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Accuracy estimation of neural networks by extreme value theory

Gero Junike, Marco Oesting

TL;DR

The paper addresses quantifying the tail of the neural network approximation error $\mathcal{E}=|f-\varphi|$ on compact domains. It adopts extreme value theory, showing exceedances above a high threshold $u$ follow a generalized Pareto distribution with scale $\sigma(u)$ and shape $\gamma$, and introduces a new estimator $\hat{\gamma}_{k,N}$ that remains negative with probability one, along with an estimator for the upper endpoint $x^{*}$. The authors provide plug-in expressions for tail quantities $P(\mathcal{E}>x)$ and $\mathbb{E}[\mathcal{E}-u\mid \mathcal{E}>u]$ using $\widehat{x^{*}}_{k,N}$ and $\hat{\gamma}_{k,N}$, and validate the approach on a financial pricing task (American put options), showing accurate tail probability and mean-excess estimates. The work offers a rigorous tail-risk assessment for neural network errors, enabling improved risk management for mispricing and other applications where large errors matter.

Abstract

Neural networks are able to approximate any continuous function on a compact set. However, it is not obvious how to quantify the error of the neural network, i.e., the remaining bias between the function and the neural network. Here, we propose the application of extreme value theory to quantify large values of the error, which are typically relevant in applications. The distribution of the error beyond some threshold is approximately generalized Pareto distributed. We provide a new estimator of the shape parameter of the Pareto distribution suitable to describe the error of neural networks. Numerical experiments are provided.

Accuracy estimation of neural networks by extreme value theory

TL;DR

The paper addresses quantifying the tail of the neural network approximation error on compact domains. It adopts extreme value theory, showing exceedances above a high threshold follow a generalized Pareto distribution with scale and shape , and introduces a new estimator that remains negative with probability one, along with an estimator for the upper endpoint . The authors provide plug-in expressions for tail quantities and using and , and validate the approach on a financial pricing task (American put options), showing accurate tail probability and mean-excess estimates. The work offers a rigorous tail-risk assessment for neural network errors, enabling improved risk management for mispricing and other applications where large errors matter.

Abstract

Neural networks are able to approximate any continuous function on a compact set. However, it is not obvious how to quantify the error of the neural network, i.e., the remaining bias between the function and the neural network. Here, we propose the application of extreme value theory to quantify large values of the error, which are typically relevant in applications. The distribution of the error beyond some threshold is approximately generalized Pareto distributed. We provide a new estimator of the shape parameter of the Pareto distribution suitable to describe the error of neural networks. Numerical experiments are provided.

Paper Structure

This paper contains 8 sections, 1 theorem, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Extreme value theory
  4. Numerical experiments
  5. Markov's Inequality
  6. Extreme value theory
  7. Conclusions
  8. Proof of Theorem \ref{['Thm1']}

Key Result

Theorem 1

Let $\gamma <0$ be the extreme value index of $\mathcal{E}$ and $x^*$ be the corresponding upper end point. Furthermore, let Then $\widetilde{\gamma}_{k,N}$ is negative with probability one and converges to $\gamma$ in probability for $N\to\infty$.

Figures (1)

  • Figure 1: Estimation of the probability of exceedance, $P(\mathcal{E}>x)$, by extreme value theory and Markov's inequality. We use the threshold $u=0.33$ U.S. cents by setting $k=270$.

Theorems & Definitions (2)

  • Theorem 1
  • proof