Consider the following regularized logistic regression problem:

$$\textbf{w}^* = \min_{\textbf{w} \in {\rm I\!R}^p} \mathcal{L}(\textbf{w}) $$

where

$$ \mathcal{L}(\textbf{w}) = \frac{1}{n} \sum_{i=1:n} \{-y_i \textbf{w}^T \textbf{x}_i + \log(1 + \exp(-\textbf{w}^T \textbf{x}_i)) \} + \lambda G(\textbf{w})$$

  1. Derive subgradient for L(w) if $G(\textbf{w}) = \sum_{i=1:p} (a|w_i|^{\frac{1}{2}} + b)^2$

  2. Derive a subgradient stochastic gradient descent algorithm for when $G(\textbf{w}) = \sum_{i=1:p}|w_i|$

  3. If $G(\textbf{w}) = \frac{1}{2} \sum_{i=1:p} w_i^2$ what is the expected runtimes for Gradient descent and Stochastic Gradient Descent in terms of $(n, \epsilon)$

I'm struggling with the calculus for these problems. I know that for the subgradient of the L1 norm we'll end up with 3 cases, but I'm not sure what form the final answer should take.

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.