We study high-dimensional linear regression under a general symmetric convex constraint. Rather than imposing a specific sparsity-inducing penalty, we start from an arbitrary sign-symmetric and permutation-invariant convex body $K\subseteq \mathbb R^p$ and construct the sparse convexification hierarchy \[ K^{(s)} = \operatorname{conv}\{v\in K:\|v\|_0\le s\}. \] We propose a penalized least-squares estimator that searches over this hierarchy and adapts to the best sparse convex approximation of the target. Under standard sub-Gaussian assumptions on the random design and noise, we prove an oracle inequality showing that the estimator adapts to the best sparse convex approximation of the target. For an $s$-sparse target, the result yields a squared-error rate governed by the effective sparse dimension $s\log(ep/s)$, the noise level $σ$, and the Euclidean diameter $d_s$ of the sparse convexification $K^{(s)}$. The method applies broadly to symmetric norm balls and can be implemented using oracle access to the Minkowski functional of $K$. As a special case, the framework yields a consistency result for the constrained Lasso.

翻译：我们研究在一般对称凸约束下的高维线性回归。不同于施加特定的稀疏诱导惩罚，我们从任意符号对称且置换不变的凸体$K\subseteq \mathbb R^p$出发，构建稀疏凸化层级\[ K^{(s)} = \operatorname{conv}\{v\in K:\|v\|_0\le s\} \]。我们提出一种惩罚最小二乘估计方法，该方法在该层级中搜索并自适应于目标的最佳稀疏凸近似。在随机设计与噪声满足标准次高斯假设的条件下，我们证明了一个预言不等式，表明该估计器能够自适应于目标的最佳稀疏凸近似。对于$s$-稀疏目标，该结果给出了由有效稀疏维数$s\log(ep/s)$、噪声水平$σ$以及稀疏凸化$K^{(s)}$的欧几里得直径$d_s$所决定的平方误差率。该方法广泛适用于对称范数球，并可通过访问$K$的闵可夫斯基泛函的原型进行实现。作为特例，该框架为约束Lasso提供了一个一致性结果。