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Lipschitz constant of L2 reg. logistic regression $sum_i log left(1 + expleft{ -t_i left(w^T x_iright)right} right) + mu |w |_2^2$

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2

$begingroup$

Let the L2 regularized logistic regression function is given by,
begin{align}
f(w) &= frac{1}{N} sum_i log left(1 + expleft{ -t_i left(w^T x_iright)right} right) + mu |w |_2^2 = frac{1}{N} sum_i f_i(w),
end{align}
where $t_i in mathbb{R}$, $w, x_i in mathbb{R}^n$, $mu in mathbb{R}$, and $f_i(w) := log left(1 + expleft{ -t_i left(w^T x_iright)right} right) + mu |w |_2^2$ .

Questions:

• How would/could I find the (small) Lipschitz constant $M$ and $nu$-strongly parameter of $f(w)$?


• How can I find the (small) Lipschitz constant $L$ of $nabla f(w)$ and $L_i$ of $nabla f_i(w)$?

---

I am sorry if this question has already been asked or it is trivial to compute (analytically?).

real-analysis functional-analysis convex-analysis lipschitz-functions

share|cite|improve this question

edited Feb 11 at 21:10

user550103

asked Feb 11 at 11:22

user550103user550103

7931315

$endgroup$

add a comment | 

2

$begingroup$

Let the L2 regularized logistic regression function is given by,
begin{align}
f(w) &= frac{1}{N} sum_i log left(1 + expleft{ -t_i left(w^T x_iright)right} right) + mu |w |_2^2 = frac{1}{N} sum_i f_i(w),
end{align}
where $t_i in mathbb{R}$, $w, x_i in mathbb{R}^n$, $mu in mathbb{R}$, and $f_i(w) := log left(1 + expleft{ -t_i left(w^T x_iright)right} right) + mu |w |_2^2$ .

Questions:

• How would/could I find the (small) Lipschitz constant $M$ and $nu$-strongly parameter of $f(w)$?


• How can I find the (small) Lipschitz constant $L$ of $nabla f(w)$ and $L_i$ of $nabla f_i(w)$?

---

I am sorry if this question has already been asked or it is trivial to compute (analytically?).

real-analysis functional-analysis convex-analysis lipschitz-functions

share|cite|improve this question

edited Feb 11 at 21:10

user550103

asked Feb 11 at 11:22

user550103user550103

7931315

$endgroup$

add a comment | 

$begingroup$

Let the L2 regularized logistic regression function is given by,
begin{align}
f(w) &= frac{1}{N} sum_i log left(1 + expleft{ -t_i left(w^T x_iright)right} right) + mu |w |_2^2 = frac{1}{N} sum_i f_i(w),
end{align}
where $t_i in mathbb{R}$, $w, x_i in mathbb{R}^n$, $mu in mathbb{R}$, and $f_i(w) := log left(1 + expleft{ -t_i left(w^T x_iright)right} right) + mu |w |_2^2$ .

Questions:

• How would/could I find the (small) Lipschitz constant $M$ and $nu$-strongly parameter of $f(w)$?


• How can I find the (small) Lipschitz constant $L$ of $nabla f(w)$ and $L_i$ of $nabla f_i(w)$?

---

I am sorry if this question has already been asked or it is trivial to compute (analytically?).

real-analysis functional-analysis convex-analysis lipschitz-functions

share|cite|improve this question

edited Feb 11 at 21:10

user550103

asked Feb 11 at 11:22

user550103user550103

7931315

$endgroup$

share|cite|improve this question

edited Feb 11 at 21:10

user550103

asked Feb 11 at 11:22

user550103user550103

7931315

share|cite|improve this question

edited Feb 11 at 21:10

user550103

asked Feb 11 at 11:22

user550103user550103

7931315

asked Feb 11 at 11:22

user550103user550103

7931315

asked Feb 11 at 11:22

user550103user550103

7931315