Lecture10 - Duality [Slide 7] - Dual Lasso Problem

Dear CS-439 Team,
In the Q&A session, I remember that with Professor Jaggi, we discussed that the information given on the dual lasso problem might include some typos. However, I couldn't find any changes on the slide or any remarks in the forum based on our discussion, hence I derived the dual lasso as shown in my writing below image.

First thing I realized that I believe that there is a typo in the definition of the conjugate of the function f in the lecture slides. I was confused and checked many sources online because we took \(x^{T}y\) instead of \(y^{T}x\). I found out that in B & V cvxbook page 221, section 5.1.6, the conjugate of function f is defined as follow:

output-onlinejpgtools.jpg

Following that, you can find my derivation for the dual lasso problem below. I would be very happy if you could check it and confirm that I actually derived it correctly.
WhatsApp Image 2021-07-05 at 01.47.45.jpeg

Hence, if you also agree that my derivation is correct, I think the slide needs to be corrected like here:
output-onlinejpgtools (1).jpg

Thanks for your confirmation,
Asli

Top comment

Hi Asli,
I answered the same question in this thread:
https://www.oknoname.com/EPFL/CS-439/topic/1986/lecture-10-slide-7/#c3
The slide indeed contains errors.

Hi Thijs,

I saw that thread actually, but I don't see why my derivation for the conjugate of the function f is different than yours. But all other dual forms that we both came up with are the same.

Hence, I only have a follow-up question regarding the conjugate of the function f. Could you please check my step for calculating it and tell me what I am doing wrong?

Thanks

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