Adaptive Experimental Design Using Shrinkage Estimators

Abstract

In multi-armed trials, adaptive designs are a popular way to increase estimation efficiency or identify optimal treatments. Several recent papers have proposed adaptive variants of the classical Neyman allocation to assign treatments in sequential trials, with the goal of minimizing the error of a Horvitz-Thompson-style estimator. However, this approach may be inefficient, because it fails to borrow information across the treatment arms. In this paper, we consider adaptivity in a sequential trial with K active treatments and a control, and suggest the use of Stein-like shrinkage estimators to obtain the final causal estimates. These estimators share information across arms, yielding provable reductions in expected squared error loss relative to estimating each causal effect in isolation. Moreover, for each of our candidate shrinkers, the risk is the expectation of ratios of Gaussian quadratic forms, and can be computed efficiently via numerical integration. Hence, we suggest a simple algorithm for sequential adaptivity: assign treatments to each new arrival by choosing the arm that will minimize the estimated shrinker loss. Through simulations, we demonstrate that this approach can yield meaningful reductions in estimation error, especially in the low signal-to-noise regime. We also characterize how our adaptive algorithm assigns treatments differently than would a sequential Neyman allocation, and suggest a method for constructing shorter confidence intervals at the trial's conclusion.