Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Online monotone density estimation and log-optimal calibration

Rohan Hore, Ruodu Wang, Aaditya Ramdas

TL;DR

The paper develops online methods for monotone density estimation on [0,1], introducing an online Grenander estimator and an expert-aggregation estimator. It proves an 𝒪(n^{1/3}) excess KL-risk bound for the online methods in well-specified settings and a pathwise 𝒪(√(n log n)) regret bound for EA relative to the offline Grenander, under mild regularity. A key application is constructing log-optimal calibrators for sequential hypothesis testing by recasting p-to-e calibration as online monotone density estimation, yielding empirically adaptive calibrators that are asymptotically log-optimal. The work combines rigorous risk/regret analysis with practical sequential testing tools, and includes simulations demonstrating finite-sample performance and adaptivity of the estimators.

Abstract

We study the problem of online monotone density estimation, where density estimators must be constructed in a predictable manner from sequentially observed data. We propose two online estimators: an online analogue of the classical Grenander estimator, and an expert aggregation estimator inspired by exponential weighting methods from the online learning literature. In the well-specified stochastic setting, where the underlying density is monotone, we show that the expected cumulative log-likelihood gap between the online estimators and the true density admits an $O(n^{1/3})$ bound. We further establish a $\sqrt{n\log{n}}$ pathwise regret bound for the expert aggregation estimator relative to the best offline monotone estimator chosen in hindsight, under minimal regularity assumptions on the observed sequence. As an application of independent interest, we show that the problem of constructing log-optimal p-to-e calibrators for sequential hypothesis testing can be formulated as an online monotone density estimation problem. We adapt the proposed estimators to build empirically adaptive p-to-e calibrators and establish their optimality. Numerical experiments illustrate the theoretical results.

Online monotone density estimation and log-optimal calibration

TL;DR

The paper develops online methods for monotone density estimation on [0,1], introducing an online Grenander estimator and an expert-aggregation estimator. It proves an 𝒪(n^{1/3}) excess KL-risk bound for the online methods in well-specified settings and a pathwise 𝒪(√(n log n)) regret bound for EA relative to the offline Grenander, under mild regularity. A key application is constructing log-optimal calibrators for sequential hypothesis testing by recasting p-to-e calibration as online monotone density estimation, yielding empirically adaptive calibrators that are asymptotically log-optimal. The work combines rigorous risk/regret analysis with practical sequential testing tools, and includes simulations demonstrating finite-sample performance and adaptivity of the estimators.

Abstract

We study the problem of online monotone density estimation, where density estimators must be constructed in a predictable manner from sequentially observed data. We propose two online estimators: an online analogue of the classical Grenander estimator, and an expert aggregation estimator inspired by exponential weighting methods from the online learning literature. In the well-specified stochastic setting, where the underlying density is monotone, we show that the expected cumulative log-likelihood gap between the online estimators and the true density admits an bound. We further establish a pathwise regret bound for the expert aggregation estimator relative to the best offline monotone estimator chosen in hindsight, under minimal regularity assumptions on the observed sequence. As an application of independent interest, we show that the problem of constructing log-optimal p-to-e calibrators for sequential hypothesis testing can be formulated as an online monotone density estimation problem. We adapt the proposed estimators to build empirically adaptive p-to-e calibrators and establish their optimality. Numerical experiments illustrate the theoretical results.
Paper Structure (35 sections, 12 theorems, 133 equations, 1 figure)

This paper contains 35 sections, 12 theorems, 133 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Risk and regret for online monotone density estimation
  3. Stochastic well-specified setting.
  4. Adversarial setting.
  5. The Grenander estimator and its properties
  6. Our contributions
  7. Online monotone density estimation.
  8. Application to sequential testing.
  9. Organization of the paper.
  10. Online monotone density estimation
  11. Online Grenander algorithm.
  12. Expert aggregation algorithm.
  13. Risk bounds for online monotone density estimation
  14. Bounding regret for the EA algorithm
  15. Proof sketch for the pathwise regret bound (Theorem \ref{['thm:pathwise_regret_monotone_density']})
  16. ...and 20 more sections

Key Result

Lemma 2.1

Fix $n\in\mathbb N$. Suppose that there exist $0<a<b<\infty$ such that $q\in\mathcal{D}_{a,b}$. Then, the constrained Grenander estimator $\hat{f}^{\mathrm{MLE}}_{n,a,b}\equiv \hat{f}^{\mathrm{MLE}}_{n,a,b}(X_1,\ldots,X_n)$ satisfies ${\mathbb{E}}_{\mathbb{Q}}\bigl[\mathrm{KL}\bigl(q\|\hat{f}^{\math

Figures (1)

  • Figure 1: Cumulative log-likelihood trajectories in well-specified and mis-specified stochastic setting---both OG and EA closely track the trajectory of true density $q$, with EA exhibiting smaller finite-sample regret. In the mis-specified setting with a structural change (rightmost), EA adapts more rapidly to the evolving density, while OG lags due to its reliance on the full history.

Theorems & Definitions (17)

  • Lemma 2.1
  • Theorem 2.2: Risk bounds for OG and EA algorithms
  • Theorem 2.3
  • Corollary 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Definition 1: Log-optimal decreasing calibrator
  • Theorem 3.1: Asymptotic log-optimality
  • Corollary 3.2: Almost sure divergence of log-wealth
  • Lemma B.1: Histogram form of the constrained Grenander
  • ...and 7 more