This video showed me how to change the world. Wow. Thank you so much Andrew McCrady. Also at the 36:41 Mark you referred to Grand Emperor Zuul as the fourth Grand Emperor of the Universe but he was actually the fifth.
This video helps me with machine learning. Thank you. (I am learning gradient descent, one condition to ensure a function is converging is the function is c-Lipschitz.)
Of course I'm a bit in doubt about the last proof, by setting the x in the boundry (0, 1/(k^2)), you're also limiting the k, because the x has to be from [0, 2]. that means k will not be any arbitrary number, while nothing in the Lipschitz theorem states that there are any limits on choosing the k.
The set of such numbers K is not bounded, specifically it’s never bounded above. If |f(x)-f(y)| is less than K|x-y| for some K, then it’s true for all M larger than K, too. The definition says there should be some number K that works for all inputs, x and y, from the domain A. So “Lipschitz” depends on both (1) the function f and (2) the domain of inputs A.
I think i couldn't convey my idea .... When i say set of Ks ...that doesn't mean set of Ks for particular value of x and y ..... It essentially the set of real numbers needed for different pairs x and y.... And for that it has to be bounded...... For example for √x in[0,1]....you cannot find such bounded set of values of K ....for all pairs x and y such that Lipschitz condition is satisfied....while for Lipschitz functions ...you can find a set of values of K for different-different pairs x and y ....such that the set is bounded ....and its upper bound can be taken as that one value of K that will work for every pair of values x and y
I think you would need to take the minimal K for each pair x and y, then Lipschitz might be equivalent to the set of all such minimal K being bounded above. In other words, the Lipschitz constant of the function over the whole domain is the supremum of the Lipschitz constants for each pair of inputs from the domain? Is that what you’re saying?
@@DrMcCrady ya .... Well let it be .... You were not wrong ...... I just wanted to say k is not necessarily same for each pair of x and y ........ that's it ...good lecture 👍
I think I struck gold with this channel
This video showed me how to change the world. Wow. Thank you so much Andrew McCrady. Also at the 36:41 Mark you referred to Grand Emperor Zuul as the fourth Grand Emperor of the Universe but he was actually the fifth.
This video helps me with machine learning. Thank you. (I am learning gradient descent, one condition to ensure a function is converging is the function is c-Lipschitz.)
Very cool!
this was so clear, thank you!
Glad it was helpful!
Made more sense than my prof explaining this to me for an hour.
Thank you so much!
Great explanation there Senior…appreciate that
Glad it was helpful!
Thank you very much for clearing my doubts.
Happy to help
Very nicely and thoroughly explained. Thanks.
Glad it was helpful!
The intuition in the beginning was very helpfull, thank you!
Glad it was helpful!
Thank you!
You're welcome!
Thanks for the vid!
This video is very clear. Thanks.
Glad it was helpful!
Fantastic video!
Thank you!
Thank you.
You're welcome!
Thanks! :)
kolos za godzine. dzkk
legend, great video
Glad it was helpful!
Thank you
Glad it was helpful!
Of course I'm a bit in doubt about the last proof, by setting the x in the boundry (0, 1/(k^2)), you're also limiting the k, because the x has to be from [0, 2]. that means k will not be any arbitrary number, while nothing in the Lipschitz theorem states that there are any limits on choosing the k.
0:47.....i think that's not correct.... K can be different for different choices of x and y .... It is just that set of K's should be bounded ...
The set of such numbers K is not bounded, specifically it’s never bounded above. If
|f(x)-f(y)| is less than K|x-y| for some K, then it’s true for all M larger than K, too.
The definition says there should be some number K that works for all inputs, x and y, from the domain A. So “Lipschitz” depends on both (1) the function f and (2) the domain of inputs A.
I think i couldn't convey my idea .... When i say set of Ks ...that doesn't mean set of Ks for particular value of x and y ..... It essentially the set of real numbers needed for different pairs x and y.... And for that it has to be bounded......
For example for √x in[0,1]....you cannot find such bounded set of values of K ....for all pairs x and y such that Lipschitz condition is satisfied....while for Lipschitz functions ...you can find a set of values of K for different-different pairs x and y ....such that the set is bounded ....and its upper bound can be taken as that one value of K that will work for every pair of values x and y
I think you would need to take the minimal K for each pair x and y, then Lipschitz might be equivalent to the set of all such minimal K being bounded above. In other words, the Lipschitz constant of the function over the whole domain is the supremum of the Lipschitz constants for each pair of inputs from the domain? Is that what you’re saying?
@@DrMcCrady ya .... Well let it be .... You were not wrong ...... I just wanted to say k is not necessarily same for each pair of x and y ........ that's it ...good lecture 👍
Thanks for the discussion, cheers!