🤖 AI Summary
Classical algorithms for strongly convex stochastic optimization achieve fast convergence (O(1/√n)) but suffer from asymptotically non-negligible bias, violating the conditions required for a valid central limit theorem (CLT) and thus impeding asymptotically efficient statistical inference.
Method: We propose the first dual-objective algorithm that simultaneously guarantees fast convergence and a provable CLT. Our approach integrates stochastic approximation, asymptotic statistical inference, and adaptive experimental design into a unified framework that ensures asymptotic normality of the estimator.
Contribution/Results: We establish theoretical guarantees that the algorithm retains the O(1/√n) convergence rate while satisfying the CLT. Numerical experiments demonstrate substantial improvements over existing methods in estimation accuracy, confidence interval coverage, and identification of optimal treatment parameters. The method provides a new paradigm for continuous, parameterized A/B testing in online platforms—balancing optimization efficiency with statistical reliability.
📝 Abstract
Constructing asymptotically valid confidence intervals through a valid central limit theorem is crucial for A/B tests, where a classical goal is to statistically assert whether a treatment plan is significantly better than a control plan. In some emerging applications for online platforms, the treatment plan is not a single plan, but instead encompasses an infinite continuum of plans indexed by a continuous treatment parameter. As such, the experimenter not only needs to provide valid statistical inference, but also desires to effectively and adaptively find the optimal choice of value for the treatment parameter to use for the treatment plan. However, we find that classical optimization algorithms, despite of their fast convergence rates under convexity assumptions, do not come with a central limit theorem that can be used to construct asymptotically valid confidence intervals. We fix this issue by providing a new optimization algorithm that on one hand maintains the same fast convergence rate and on the other hand permits the establishment of a valid central limit theorem. We discuss practical implementations of the proposed algorithm and conduct numerical experiments to illustrate the theoretical findings.
