Optimal e-variables under constraints

TL;DR

This paper develops a method to construct optimal e-variables under various structural constraints by transforming unconstrained solutions, simplifying constrained inference problems.

Contribution

It introduces an 'optimize-then-constrain' approach that allows deriving constrained log-optimal e-variables from unconstrained solutions through post-processing.

Findings

Constrained optimal e-variables can be obtained by transforming unconstrained solutions.

The approach simplifies the process of imposing constraints like privacy or boundedness.

The method applies to a range of structural constraints in inference.

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.