this ass
I SWEAR why is this shit so fucking complicated T_T this shit couldve been 20 lines (AND THEY ARE ALSO currently WRONG omfg)
question
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Linear Regression Implementation Download the advertising dataset. This is a dataset that gives us the total sales for different products, after marketing them on Television, Radio and Newspaper.
- Implement the Linear regression algorithm to predict the sales value, given three inputs (Television, Radio and Newspaper).
- Calculate the cost function after each iteration, draw cost function as a function of the iteration
- Try the program with various values for the learning rate alpha and number of iteration.’
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Solve the problem 1 with normal equation method
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Redo exercise 1 with regularization
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Redo exercise 2 with regularization
plan for 1
what the fuck is a Linear regression?
it is to my knowledge: fit a thing to a linear equation, with the Minimal Error.
the data
lets say the data has m rows. and its linear regression. pretty cool how you can train on all the data at the same time.
this is still basic ml after all
error function, cost function: the thig we want reduced
this is the cost function, also loss function:
E = \frac{1}{2m} \sum^m_{i = 1}\left(h(x^{(i)})- y^{(i)}\right)^2
with h(x) being our model (we are trying to find a and b and make our guess h(x) as close to y as possible)
this comes from the error function here:
e = \frac{1}{2}\left(h(x^{(i)})-y^{(i)}\right)^2
and the cost function from the error function:
E = \frac{1}{m}\sum^m_{i=1}e^{(i)}
as were dealing with linear fitting (h(x) = ax + b) the thing turns to this:
E = \frac{1}{2m} \sum_{i=1}^{m} \left( ax_i + b - y_i \right)^2
chatgpt-ed this shit it calls this strat Mean Squared Error (MSE)
gradient descent: we are taking the derivative of this shit
partial derivative of course.
its newton raphson all over again:
you have to change every coefficients so they yk fit the thing every time duh
and how you change goes like this:
a:= a - \alpha \frac{\partial E}{\partial a}
yes. partial deriv that fucking series wrt a. fucking insane innit.
however they did the calculations:
for our case h(x) = ax + b we have
\begin{align*}
&\frac{\partial}{\partial a}{E(a, b)} &&= \\
&\frac{\partial E}{\partial a} &&= \frac{1}{m}\sum^m_{i=1}(ax^{(i)}+b - y^{(i)})\cdot x^{(i)}
\end{align*}
yeah also for the case of b:
b:= b - \alpha \frac{\partial E}{\partial b}
\frac{\partial E}{\partial b} = \frac{1}{m}\sum^m_{i=1}(ax^{(i)}+b - y^{(i)})
note that that \alpha is important. its called learning rate.
matrix rep. we go big
but now when the columns are many and the rows are also many, what do we do?
it is very convenient that the linear can be represented by a vector of x times a vector of coefficients (weights):
\theta = \begin{vmatrix}\theta_0 \\ \theta_1 \\ \cdots \\ \theta_n \end{vmatrix} \text{ and }x = \begin{vmatrix}1 \\ x_1 \\ \cdots \\ x_n \end{vmatrix}\\
\implies h(x) = \theta^Tx
cost fucntion here:
E = \frac{1}{2m} \sum^m_{i = 1}\left(\theta^Tx^{(i)}- y^{(i)}\right)^2
and a very generalized gradient descent!!!
\theta_j:= \theta_j - \alpha \frac{\partial E}{\partial \theta_j}\\
\frac{\partial E}{\partial \theta_j} = \frac{1}{m}\sum^m_{i=1}(\theta^Tx^{(i)} - y^{(i)})\cdot x^{(i)}_j
thats insane! also very codable! we already has dot product of 2 vectors np.dot, and that is for \theta^Tx^{(i)}. the entire table of x across m versions is just a 2d table.
so we could plug this into a
into a chatgpt LMAO aint no way doing this by hand
ok when are we done?
so its an iterative method where you find a and b and all the other things at the same time using that. and again. and again.
the way we could turn off is jus check if E or e is small enough or we are looping too much
any way to not ruin our computers?
this does NOT harm your computers. it requires NOTHING to run.
but see this. the m can be anything. it can be one. 64. the whole table.
that is Stochastic, Mini-batch and Batch. crazy. think about it.
plan for 2
when i calculate \frac{\partial E}{\partial \theta} = 0 directly
\theta = (X^T X)^{-1} X^T Y
???