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(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. 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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} \ No newline at end of file +\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}