Optimal e-variables under constraints

2026-04-23

AI Summary

This work addresses the challenge of constructing log-optimal e-variables under structural constraints—such as differential privacy, quantization, boundedness, or moment restrictions—in composite hypothesis testing. We propose a “optimize-then-constrain” principle: first compute the unconstrained log-optimal e-variable, then apply a suitable transformation to enforce the desired constraint, thereby circumventing the need to recompute the least favorable distribution pair under restrictions. This approach reveals that constrained optimal e-variables can be obtained via post-processing of their unconstrained counterparts, substantially simplifying growth-rate optimization under constraints. Leveraging likelihood ratios, least favorable distributions, and information-theoretic tools, we efficiently construct log-optimal e-variables that maintain validity and safety at all times across a range of common constraints, significantly reducing computational complexity.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

E-variables enable safe and anytime-valid inference, with log-optimal e-variables given by the likelihood ratio of the least favorable distributions (LFDs) when they exist in composite settings. While this unconstrained theory is well understood, one may need/wish to impose additional structural constraints, including differential privacy, quantization, boundedness, or moment restrictions. We show that under these constraints, log-optimal constrained e-variables can often be constructed by a simple \emph{optimize-then-constrain} principle: first compute the unconstrained log-optimal e-variable, then impose the constraint via an appropriate transformation. Thus, the constrained growth-rate optimization problem does not require solving for a different LFD pair; the constrained optimal solution is just a post-processing of the unconstrained optimal solution.

Problem

• e-variables
• constraints
• log-optimal
• likelihood ratio
• structural constraints

Innovation

• e-variables
• log-optimal
• least favorable distributions
• optimize-then-constrain
• structural constraints