= Square-root Lasso
{c}
The <Square-root Lasso> minimizes
$$
\frac{\|Y-X\beta\|_2}{\sqrt n}+\lambda\|\beta\|_1.
$$
Its residual <norm> is unsquared. For errors $\varepsilon=\sigma Z\ne0$, the magnitude of the loss <gradient> at the true <regression coefficients> is governed by
$$
\frac{X^{\mathsf T}\varepsilon}{\sqrt n\|\varepsilon\|_2}
=\frac{X^{\mathsf T}Z}{\sqrt n\|Z\|_2}.
$$
The unknown noise <standard deviation> $\sigma$ cancels. This makes a penalty calibrated to that <score function> pivotal with respect to noise scale.<Square-Root Lasso: Pivotal Recovery of Sparse Signals via Conic Programming by Alexandre Belloni and others>[Section 1, equations (8)–(11), printed pp. 2–4]{ref}
For normalized columns and <Gaussian noise>, a typical theoretical penalty has order $\sqrt{\log(p)/n}$ without an extra factor of $\sigma$. Prediction guarantees still require assumptions on the number of nonzero <regression coefficients>, the <design matrix> and the noise. Pivotal tuning does not remove those assumptions. At an exact zero residual the <Euclidean norm> is nondifferentiable, so its <subdifferential> must replace the displayed nonzero-residual <gradient>.
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