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Suppose $Y=X{\beta }^0+\epsilon$ and $\hat{\beta }$ minimizes the Lasso objective $\Vert Y-X\beta {\Vert }_2^2/(2n)+\lambda \Vert \beta {\Vert }_1$. With $\delta =\hat{\beta }-{\beta }^0$, comparing its value with the value at ${\beta }^0$ givesIndeed, expand $\Vert \epsilon -X\delta {\Vert }_2^2$ and cancel $\Vert \epsilon {\Vert }_2^2$. This deterministic inequality needs neither a normal distribution nor a full-rank design matrix.^[ref]

The noise term is at most $\Vert X^{\mathsf{T}}\epsilon /n{\Vert }_{\infty }\Vert \delta {\Vert }_1$ by Holder inequality. Controlling that term probabilistically leads to a Lasso cone condition; a Compatibility condition for the Lasso then converts coefficient error into prediction error. The stronger prediction inequality from Lasso stationarity uses subgradient optimality and has a different leading constant.

In $Y=X{\beta }^0+\epsilon$, put $\hat{\Sigma }=X^{\mathsf{T}}X/n$ and choose a matrix $M$ from $X$ alone. The corrected Lasso estimatehas the exact decompositionIf $\eta =\Vert I-M\hat{\Sigma }{\Vert }_{\max }$, every remainder coordinate is bounded by $\sqrt{n}\eta \Vert \hat{\beta }-{\beta }^0{\Vert }_1$. Thus a sufficiently accurate approximate inverse of a Gram matrix and a small coefficient L1 norm error make the remainder negligible.^[ref]

Given $X$, Gaussian noise $\epsilon \sim N(0,{\sigma }^{2}I)$ makes the first term exactly normal with covariance matrix ${\sigma }^{2}M\hat{\Sigma }{M}^{\mathsf{T}}$. For coordinate $j$, let $v_j=\sigma \sqrt{(M\hat{\Sigma }{M}^{\mathsf{T}}{)}_{jj}}>0$. If the remainder divided by $v_j$ tends to zero in probability, then $\sqrt{n}({\tilde{\beta }}_j-{\beta }_j^0)/v_j$ converges to a standard normal distribution. Replacing $\sigma$ requires a consistent estimator; non-Gaussian errors require a corresponding central limit theorem.

For response vector $Y$, design matrix $X$ and $\lambda \ge 0$, the Lasso estimates regression coefficients byAn equivalent constrained formulation minimizes the squared residual norm subject to $\Vert \beta {\Vert }_1\le t$, with an appropriate matching constraint radius. The L1 norm has a subdifferential interval at zero, allowing exact zero regression coefficients rather than only small ones. This combines estimation with variable selection.^[ref]

For an orthonormalized design matrix, $X^{\mathsf{T}}X/n=I$, put $z=X^{\mathsf{T}}Y/n$. Expanding the objective separates its coordinates and givesthe soft-thresholding operator. Thus $z=(2,0.4)$ and $\lambda =0.5$ give $\hat{\beta }=(1.5,0)$. With correlated columns, the Karush-Kuhn-Tucker conditions for the Lasso replace this independent-coordinate calculation. An unpenalized regression intercept is ordinarily handled by centering.

Sara van de Geer develops the mathematical theory of high-dimensional statistics. Her Statistical Theory for High-Dimensional Models by Sara van de Geer studies prediction bounds for the Lasso and Square-root Lasso, and inference after correcting shrinkage bias.^[ref]

The distinction is useful: an estimator that predicts well may still bias an individual coefficient toward zero. A de-sparsified estimator adds a correction whose leading error can have an approximately normal distribution. Bounds on the remaining error, together with sparsity and design assumptions, then justify confidence intervals. The same notes connect generic chaining and concentration inequalities with control of random-process suprema.^[ref]