[Paper Review] Benign overfitting in the large deviation regime

Geoffrey Chinot, Matthieu Lerasle|arXiv (Cornell University)|Mar 12, 2020

Statistical Methods and Inference28 references9 citations

TL;DR

This paper establishes that benign overfitting can achieve prediction risk bounds converging to zero under the large deviation regime, where bounds hold with high probability $1 - e^{-\zeta n}$. By introducing a novel localization analysis of the minimal-norm interpolating estimator, the authors derive tight risk bounds for quadratic, Huber, and absolute losses, extending prior results in robust and standard statistics.

ABSTRACT

We investigate the benign overfitting phenomenon in the large deviation regime where the bounds on the prediction risk hold with probability $1-e^{-\zeta n}$, for some absolute constant $\zeta$. We prove that these bounds can converge to $0$ for the quadratic loss. We obtain this result by a new analysis of the interpolating estimator with minimal Euclidean norm, relying on a preliminary localization of this estimator with respect to the Euclidean norm. This new analysis complements and strengthens particular cases obtained in previous works for the square loss and is extended to other loss functions. To illustrate this, we also provide excess risk bounds for the Huber and absolute losses, two widely spread losses in robust statistics.

Motivation & Objective

• To understand the conditions under which benign overfitting leads to vanishing prediction risk in the large deviation regime.
• To extend existing risk bounds for the quadratic loss to other robust loss functions such as Huber and absolute loss.
• To develop a refined analysis of the minimal-norm interpolating estimator that improves upon prior approaches.

Proposed method

• Introduces a new localization technique for the minimal-norm interpolating estimator with respect to the Euclidean norm.
• Analyzes the estimator in the large deviation regime, where high-probability bounds hold with probability $1 - e^{-\zeta n}$.
• Derives prediction risk bounds for the quadratic loss and extends the analysis to Huber and absolute losses.
• Uses concentration and localization arguments to control the estimator's behavior in high-dimensional settings.
• Applies tools from statistical learning theory and random matrix theory to establish convergence of risk bounds.

Experimental results

Research questions

• RQ1 Can benign overfitting lead to prediction risk converging to zero under large deviation probability bounds?
• RQ2 How does the minimal-norm interpolating estimator behave when localized in the Euclidean norm under high-probability regimes?
• RQ3 To what extent can the analysis of the quadratic loss be extended to robust losses like Huber and absolute loss?
• RQ4 What are the conditions under which the risk bounds for these losses converge to zero in the large deviation regime?

Key findings

• The prediction risk for the quadratic loss converges to zero under the large deviation regime with probability $1 - e^{-\zeta n}$.
• The proposed localization technique enables tighter control of the minimal-norm interpolating estimator, improving upon prior bounds.
• Excess risk bounds are established for the Huber and absolute losses, extending the applicability of benign overfitting to robust statistics.
• The analysis confirms that benign overfitting is not limited to the quadratic loss but holds under broader loss functions.