论文标题
元素估计误差的元素估计误差
Element-wise Estimation Error of Generalized Fused Lasso
论文作者
论文摘要
本文的主要结果是，我们为任何一般凸损耗函数$ρ$限制了Fused Lasso估计器的元素WISE错误。然后，当$ρ$是正方形损耗函数（对于平均回归），或者是分位数损耗函数（对于分位数回归）时，我们将重点放在特殊情况下。即使通常研究了通常的融合Lasso估计器及其分位版本的错误界限；我们的界限似乎是新的。这是因为所有先前的作品都构成了全局损耗函数，例如平方误差之和，或者在Padilla和Chatterjee（2021）中的分位数回归的情况下（2021年）的Huber损失之和。显然，元素明智的界限比全局损耗误差界限强，因为它揭示了每个点损失的局部行为。我们的元素明智错误绑定也对调谐参数$λ$具有干净，明确的依赖性，该$λ$为用户提供了$λ$的不错选择。此外，我们的界限是无染色的，具有显式常数，并且能够恢复几乎所有已知的融合套索结果（均值和分位数回归），在某些情况下进行了其他改进。
The main result of this article is that we obtain an elementwise error bound for the Fused Lasso estimator for any general convex loss function $ρ$. We then focus on the special cases when either $ρ$ is the square loss function (for mean regression) or is the quantile loss function (for quantile regression) for which we derive new pointwise error bounds. Even though error bounds for the usual Fused Lasso estimator and its quantile version have been studied before; our bound appears to be new. This is because all previous works bound a global loss function like the sum of squared error, or a sum of Huber losses in the case of quantile regression in Padilla and Chatterjee (2021). Clearly, element wise bounds are stronger than global loss error bounds as it reveals how the loss behaves locally at each point. Our element wise error bound also has a clean and explicit dependence on the tuning parameter $λ$ which informs the user of a good choice of $λ$. In addition, our bound is nonasymptotic with explicit constants and is able to recover almost all the known results for Fused Lasso (both mean and quantile regression) with additional improvements in some cases.
