Robust density estimation over star-shaped density classes

TL;DR

This paper develops a new method for robust density estimation within star-shaped classes under adversarial data corruption, providing minimax bounds that account for corruption fraction and density class complexity.

Contribution

It introduces a novel criterion for comparing densities under data corruption and constructs an estimator with proven minimax bounds for star-shaped density classes.

Findings

Established a new comparison criterion for densities with corrupted data.

Derived minimax upper and lower bounds for density estimation under corruption.

Provided explicit bounds involving density class complexity and corruption fraction.

Abstract

We establish a novel criterion for comparing the performance of two densities, $g_1$ and $g_2$, within the context of corrupted data. Utilizing this criterion, we propose an algorithm to construct a density estimator within a star-shaped density class, $\mathcal{F}$, under conditions of data corruption. We proceed to derive the minimax upper and lower bounds for density estimation across this star-shaped density class, characterized by densities that are uniformly bounded above and below (in the sup norm), in the presence of adversarially corrupted data. Specifically, we assume that a fraction $\epsilon \leq \frac{1}{3}$ of the $N$ observations are arbitrarily corrupted. We obtain the minimax upper bound $\max\{ \tau_{\overline{J}}^2, \epsilon \} \wedge d^2$. Under certain conditions, we obtain the minimax risk, up to proportionality constants, under the squared $L_2$ loss as $$ \max\left\{ \tau^{*2} \wedge d^2, \epsilon \wedge d^2 \right\}, $$ where $\tau^* := \sup\left\{ \tau : N\tau^2 \leq \log \mathcal{M}_{\mathcal{F}}^{\text{loc}}(\tau, c) \right\}$ for a sufficiently large constant $c$. Here, $\mathcal{M}_{\mathcal{F}}^{\text{loc}}(\tau, c)$ denotes the local entropy of the set $\mathcal{F}$, and $d$ is the $L_2$ diameter of $\mathcal{F}$.