Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Between Resolution Collapse and Variance Inflation: Weighted Conformal Anomaly Detection in Low-Data Regimes

Oliver Hennhöfer, Christine Preisach

Abstract

Standard conformal anomaly detection provides marginal finite-sample guarantees under the assumption of exchangeability . However, real-world data often exhibit distribution shifts, necessitating a weighted conformal approach to adapt to local non-stationarity. We show that this adaptation induces a critical trade-off between the minimum attainable p-value and its stability. As importance weights localize to relevant calibration instances, the effective sample size decreases. This can render standard conformal p-values overly conservative for effective error control, while the smoothing technique used to mitigate this issue introduces conditional variance, potentially masking anomalies. We propose a continuous inference relaxation that resolves this dilemma by decoupling local adaptation from tail resolution via continuous weighted kernel density estimation. While relaxing finite-sample exactness to asymptotic validity, our method eliminates Monte Carlo variability and recovers the statistical power lost to discretization. Empirical evaluations confirm that our approach not only restores detection capabilities where discrete baselines yield zero discoveries, but outperforms standard methods in statistical power while maintaining valid marginal error control in practice.

Between Resolution Collapse and Variance Inflation: Weighted Conformal Anomaly Detection in Low-Data Regimes

Abstract

Standard conformal anomaly detection provides marginal finite-sample guarantees under the assumption of exchangeability . However, real-world data often exhibit distribution shifts, necessitating a weighted conformal approach to adapt to local non-stationarity. We show that this adaptation induces a critical trade-off between the minimum attainable p-value and its stability. As importance weights localize to relevant calibration instances, the effective sample size decreases. This can render standard conformal p-values overly conservative for effective error control, while the smoothing technique used to mitigate this issue introduces conditional variance, potentially masking anomalies. We propose a continuous inference relaxation that resolves this dilemma by decoupling local adaptation from tail resolution via continuous weighted kernel density estimation. While relaxing finite-sample exactness to asymptotic validity, our method eliminates Monte Carlo variability and recovers the statistical power lost to discretization. Empirical evaluations confirm that our approach not only restores detection capabilities where discrete baselines yield zero discoveries, but outperforms standard methods in statistical power while maintaining valid marginal error control in practice.
Paper Structure (46 sections, 3 theorems, 36 equations, 2 figures, 7 tables)

This paper contains 46 sections, 3 theorems, 36 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Prior Works
  3. Preliminaries
  4. Weighted Conformal Anomaly Detection
  5. 1. The Deterministic Estimator.
  6. 2. The Randomized Estimator.
  7. False Discovery Rate Control
  8. Multiple Testing.
  9. The Dilemma
  10. Failure Mode A: Resolution Collapse (Discretization).
  11. Failure Mode B: Variance Inflation (Randomization).
  12. Effective sample size
  13. Continuous Weighted Conformal Inference
  14. Covariate Shift Adaptation
  15. Weighted Kernel Density Estimation
  16. ...and 31 more sections

Key Result

Theorem 1

Assume (i) $F_0^w$ is continuous; (ii) $\sup_z|\hat{w}(z)-w(z)|\xrightarrow{p}0$ and standard regularity conditions ensuring weighted KDE consistency; (iii) $K$ is bounded with $\int K=1$, the bandwidth $h_N\to0$, and $N h_N/\log N\to\infty$. Let $Z\sim Q_0$ be independent of the calibration data us Moreover, for each fixed $u\in[0,1]$, i.e., $\hat{p}$ is asymptotically (marginally) super-uniform

Figures (2)

  • Figure 1: The Resolution--Variance Dilemma. Under distribution shift, high importance weights create large steps in the conservative estimator, imposing a resolution floor of minimum attainable $p$-values that prevents rejection even for extreme scores. Standard randomization resolves the floor but introduces variance inflation via noise that can mask the signal. The proposed Weighted KDE decouples resolution from sample size, enabling rejection below the floor without stochastic noise.
  • Figure 2: Statistical power of weighted and randomized conformal methods with WCS pruning strategies, including the weighted KDE-based approach. Labels refer to the WCS pruning method. Results for Musk are omitted due to ceiling performance across all strategies. Error bars denote mean $\pm$ standard error over 20 randomized trials.

Theorems & Definitions (5)

  • Theorem 1: Consistency and asymptotic marginal validity of KDE $p$-values
  • Proposition 1: Resolution Collapse through Discreteness
  • Proposition 2: Conditional Variance through Randomness
  • proof
  • proof : Proof sketch of Theorem \ref{['thm:asymp_valid']}