Problem 1 Show that Eqn. (8) Lecture 3 holds, i.e., the...

Problem 1 Show that Eqn. (8) Lecture 3 holds, i.e., the partial derivative of $J(\beta)$ with respect to $\beta_j$ is $\frac{\partial}{\partial \beta_j} J(\beta) = \frac{1}{N} \sum_{i=1}^{N} (\sigma(\beta^T x_i) - y_i) x_{ij}$

Transcript

00:01 Here we have been given a function z equal to x plus y times e raised to y and x equal to u square plus v square y equal to u square minus v square we need to evaluate the partial derivative of z with respect to u and v in terms of u and v only using chain rule okay so i will write first the formula for the chain rule okay so for this case partial derivative with respect to u will be given by partial derivative of z with respect to x then x with respect to u plus z with respect to y then y with respect to u okay similarly with respect to v it will be del z divided by del x del x divided by del u sorry del v plus del z divided by del y del y divided by del y divided by del v okay so let us first evaluate all those values here so del j divided by del x okay so it will be what so first of all it will be just now x plus y derivative will become what e raise to y will be as it is since it is a constant now x plus y derivative x derivative is just one so this is just e raise to y so del z divided by del y if i find out so with respect to why now now we have to apply the product tool okay so x plus y derivative will be 1 times e raised to y plus e raise to y derivative is erase to y only times x plus y okay so this will give us e raise to y x plus y plus x plus y plus one okay so this is what we are getting now we have to find out del x by del u as well correct so let us find out del x divided by del u so that will be just two times u and del x divided by del v will again be two times v del y divided by del u it will be 2 u and del y divided by del v it will be minus 2v okay so just put all those formula in the formula above so del j divided by del u okay it will be what del j divided by del x that is e raised to y times del x divided by del u so that is 2 times u plus del y divided by sorry del j divided by del y which is e raised to y x plus y plus one and del y with respect to u that is to u so this we are getting okay so e raise to y can take common or now we have to find out only in terms of u and v so i can substitute y equal to u square minus v square so e to u square minus v squared times 2 u plus or this can be taken out to 2 u plus x plus y so put the x value and y value also so we will get u square plus v square plus u square minus v square plus 1 okay and this times to u so this is what we are getting now.

03:48 So, e raised to u square minus v square, 2u also common that we can take here.

03:55 So this will become u plus this v and v gets cancelled.

03:59 So 2u square plus 1.

04:03 So this reduces to 2 u time e raised to u square minus v square 2 plus 2 u square.

04:14 Okay...

Question

Problem 1 Show that Eqn. (8) Lecture 3 holds, i.e., the partial derivative of $J(\beta)$ with respect to $\beta_j$ is $\frac{\partial}{\partial \beta_j} J(\beta) = \frac{1}{N} \sum_{i=1}^{N} (\sigma(\beta^T x_i) - y_i) x_{ij}$

Please give Ace some feedback