Interpretation of positive/negative and magnitude of coefficients in Lasso I am doing a multiple linear regression, in order to see whether the predictors are positively (positive $\beta$s) or negatively correlated (negative $\beta$s) with the response.
Since I have a lot of predictors, I use Lasso to select variables. With cross-validation, I choose my optimal $\lambda$, but I am left with coefficients ($\beta$s) with no standard errors. I read that the significance testing for penalized regression is still under research.
So my questions are, 
1) How do I interpret positive, negative, and 0 coefficients for Lasso?
2) In addition, can I compare the magnitude of predictors (all have the same unit/dimension) to say which has is more important over another on the response?
 A: To be clear, in my opinion, the best way to interpret the coefficients in lasso is just to interpret whether they're zero or not. All the same, here's some progress towards interpreting the sign and magnitude of the coefficients.
From the KKT conditions, we see that $\mathrm{sgn}(\hat{\beta}_j) = \mathrm{sgn}(x_j^T ( y - X \hat{\beta}))$, so that the sign of the lasso estimate gives the sign of the correlation of the feature with the residual. (In fact, modifying LARS so that this condition is enforced is exactly what produces the LASSO.)
It could be interesting to note that $|\hat\beta_j| \geq |\hat\beta_k|$ is equivalent to $$\|\hat{r}_{-(j,k)} - (x_j \hat{\beta}_j + x_k \hat{\beta}_k)\|_2^2 \leq \|\hat{r}_{-(j,k)} - (x_k \hat{\beta}_j + x_j \hat{\beta}_k)\|_2 ^2,$$ where $\hat{r}_{-(j,k)} = y - \sum_{l \neq j,k} x_l \hat\beta_l$. Since $\|x\|_2 = 1$ for all features $x$, this suggests that $|\hat\beta_j| \geq |\hat\beta_k|$ means the $j$th covariate is more correlated with the partial residual than the $k$th.
Unfortunately, at the moment, I don't have anything more actionable.
