On the Non-Asymptotic Properties of Regularized M-estimators

TL;DR

This paper develops a non-asymptotic analysis framework for regularized M-estimators with time-dependent data, revealing how mixing structures influence estimator variance and proposing a data-driven tuning method.

Contribution

It introduces a general non-asymptotic concentration bound for regularized M-estimators under mixing conditions, including a novel complexity measure and a practical tuning parameter selection method.

Findings

Variance is affected by the mixing decay rate.

The proposed tuning method achieves near-optimal concentration bounds.

Mixing structure influences the asymptotic behavior of estimators.

Abstract

We propose a general framework for regularization in M-estimation problems under time dependent (absolutely regular-mixing) data which encompasses many of the existing estimators. We derive non-asymptotic concentration bounds for the regularized M-estimator. Our results exhibit a variance-bias trade-off, with the variance term being governed by a novel measure of the complexity of the parameter set. We also show that the mixing structure affect the variance term by scaling the number of observations; depending on the decay rate of the mixing coefficients, this scaling can even affect the asymptotic behavior. Finally, we propose a data-driven method for choosing the tuning parameters of the regularized estimator which yield the same (up to constants) concentration bound as one that optimally balances the (squared) bias and variance terms. We illustrate the results with several canonical examples.