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Consider the standard linear regression model $\y = \Xmat \betastar + w$,where $\y \in \real^\numobs$ is an observation vector, $\Xmat \in\real^{\numobs \times \pdim}$ is a design matrix, $\betastar \in \real^\pdim$is the unknown regression vector, and $w \sim \mathcal{N}(0, \sigma^2 I)$ isadditive Gaussian noise. This paper studies the minimax rates of convergencefor estimation of $\betastar$ for $\ell_\rpar$-losses and in the$\ell_2$-prediction loss, assuming that $\betastar$ belongs to an$\ell_{\qpar}$-ball $\Ballq(\myrad)$ for some $\qpar \in [0,1]$. We show thatunder suitable regularity conditions on the design matrix $\Xmat$, the minimaxerror in $\ell_2$-loss and $\ell_2$-prediction loss scales as $\Rq\big(\frac{\log \pdim}{n}\big)^{1-\frac{\qpar}{2}}$. In addition, we providelower bounds on minimax risks in $\ell_{\rpar}$-norms, for all $\rpar \in [1,+\infty], \rpar \neq \qpar$. Our proofs of the lower bounds areinformation-theoretic in nature, based on Fano's inequality and results on themetric entropy of the balls $\Ballq(\myrad)$, whereas our proofs of the upperbounds are direct and constructive, involving direct analysis of least-squaresover $\ell_{\qpar}$-balls. For the special case $q = 0$, a comparison with$\ell_2$-risks achieved by computationally efficient $\ell_1$-relaxationsreveals that although such methods can achieve the minimax rates up to constantfactors, they require slightly stronger assumptions on the design matrix$\Xmat$ than algorithms involving least-squares over the $\ell_0$-ball.

Minimax rates of estimation for high-dimensional linear regression over $\ell_q$-balls