Compute psi_vec

Arguments

y_vec

numeric vector

l_type

character string, type of loss function.

• "L1": l-1 loss(absolute value loss)

• "L2": l-2 loss(squared error loss)

• "Huber": Huber loss. Its parameter is given in l_param.

Default value is "L1".

l_param

numeric vector containing necessary parameters of the corresponding loss function. The default value is NULL.

Value

the psi_vec

Details

In the algorithm, when all $\mu$ are shrinked to the same value, we have $$\mu_0 = \mathrm{argmin}_{\mu} \sum_i\rho(y_i - \mu)$$ and $$psi\_vec = (\psi(y_1 - \mu_0), \cdots, \psi(y_n - \mu_0))$$ where $\psi = \partial \rho$.

Note

there is a SIGN different between this psi_vec and $\partial \rho$ w.r.t $\mu$

for non-differential loss like L1 at 0, the derivative at 0+ is returned.