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For Lasso loss $\Vert Y-X\beta {\Vert }_2^2/(2n)+\lambda \Vert \beta {\Vert }_1$ and $Y=X{\beta }^0+\eta$, stationarity with a penalty subgradient gives $\Vert X\delta {\Vert }_2^2/n\le {\eta }^{T}X\delta /n+\lambda (\Vert {\beta }^0{\Vert }_1-\Vert \hat{\beta }{\Vert }_1)$, $\delta =\hat{\beta }-{\beta }^0$. This follows by multiplying the Karush-Kuhn-Tucker conditions by $\delta$ and using $z^T\hat{\beta }=\Vert \hat{\beta }{\Vert }_1$ and $z^T{\beta }^0\le \Vert {\beta }^0{\Vert }_1$.

If every nonzero active coefficient has the sign of its residual score along the least angle regression path, the LAR active correlation invariant and inactive bounds give the Karush-Kuhn-Tucker conditions for the Lasso. A unique Lasso solution then equals the LAR path. Zero coefficients use a subgradient interval; literal sign equality with sign(0)=0 is not appropriate for a newly entering coefficient at a positive knot.

For $Y=X{\beta }^0+\eta$, the Lasso objective $\Vert Y-X\beta {\Vert }_2^2/(2n)+\lambda \Vert \beta {\Vert }_1$ has this prediction bound whenever $\Vert X^T\eta {\Vert }_{\infty }/n\le \lambda$. Compare its value at $\hat{\beta }$ and ${\beta }^0$, apply Holder inequality to the score, and use the triangle inequality. No restricted eigenvalue condition is required.

The subdifferential of the L1 norm consists of vectors $z$ with $z_j=\mathrm{sgn}({\beta }_j)$ when ${\beta }_j\ne 0$ and $z_j\in [-1,1]$ when ${\beta }_j=0$. These coordinate conditions give the Karush-Kuhn-Tucker conditions for Lasso and Square-root Lasso.

A Lasso minimizer with positive residual scale ${\hat{\sigma }}_{\lambda }=\Vert Y-X{\hat{\beta }}_{\lambda }{\Vert }_2/\sqrt{n}$ also minimizes the Square-root Lasso objective at the displayed tuning parameter. Divide its Karush-Kuhn-Tucker conditions by the residual scale and use the subgradient optimality condition.

Variable selection chooses which predictor variables enter a statistical model. The Lasso produces exact zero slopes along its Lasso regularization path; ridge regression generally shrinks without excluding variables. Selection for prediction should use predictive cross-validation or another stated criterion, rather than equating a coefficient p-value with practical usefulness.