Gaussian Sequence Model: Sample Complexities of Testing, Estimation and LFHT

2025-07-22

AI Summary

This paper investigates the interplay between goodness-of-fit testing and parameter estimation in the Gaussian sequence model over a convex compact parameter set Γ, and characterizes the statistical limits of likelihood-free hypothesis testing (LFHT) on ℓₚ-balls. Methodologically, it integrates tools from Gaussian analysis, orthogonal symmetry, quadratic convexity theory, and information-theoretic lower bound techniques. Its key contributions are twofold: First, it establishes, for the first time, a tight lower bound showing that the testing complexity is at least the square root of the estimation complexity—over orthogonally symmetric convex sets—thereby extending the testing–estimation equivalence to broader convexly constrained settings. Second, it fully characterizes the sample-size trade-off curve for LFHT on ℓₚ-balls, uncovering a novel statistical compensation mechanism between observed and simulated samples. The results demonstrate that quadratic convexity is essential for tightness of the lower bound, providing a theoretical benchmark for likelihood-free inference.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study the Gaussian sequence model, i.e. $X sim N(mathbfθ, I_infty)$, where $mathbfθ in Γsubset ell_2$ is assumed to be convex and compact. We show that goodness-of-fit testing sample complexity is lower bounded by the square-root of the estimation complexity, whenever $Γ$ is orthosymmetric. We show that the lower bound is tight when $Γ$ is also quadratically convex, thus significantly extending validity of the testing-estimation relationship from [GP24]. Using similar methods, we also completely characterize likelihood-free hypothesis testing (LFHT) complexity for $ell_p$-bodies, discovering new types of tradeoff between the numbers of simulation and observation samples.

Problem

• Characterize sample complexity in Gaussian sequence model
• Relate testing and estimation complexities for convex sets
• Discover tradeoffs in likelihood-free hypothesis testing

Innovation

• Gaussian sequence model for testing and estimation
• Lower bound tight for quadratically convex sets
• Characterize LFHT complexity for ell_p-bodies