Completed HW5
This commit is contained in:
parent
2f85e20ee5
commit
a485e61ecd
BIN
hw5/hw5.pdf
Normal file
BIN
hw5/hw5.pdf
Normal file
Binary file not shown.
93
hw5/regularization.py
Normal file
93
hw5/regularization.py
Normal file
@ -0,0 +1,93 @@
|
||||
import math
|
||||
from math import sin, pi
|
||||
from random import random
|
||||
|
||||
|
||||
def f(x):
|
||||
return sin(pi * x)
|
||||
|
||||
|
||||
def generate_training_examples(n=2):
|
||||
xs = [random() * 2 - 1 for _ in range(n)]
|
||||
return [(x, f(x)) for x in xs]
|
||||
|
||||
|
||||
def fit_without_reg(examples):
|
||||
"""Computes values of w0 and w1 that minimize the sum-of-squared-errors cost function
|
||||
|
||||
Args:
|
||||
- examples: a list of two (x, y) tuples, where x is the feature and y is the label
|
||||
"""
|
||||
w0 = 0
|
||||
w1 = 0
|
||||
|
||||
x1, y1 = examples[0]
|
||||
x2, y2 = examples[1]
|
||||
|
||||
w1 = (y2 - y1) / (x2 - x1)
|
||||
w0 = y1 - w1 * x1
|
||||
|
||||
return w0, w1
|
||||
|
||||
|
||||
def fit_with_reg(examples, lambda_hp, learning_rate=0.01, max_iter=1000, tol=1e-6):
|
||||
"""Computes values of w0 and w1 that minimize the regularized sum-of-squared-errors cost function
|
||||
|
||||
Args:
|
||||
- examples: a list of two (x, y) tuples, where x is the feature and y is the label
|
||||
- lambda_hp: a float representing the value of the lambda hyperparameter; a larger value means more regularization
|
||||
"""
|
||||
w0 = 0
|
||||
w1 = 0
|
||||
for _ in range(max_iter):
|
||||
# Compute the sum of squared errors
|
||||
sse = sum((w0 + w1 * x - y) ** 2 for x, y in examples)
|
||||
|
||||
# Compute the regularization term
|
||||
reg_term = lambda_hp * (w0 ** 2 + w1 ** 2)
|
||||
|
||||
# Compute the total cost function
|
||||
cost = sse + reg_term
|
||||
|
||||
# Compute the partial derivatives of the cost function with respect to w0 and w1
|
||||
d_w0 = 2 * sum(w0 + w1 * x - y for x, y in examples) + 2 * lambda_hp * w0
|
||||
d_w1 = 2 * sum((w0 + w1 * x - y) * x for x, y in examples) + 2 * lambda_hp * w1
|
||||
|
||||
# Update w0 and w1 using gradient descent
|
||||
w0_new = w0 - learning_rate * d_w0
|
||||
w1_new = w1 - learning_rate * d_w1
|
||||
|
||||
# Check for convergence
|
||||
if abs(w0_new - w0) < tol and abs(w1_new - w1) < tol:
|
||||
break
|
||||
|
||||
w0, w1 = w0_new, w1_new
|
||||
|
||||
return w0, w1
|
||||
|
||||
|
||||
def test_error(w0, w1):
|
||||
n = 100
|
||||
xs = [i/n for i in range(-n, n + 1)]
|
||||
return sum((w0 + w1 * x - f(x)) ** 2 for x in xs) / len(xs)
|
||||
|
||||
def main():
|
||||
print("Starting...")
|
||||
no_reg_errors=[]
|
||||
reg_erros = []
|
||||
exp_size = 1000
|
||||
for _ in range(exp_size):
|
||||
|
||||
train_examples = generate_training_examples(n=2)
|
||||
|
||||
w0, w1 = fit_without_reg(train_examples)
|
||||
w0_r, w1_r = fit_with_reg(train_examples, lambda_hp=1)
|
||||
|
||||
no_reg_errors.append(test_error(w0, w1))
|
||||
reg_erros.append(test_error(w0_r, w1_r))
|
||||
|
||||
print(f"Avg loss, regularization: {sum(no_reg_errors)/(len(no_reg_errors))}")
|
||||
print(f"Avg loss, no regularization: {sum(reg_erros)/(len(reg_erros))}")
|
||||
|
||||
if __name__=="__main__":
|
||||
main()
|
||||
2
main.aux
2
main.aux
@ -20,4 +20,6 @@
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {4}Homework 4}{15}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Inference in Bayesian Networks}{15}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Gradient Descent}{15}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {5}Homework 5}{16}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Regularization}{16}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{17}
|
||||
|
||||
@ -1,8 +1,8 @@
|
||||
# Fdb version 4
|
||||
["pdflatex"] 1714460109.50694 "d:/uzair/projects/sfsu/csc-665-homework/main.tex" "main.pdf" "main" 1714460116.17505 0
|
||||
"C:/Users/uzair/AppData/Local/MiKTeX/fonts/map/pdftex/pdftex.map" 1714457440 80909 eab91d9745dd2edfd62a31d53cd5fe15 ""
|
||||
"C:/Users/uzair/AppData/Local/MiKTeX/fonts/pk/ljfour/jknappen/ec/dpi600/tcrm1000.pk" 1712556658 11548 a6bc95c64a58583bcc706361e95fcb96 ""
|
||||
"C:/Users/uzair/AppData/Local/MiKTeX/miktex/data/le/pdftex/pdflatex.fmt" 1712556641 24236005 6c179fc8e955800dbf130a4c2d55d82a ""
|
||||
["pdflatex"] 1715927540.38441 "d:/uzair/projects/sfsu/csc-665-homework/main.tex" "main.pdf" "main" 1715927541.49816 0
|
||||
"C:/Users/uzair/AppData/Local/MiKTeX/fonts/map/pdftex/pdftex.map" 1715922414 80909 eab91d9745dd2edfd62a31d53cd5fe15 ""
|
||||
"C:/Users/uzair/AppData/Local/MiKTeX/fonts/pk/ljfour/jknappen/ec/dpi600/tcrm1000.pk" 1715721639 11548 a903b62e24e4c2fe5a80812a4a494d70 ""
|
||||
"C:/Users/uzair/AppData/Local/MiKTeX/miktex/data/le/pdftex/pdflatex.fmt" 1715720603 24240354 70429886933c6c598c1dc6fc281f7cf2 ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/jknappen/ec/tcrm1000.tfm" 993062508 1436 c7f957a372ef2fbe93c0982f96625e12 ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1233951848 1004 54797486969f23fa377b128694d548df ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1233951854 916 f87d7c45f9c908e672703b83b72241a3 ""
|
||||
@ -44,25 +44,24 @@
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/cancel/cancel.sty" 1388370247 7592 dd751af313a16a0308545d5bfd7aaaa2 ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/epstopdf-pkg/epstopdf-base.sty" 1623003186 13886 d1306dcf79a944f6988e688c1785f9ce ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/graphics-cfg/graphics.cfg" 1465894292 1224 978390e9c2234eab29404bc21b268d1e ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/graphics-def/pdftex.def" 1663918690 19448 1e988b341dda20961a6b931bcde55519 ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/graphics-def/pdftex.def" 1713599642 19440 9da9dcbb27470349a580fca7372d454b ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/graphics/graphics.sty" 1665067579 18387 8f900a490197ebaf93c02ae9476d4b09 ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/graphics/graphicx.sty" 1665067579 8010 a8d949cbdbc5c983593827c9eec252e1 ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/graphics/keyval.sty" 1665067579 2671 7e67d78d9b88c845599a85b2d41f2e39 ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/graphics/trig.sty" 1665067579 4023 293ea1c16429fc0c4cf605f4da1791a9 ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/l3backend/l3backend-pdftex.def" 1710412547 29918 d08e4795c0778dfb000e82d9225c6ce2 ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/l3backend/l3backend-pdftex.def" 1714423267 29785 470380cd55470b90fdb46d40220ba304 ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/mathtools/mathtools.sty" 1710159379 62672 9ff036bc89365461cc2bd482cc1e4879 ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/mathtools/mhsetup.sty" 1710159379 5582 a43dedf8e5ec418356f1e9dfe5d29fc3 ""
|
||||
"C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/tools/calc.sty" 1700599895 10214 547fd4d29642cb7c80bf54b49d447f01 ""
|
||||
"d:/uzair/projects/sfsu/csc-665-homework/main.tex" 1714460101 23856 14667251c3432ebf46f977146e5a21e8 ""
|
||||
"hw2/gameTree.png" 1712556571 51736 b13904e696801b947e455a901505f683 ""
|
||||
"hw3/bayesian_network_1_hw3.png" 1712533911 42320 08df33c7ebceab0a98b148f3e43f6efe ""
|
||||
"hw3/motorcade.png" 1712535186 144191 cdb52bfe315df9dd25d1d06a6414fa5c ""
|
||||
"hw4/Gemini_Chart_Image_5v3caz5v3caz5v3c.png" 1714457302 318397 3e525423aa7c6edf007c999a2ae91ab1 ""
|
||||
"hw4/Picture1.png" 1714460025 5619 c6261e04afc55b58bedd9e8fe858bb3b ""
|
||||
"main.aux" 1714460115 2151 9484a2fc0016201d7b4e5a7ff65abe15 "pdflatex"
|
||||
"main.tex" 1714460101 23856 14667251c3432ebf46f977146e5a21e8 ""
|
||||
"titlePage.aux" 1714460110 456 c979711f53deacf2b09031977e556db8 "pdflatex"
|
||||
"titlePage.tex" 1712556571 407 ccdbd50244b3194dbad72dd1c7995bf0 ""
|
||||
"d:/uzair/projects/sfsu/csc-665-homework/main.tex" 1715927539 25340 6bbd1cec5abb985a0a933a737359bbb0 ""
|
||||
"hw2/gameTree.png" 1715628443 51736 b13904e696801b947e455a901505f683 ""
|
||||
"hw3/bayesian_network_1_hw3.png" 1715628443 42320 08df33c7ebceab0a98b148f3e43f6efe ""
|
||||
"hw3/motorcade.png" 1715628443 144191 cdb52bfe315df9dd25d1d06a6414fa5c ""
|
||||
"hw4/Picture1.png" 1715628443 5619 c6261e04afc55b58bedd9e8fe858bb3b ""
|
||||
"main.aux" 1715927541 2358 1cb49ec657da2fa194a2062e36b8235c "pdflatex"
|
||||
"main.tex" 1715927539 25340 6bbd1cec5abb985a0a933a737359bbb0 ""
|
||||
"titlePage.aux" 1715927540 456 c979711f53deacf2b09031977e556db8 "pdflatex"
|
||||
"titlePage.tex" 1715628444 407 ccdbd50244b3194dbad72dd1c7995bf0 ""
|
||||
(generated)
|
||||
"main.aux"
|
||||
"main.log"
|
||||
|
||||
7
main.fls
7
main.fls
@ -112,13 +112,6 @@ INPUT .\hw3\motorcade.png
|
||||
INPUT .\hw3\motorcade.png
|
||||
INPUT .\hw3\motorcade.png
|
||||
INPUT .\hw3\motorcade.png
|
||||
INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
|
||||
INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
|
||||
INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
|
||||
INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
|
||||
INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
|
||||
INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
|
||||
INPUT .\hw4\Gemini_Chart_Image_5v3caz5v3caz5v3c.png
|
||||
INPUT .\hw4\Picture1.png
|
||||
INPUT .\hw4\Picture1.png
|
||||
INPUT .\hw4\Picture1.png
|
||||
|
||||
107
main.log
107
main.log
@ -1,4 +1,4 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.26 (MiKTeX 24.3.31) (preloaded format=pdflatex 2024.4.7) 29 APR 2024 23:55
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.26 (MiKTeX 24.4) (preloaded format=pdflatex 2024.5.14) 16 MAY 2024 23:32
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
file:line:error style messages enabled.
|
||||
@ -6,20 +6,20 @@ entering extended mode
|
||||
**d:/uzair/projects/sfsu/csc-665-homework/main.tex
|
||||
(d:/uzair/projects/sfsu/csc-665-homework/main.tex
|
||||
LaTeX2e <2023-11-01> patch level 1
|
||||
L3 programming layer <2024-03-14>
|
||||
L3 programming layer <2024-04-11>
|
||||
(C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/base\article.cls
|
||||
Document Class: article 2023/05/17 v1.4n Standard LaTeX document class
|
||||
(C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/base\size10.clo
|
||||
File: size10.clo 2023/05/17 v1.4n Standard LaTeX file (size option)
|
||||
)
|
||||
\c@part=\count188
|
||||
\c@section=\count189
|
||||
\c@subsection=\count190
|
||||
\c@subsubsection=\count191
|
||||
\c@paragraph=\count192
|
||||
\c@subparagraph=\count193
|
||||
\c@figure=\count194
|
||||
\c@table=\count195
|
||||
\c@part=\count189
|
||||
\c@section=\count190
|
||||
\c@subsection=\count191
|
||||
\c@subsubsection=\count192
|
||||
\c@paragraph=\count193
|
||||
\c@subparagraph=\count194
|
||||
\c@figure=\count195
|
||||
\c@table=\count196
|
||||
\abovecaptionskip=\skip48
|
||||
\belowcaptionskip=\skip49
|
||||
\bibindent=\dimen140
|
||||
@ -40,14 +40,14 @@ Package: amsbsy 1999/11/29 v1.2d Bold Symbols
|
||||
) (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/amsmath\amsopn.sty
|
||||
Package: amsopn 2022/04/08 v2.04 operator names
|
||||
)
|
||||
\inf@bad=\count196
|
||||
\inf@bad=\count197
|
||||
LaTeX Info: Redefining \frac on input line 234.
|
||||
\uproot@=\count197
|
||||
\leftroot@=\count198
|
||||
\uproot@=\count198
|
||||
\leftroot@=\count199
|
||||
LaTeX Info: Redefining \overline on input line 399.
|
||||
LaTeX Info: Redefining \colon on input line 410.
|
||||
\classnum@=\count199
|
||||
\DOTSCASE@=\count266
|
||||
\classnum@=\count266
|
||||
\DOTSCASE@=\count267
|
||||
LaTeX Info: Redefining \ldots on input line 496.
|
||||
LaTeX Info: Redefining \dots on input line 499.
|
||||
LaTeX Info: Redefining \cdots on input line 620.
|
||||
@ -60,20 +60,20 @@ LaTeX Info: Redefining \Bigg on input line 725.
|
||||
\big@size=\dimen143
|
||||
LaTeX Font Info: Redeclaring font encoding OML on input line 743.
|
||||
LaTeX Font Info: Redeclaring font encoding OMS on input line 744.
|
||||
\macc@depth=\count267
|
||||
\macc@depth=\count268
|
||||
LaTeX Info: Redefining \bmod on input line 905.
|
||||
LaTeX Info: Redefining \pmod on input line 910.
|
||||
LaTeX Info: Redefining \smash on input line 940.
|
||||
LaTeX Info: Redefining \relbar on input line 970.
|
||||
LaTeX Info: Redefining \Relbar on input line 971.
|
||||
\c@MaxMatrixCols=\count268
|
||||
\c@MaxMatrixCols=\count269
|
||||
\dotsspace@=\muskip16
|
||||
\c@parentequation=\count269
|
||||
\dspbrk@lvl=\count270
|
||||
\c@parentequation=\count270
|
||||
\dspbrk@lvl=\count271
|
||||
\tag@help=\toks18
|
||||
\row@=\count271
|
||||
\column@=\count272
|
||||
\maxfields@=\count273
|
||||
\row@=\count272
|
||||
\column@=\count273
|
||||
\maxfields@=\count274
|
||||
\andhelp@=\toks19
|
||||
\eqnshift@=\dimen144
|
||||
\alignsep@=\dimen145
|
||||
@ -94,20 +94,20 @@ Package: keyval 2022/05/29 v1.15 key=value parser (DPC)
|
||||
\KV@toks@=\toks22
|
||||
) (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/tools\calc.sty
|
||||
Package: calc 2023/07/08 v4.3 Infix arithmetic (KKT,FJ)
|
||||
\calc@Acount=\count274
|
||||
\calc@Bcount=\count275
|
||||
\calc@Acount=\count275
|
||||
\calc@Bcount=\count276
|
||||
\calc@Adimen=\dimen150
|
||||
\calc@Bdimen=\dimen151
|
||||
\calc@Askip=\skip53
|
||||
\calc@Bskip=\skip54
|
||||
LaTeX Info: Redefining \setlength on input line 80.
|
||||
LaTeX Info: Redefining \addtolength on input line 81.
|
||||
\calc@Ccount=\count276
|
||||
\calc@Ccount=\count277
|
||||
\calc@Cskip=\skip55
|
||||
) (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/mathtools\mhsetup.sty
|
||||
Package: mhsetup 2021/03/18 v1.4 programming setup (MH)
|
||||
)
|
||||
\g_MT_multlinerow_int=\count277
|
||||
\g_MT_multlinerow_int=\count278
|
||||
\l_MT_multwidth_dim=\dimen152
|
||||
\origjot=\skip56
|
||||
\l_MT_shortvdotswithinadjustabove_dim=\dimen153
|
||||
@ -130,8 +130,8 @@ LaTeX Font Info: Redeclaring math symbol \hbar on input line 98.
|
||||
LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
|
||||
(Font) U/euf/m/n --> U/euf/b/n on input line 106.
|
||||
)) (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/l3backend\l3backend-pdftex.def
|
||||
File: l3backend-pdftex.def 2024-03-14 L3 backend support: PDF output (pdfTeX)
|
||||
\l__color_backend_stack_int=\count278
|
||||
File: l3backend-pdftex.def 2024-04-11 L3 backend support: PDF output (pdfTeX)
|
||||
\l__color_backend_stack_int=\count279
|
||||
\l__pdf_internal_box=\box54
|
||||
) (main.aux (titlePage.aux))
|
||||
\openout1 = `main.aux'.
|
||||
@ -161,19 +161,19 @@ File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration
|
||||
)
|
||||
Package graphics Info: Driver file: pdftex.def on input line 107.
|
||||
(C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/graphics-def\pdftex.def
|
||||
File: pdftex.def 2022/09/22 v1.2b Graphics/color driver for pdftex
|
||||
File: pdftex.def 2024/04/13 v1.2c Graphics/color driver for pdftex
|
||||
(C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/context/base/mkii\supp-pdf.mkii
|
||||
[Loading MPS to PDF converter (version 2006.09.02).]
|
||||
\scratchcounter=\count279
|
||||
\scratchcounter=\count280
|
||||
\scratchdimen=\dimen160
|
||||
\scratchbox=\box55
|
||||
\nofMPsegments=\count280
|
||||
\nofMParguments=\count281
|
||||
\nofMPsegments=\count281
|
||||
\nofMParguments=\count282
|
||||
\everyMPshowfont=\toks23
|
||||
\MPscratchCnt=\count282
|
||||
\MPscratchCnt=\count283
|
||||
\MPscratchDim=\dimen161
|
||||
\MPnumerator=\count283
|
||||
\makeMPintoPDFobject=\count284
|
||||
\MPnumerator=\count284
|
||||
\makeMPintoPDFobject=\count285
|
||||
\everyMPtoPDFconversion=\toks24
|
||||
))) (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/epstopdf-pkg\epstopdf-base.sty
|
||||
Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf
|
||||
@ -238,39 +238,34 @@ Underfull \hbox (badness 10000) in paragraph at lines 450--451
|
||||
[]
|
||||
|
||||
[14 <./hw3/motorcade.png>]
|
||||
<hw4/Gemini_Chart_Image_5v3caz5v3caz5v3c.png, id=69, 642.4pt x 481.8pt>
|
||||
File: hw4/Gemini_Chart_Image_5v3caz5v3caz5v3c.png Graphic file (type png)
|
||||
<use hw4/Gemini_Chart_Image_5v3caz5v3caz5v3c.png>
|
||||
Package pdftex.def Info: hw4/Gemini_Chart_Image_5v3caz5v3caz5v3c.png used on input line 490.
|
||||
(pdftex.def) Requested size: 321.19922pt x 240.8994pt.
|
||||
|
||||
Overfull \hbox (1.19925pt too wide) in paragraph at lines 490--491
|
||||
[]
|
||||
[]
|
||||
|
||||
<hw4/Picture1.png, id=70, 305.64188pt x 191.9672pt>
|
||||
<hw4/Picture1.png, id=69, 305.64188pt x 191.9672pt>
|
||||
File: hw4/Picture1.png Graphic file (type png)
|
||||
<use hw4/Picture1.png>
|
||||
Package pdftex.def Info: hw4/Picture1.png used on input line 512.
|
||||
(pdftex.def) Requested size: 91.69325pt x 57.59059pt.
|
||||
[15 <./hw4/Gemini_Chart_Image_5v3caz5v3caz5v3c.png>] [16 <./hw4/Picture1.png>] (main.aux (titlePage.aux))
|
||||
[15 <./hw4/Picture1.png>]
|
||||
Overfull \hbox (2.23077pt too wide) detected at line 531
|
||||
[][]
|
||||
[]
|
||||
|
||||
[16] (main.aux (titlePage.aux))
|
||||
***********
|
||||
LaTeX2e <2023-11-01> patch level 1
|
||||
L3 programming layer <2024-03-14>
|
||||
L3 programming layer <2024-04-11>
|
||||
***********
|
||||
)
|
||||
Here is how much of TeX's memory you used:
|
||||
3263 strings out of 474424
|
||||
51441 string characters out of 5741740
|
||||
1940496 words of memory out of 5000000
|
||||
25593 multiletter control sequences out of 15000+600000
|
||||
3256 strings out of 474370
|
||||
51092 string characters out of 5740535
|
||||
1940754 words of memory out of 5000000
|
||||
25640 multiletter control sequences out of 15000+600000
|
||||
562573 words of font info for 53 fonts, out of 8000000 for 9000
|
||||
1141 hyphenation exceptions out of 8191
|
||||
65i,19n,72p,462b,279s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
63i,19n,70p,462b,279s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
<C:\Users\uzair\AppData\Local\MiKTeX\fonts/pk/ljfour/jknappen/ec/dpi600\tcrm1000.pk><C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx10.pfb><C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx12.pfb><C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmex10.pfb><C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi10.pfb><C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi7.pfb><C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr10.pfb><C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr12.pfb><C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr7.pfb><C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy10.pfb><C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy7.pfb><C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmti10.pfb><C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/latxfont/line10.pfb>
|
||||
Output written on main.pdf (17 pages, 554351 bytes).
|
||||
Output written on main.pdf (17 pages, 393219 bytes).
|
||||
PDF statistics:
|
||||
133 PDF objects out of 1000 (max. 8388607)
|
||||
131 PDF objects out of 1000 (max. 8388607)
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
26 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
21 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
BIN
main.synctex.gz
BIN
main.synctex.gz
Binary file not shown.
153
main.tex
153
main.tex
@ -422,52 +422,52 @@ See attached submission.py file
|
||||
\subsection{Bayesian Networks}
|
||||
\begin{itemize}
|
||||
\item[a.] Given the following Bayesian network, write down a factorization of the joint probability distribution P(A,B, C,D, E) as a product of local conditional distributions, one for each random variable.
|
||||
\begin{center}
|
||||
\includegraphics[scale=0.3]{hw3/bayesian_network_1_hw3.png}
|
||||
\end{center}
|
||||
\begin{center}
|
||||
\includegraphics[scale=0.3]{hw3/bayesian_network_1_hw3.png}
|
||||
\end{center}
|
||||
|
||||
$$
|
||||
P(A, B, C, D, E) = \ \boxed{P(A) \cdot P(B) \cdot P(C | A, B) \cdot P(D | B) \cdot P(E | C)}
|
||||
$$
|
||||
$$
|
||||
P(A, B, C, D, E) = \ \boxed{P(A) \cdot P(B) \cdot P(C | A, B) \cdot P(D | B) \cdot P(E | C)}
|
||||
$$
|
||||
|
||||
\item[b.] For the same Bayesian network above, list all the random variables that are independent of A, assuming that none of the variables have been observed.
|
||||
\begin{itemize}
|
||||
\item B
|
||||
\end{itemize}
|
||||
\begin{itemize}
|
||||
\item B
|
||||
\end{itemize}
|
||||
|
||||
\item[c.] For the same Bayesian network above, assume that E is now observed. List all pairs of random variables (not including E) that are conditionally independent given E.
|
||||
\begin{itemize}
|
||||
\item (A, B)
|
||||
\item (A, D)
|
||||
\item (B, D)
|
||||
\end{itemize}
|
||||
\begin{itemize}
|
||||
\item (A, B)
|
||||
\item (A, D)
|
||||
\item (B, D)
|
||||
\end{itemize}
|
||||
|
||||
\item[d.] Consider the following model for traffic jams in a small town, which can be caused by either a car accident or by a visit from the president (and the accompanying security motorcade).
|
||||
\begin{center}
|
||||
\includegraphics[scale=0.3]{hw3/motorcade.png}
|
||||
\end{center}
|
||||
\begin{center}
|
||||
\includegraphics[scale=0.3]{hw3/motorcade.png}
|
||||
\end{center}
|
||||
|
||||
Compute P(Accident = 1 | Traffic = 1) and P(Accident = 1 | Traffic = 1, President = 1). \\
|
||||
Compute P(Accident = 1 | Traffic = 1) and P(Accident = 1 | Traffic = 1, President = 1). \\
|
||||
|
||||
Let:
|
||||
\begin{itemize}
|
||||
\item A = event of accident
|
||||
\item P = event of president's visit
|
||||
\item T = event of traffic jam
|
||||
\end{itemize}
|
||||
Let:
|
||||
\begin{itemize}
|
||||
\item A = event of accident
|
||||
\item P = event of president's visit
|
||||
\item T = event of traffic jam
|
||||
\end{itemize}
|
||||
|
||||
We're given:
|
||||
\begin{itemize}
|
||||
\item P(A) = prior probability of an accident
|
||||
\item P(P) = prior probability of the president's visit
|
||||
\item P(T | A) = probability of a traffic jam given an accident
|
||||
\item P(T | P) = probability of a traffic jam given the president's visit
|
||||
\end{itemize}
|
||||
We're given:
|
||||
\begin{itemize}
|
||||
\item P(A) = prior probability of an accident
|
||||
\item P(P) = prior probability of the president's visit
|
||||
\item P(T | A) = probability of a traffic jam given an accident
|
||||
\item P(T | P) = probability of a traffic jam given the president's visit
|
||||
\end{itemize}
|
||||
|
||||
$$
|
||||
P(\text{Accident} = 1 | \text{Traffic} = 1) \rightarrow P(A | T) = \frac{P(T | A) \cdot P(A)}{P(T)}
|
||||
$$
|
||||
|
||||
$$
|
||||
P(\text{Accident} = 1 | \text{Traffic} = 1) \rightarrow P(A | T) = \frac{P(T | A) \cdot P(A)}{P(T)}
|
||||
$$
|
||||
|
||||
\end{itemize}
|
||||
|
||||
\newpage
|
||||
@ -479,40 +479,79 @@ See attached submission.py file
|
||||
\begin{itemize}
|
||||
\item[a - c. ] Refer to bn.py
|
||||
|
||||
\item[d. ] Likelihood weighting gets more accurate as we use more samples. With more samples, the estimated probabilities get closer to the real chances we calculated before. This shows that using a bigger sample size is helpful for getting better results.
|
||||
\item[d. ] Likelihood weighting gets more accurate as we use more samples. With more samples, the estimated probabilities get closer to the real chances we calculated before. This shows that using a bigger sample size is helpful for getting better results.
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Gradient Descent}
|
||||
|
||||
\begin{itemize}
|
||||
\item[a. ]
|
||||
\begin{center}
|
||||
\includegraphics[scale=0.5]{hw4/Gemini_Chart_Image_5v3caz5v3caz5v3c.png}
|
||||
\end{center}
|
||||
\item[a. ]
|
||||
% \begin{center}
|
||||
% \includegraphics[scale=0.5]{hw4/Gemini_Chart_Image_5v3caz5v3caz5v3c.png}
|
||||
% \end{center}
|
||||
|
||||
|
||||
Hinge loss is a reasonable cost function for binary classification because it focuses on creating a good margin between classes. It only penalizes models when their predictions fall close to the wrong class, and ignores classified points even with a small margin. This makes the model prioritize a clear separation between positive and negative classes.
|
||||
Hinge loss is a reasonable cost function for binary classification because it focuses on creating a good margin between classes. It only penalizes models when their predictions fall close to the wrong class, and ignores classified points even with a small margin. This makes the model prioritize a clear separation between positive and negative classes.
|
||||
|
||||
\item[b. ]
|
||||
|
||||
$$
|
||||
\frac{\partial c(h(x), y)}{\partial w_0} = 0
|
||||
$$
|
||||
\item[b. ]
|
||||
|
||||
$$
|
||||
\frac{\partial c(h(x), y)}{\partial w_1} = \begin{cases}
|
||||
-y, & \text{if } w_0 + w_1x < 1 \\
|
||||
0, & \text{if } w_0 + w_1x \geq 1
|
||||
\end{cases}
|
||||
$$
|
||||
$$
|
||||
\frac{\partial c(h(x), y)}{\partial w_0} = 0
|
||||
$$
|
||||
|
||||
$$
|
||||
\frac{\partial c(h(x), y)}{\partial w_1} = \begin{cases}
|
||||
-y, & \text{if } w_0 + w_1x < 1 \\
|
||||
0, & \text{if } w_0 + w_1x \geq 1
|
||||
\end{cases}
|
||||
$$
|
||||
|
||||
\item[c. ]
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[scale=0.3]{hw4/Picture1.png}
|
||||
\end{center}
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[scale=0.3]{hw4/Picture1.png}
|
||||
\end{center}
|
||||
|
||||
|
||||
\end{itemize}
|
||||
|
||||
\end{document}
|
||||
\newpage
|
||||
|
||||
\section{Homework 5}
|
||||
|
||||
\subsection{Regularization}
|
||||
|
||||
\begin{itemize}
|
||||
\item[a.]
|
||||
$$
|
||||
\begin{aligned}
|
||||
\frac{d}{dw_0} C(w) & = \sum_{i=1}^{n} 2(y_i - (w_0 + w_1x_i))(-1) = \boxed{-2\sum_{i=1}^{n} (y_i - (w_0 + w_1x_i))} \\
|
||||
\frac{d}{dw_1} C(w) & = \sum_{i=1}^{n} 2(y_i - (w_0 + w_1x_i))(-x_i) = \boxed{-2\sum_{i=1}^{n} x_i(y_i - (w_0 + w_1x_i))}
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
\item[b. ]
|
||||
$$
|
||||
\begin{aligned}
|
||||
\frac{d}{dw_0} \tilde{C}(w) &= \sum_{i=1}^{n} 2(y_i - (w_0 + w_1x_i))(-1) + 2\lambda w_0 \\
|
||||
&= 2\sum_{i=1}^{n} (w_0 + w_1x_i - y_i) + 2\lambda w_0 ] \\
|
||||
&= \boxed{2\sum_{i=1}^{n} (w_0 + w_1x_i - y_i + \lambda w_0)}
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
and
|
||||
|
||||
$$
|
||||
\begin{aligned}
|
||||
\frac{d}{dw_1} \tilde{C}(w) &= \sum_{i=1}^{n} 2(y_i - (w_0 + w_1x_i))(-x_i) + 2\lambda w_1 \\
|
||||
&= \boxed{2\sum_{i=1}^{n} x_i(y_i - (w_0 + w_1x_i)) + 2\lambda w_1}
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
\item[c-d.] Refer to regularization.py
|
||||
\item[e. ] Avg loss, regularization: 1.715104320688656 \\
|
||||
Avg loss, no regularization: 0.4384015750703455
|
||||
|
||||
\end{itemize}
|
||||
|
||||
\end{document}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user