1x
(i)
1 + w2x
(i)
2 + w3x
(i)
3 , i = 1, . . . , n.
We can write this more succintly by letting w = (w0, w1, w2, w3)T and X(i) = (1, x
(i)
1 , x
(i)
2 , x
(i)
3 )
T , so that
yˆ(i) = wTX(i). The mean-loss achieved by our model (w) on a given dataset of n observations is then
Lc(y, yˆ) = 1
n
n∑
i=1
Lc(y(i), yˆ(i)) = 1
n
n∑
i=1
[√
1
c2
(y(i) − 〈w(t), X(i)〉)2 + 1− 1
]
,
Compute the following derivatives:
∂Lc(y(i)yˆ(i))
∂wk
, k = 0, 1, 2, 3.
You must show your working for full marks.
Question 4. (Gradient Descent Psuedocode)
For some value of c > 0, we have
∂Lc(y(i), yˆ(i))
∂wk
=
x
(i)
k (〈w(t), X(i)〉 − y(i))
2
√
(〈w(t), X(i)〉 − y(i))2 + 4 , k = 0, 1, 2, 3.
Using this result, write down the gradient descent updates for w0, w1, w2, w3 (using pseudocode), as-
suming a step size of η. Note that in gradient descent we consider the loss over the entire dataset, not
just at a single observation. Further, provide pseudocode for stochastic gradient descent updates. Here,
w(t) denotes the weight vector at the t-th iteration of gradient descent.
Question 5. (Gradient Descent Implementation)
In this section, you will implement gradient descent from scratch on the generated dataset using the
gradients computed in Question 3, and the pseudocode in Question 4.
(a) Initialise your weight vector to w(0) = [1, 1, 1, 1]T . Consider step sizes
η ∈ {10, 5, 2, 1, 0.5, 0.25, 0.1, 0.05, 0.01}
(a total of 9 step sizes). For each step-size, generate a plot of the loss achieved at each iteration of
gradient descent. You should use 400 iterations in total (which will require you to loop over your
training data in order). Generate a 3×3 grid of plots showing performance for each step-size. Add
a screen shot of the python code written for this question in your report (as well as pasting the
code in to your solutions.py file). The following code may help with plotting:
Page 2
1 fig, ax = plt.subplots(3,3, figsize=(10,10))
2 nIter = 400
3 alphas = [10,5,2, 1,0.5, 0.25,0.1, 0.05, 0.01]
4 for i, ax in enumerate(ax.flat):
5 # losses is a list of 9 elements. Each element is an array of length nIter
storing the loss at each iteration for
6 # that particular step size
7 ax.plot(losses[i])
8 ax.set_title(f"step size: {alphas[i]}") # plot titles
9 plt.tight_layout() # plot formatting
10 plt.show()
11
(b) From your results in the previous part, choose an appropriate step size (and state your choice), and
explain why you made this choice.
(c) For this part, take η = 0.3, re-run GD under the same parameter initilisations and number of
iterations. On a single plot, plot the progression of each of the four weights over the iterations.
Print out the final weight vector. Finally, run your model on the train and test set, and print the
achieved losses.
Question 6. (Stochastic Gradient Descent Implementation)
We will now re-run the analysis in question 5, but using stochastic gradient descent (SGD) instead. In
SGD, we update the weights after seeing a single observation.
(a) Use the same settings as in 5 (a). For each step-size, generate a plot of the loss achieved at each
iteration of stochastic gradient descent, which is to be run for 6 Epochs. An epoch is one pass
over the entire training dataset, so in total there should be 6 × ntrain updates, where ntrain is the
size of your training data. Be sure to run these updates in the same ordering that the data is
stored. Generate a 3 × 3 grid of plots showing performance for each step-size. Add a screen shot
of the python code written for this question in your report (as well as pasting the code in to your
solutions.py file).
(b) From your results in the previous part, choose an appropriate step size (and state your choice), and
explain why you made this choice.
(c) Take η = 0.4 and re-run SGD under the same parameter initilisations and number of epochs. On
a single plot, plot the progression of each of the four weights over the iterations. Print yoru final
model. Finally, run your model on the train and test set, and record the achieved losses.
Question 7. Results Analysis
In a few lines, comment on your results in Questions 5 and 6. Explain the importance of the step-size
in both GD and SGD. Explain why one might have a preference for GD or SGD. Explain why the GD
paths look much smoother than the SGD paths.
Points Allocation There are a total of 10 marks, the available marks are:
• Question 1: 1 mark (split 0.5/0.5)
• Question 2: 0.5 marks
• Question 3: 1.5 marks
• Question 4: 1 mark
• Question 5: 2.5 marks (split 1/0.5/1)
Page 3
• Question 6: 2.5 marks (split 1/0.5/1)
• Question 7: 1 mark

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 101H1 IB9520 FINM 3008 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