Element-wise Estimation Error of Generalized Fused Lasso

TL;DR

This paper derives new elementwise error bounds for the Fused Lasso estimator applicable to general convex loss functions, providing more detailed local error information than previous global bounds, with explicit constants and practical tuning guidance.

Contribution

It introduces the first elementwise error bounds for Fused Lasso with general convex loss, including new bounds for mean and quantile regression, improving upon existing global error bounds.

Findings

Derived nonasymptotic elementwise error bounds with explicit constants.

Provided bounds for both mean and quantile regression cases.

Bounded the error dependence on the tuning parameter λ explicitly.

Abstract

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 $\rho$. We then focus on the special cases when either $\rho$ 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 $\lambda$ which informs the user of a good choice of $\lambda$. 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.