What is a reason that in Lasso Regression we can force all coefficients positive & intercept =0?

Question

I have a regression problem where I need all coefficents to be positive and the intercept to be zero. I can do this in sklearn but i don't understand how the algoritm can force this conditions through the loss function. Any references will be appreciated.

\begin{align}\hat{\beta}^\text{lasso}&=\underset{\beta}{ \arg\min}\sum_{i=1}^N \left(y_i-\beta_0-\sum_{j=1}^Px_{ij}\beta_j\right)^2\\&\textrm{subject to}\sum_{j=1}^P|\beta_j|\leq t. \end{align}

Answer 1

Lasso does neither of those things. It forces the coefficients of the least important predictors to be zero, but not necessarily the intercept (this would be problem dependent). At the same time, since the loss function contains the absolute value of the coefficients, their sign is irrelevant and only their magnitude has an effect on the result.

Answer 2

Forcing the intercept to be zero is straightforward: leave it out. Don’t estimate an intercept or let it enter the regression equation whatsoever. Estimate $\mathbb E\left[Y\vert X\right]=\beta_1X_1+\dots+\beta_pX_p$, rather than $=\beta_0+\beta_1X_1+\dots+\beta_pX_p$.

Regarding the Python function argument allowing for the coefficients all to be positive, I suspect that this is just another constraint on the optimization. You minimize square loss subject to the $L_1$ norm of the parameter vector being smaller than some value and subject to all components of the parameter vector being non-negative.