In this paper, we consider a high-dimensional statistical estimation problem in which the number of parameters is comparable or larger than the sample size. We present a unified analysis of the performance guarantees of exponential weighted aggregation and penalized estimators with a general class of data losses and priors which encourage objects which conform to some notion of simplicity/complexity. More precisely, we show that these two estimators satisfy sharp oracle inequalities for prediction ensuring their good theoretical performances. We also highlight the differences between them. When the noise is random, we provide oracle inequalities in probability using concentration inequalities. These results are then applied to several instances including the Lasso, the group Lasso, their analysis-type counterparts, the $$\ell _\infty $$ ℓ ∞ and the nuclear norm penalties. All our estimators can be efficiently implemented using proximal splitting algorithms.

中文翻译：

---

低复杂度先验的尖锐预言不等式

在本文中，我们考虑了一个高维统计估计问题，其中参数的数量与样本量相当或大于样本量。我们对指数加权聚合和惩罚估计器的性能保证进行了统一分析，这些估计器具有一般类别的数据丢失和先验，这些数据损失和先验鼓励符合某些简单/复杂概念的对象。更准确地说，我们表明这两个估计量满足用于预测的尖锐预言不等式，确保它们具有良好的理论性能。我们还强调了它们之间的差异。当噪声是随机的时，我们使用浓度不等式提供概率的预言机不等式。然后将这些结果应用于多个实例，包括套索、组套索、它们的分析类型对应物、$$\ell _\infty $$ ℓ ∞ 和核规范惩罚。我们所有的估计器都可以使用近端分裂算法有效地实现。