The past years have seen seen the development and deployment of machine-learning algorithms to estimate personalized treatment-assignment policies from randomized controlled trials. Yet such algorithms for the assignment of treatment typically optimize expected outcomes without taking into account that treatment assignments are frequently subject to hypothesis testing. In this article, we explicitly take significance testing of the effect of treatment-assignment policies into account, and consider assignments that optimize the probability of finding a subset of individuals with a statistically significant positive treatment effect. We provide an efficient implementation using decision trees, and demonstrate its gain over selecting subsets based on positive (estimated) treatment effects. Compared to standard tree-based regression and classification tools, this approach tends to yield substantially higher power in detecting subgroups with positive treatment effects. INTRODUCTION Recent years have seen the development of machine-learning algorithms that estimate heterogeneous causal effects from randomized controlled trials. While the estimation of average effects - for example, how effective a vaccine is overall, whether a conditional cash transfer reduces poverty, or which ad leads to more clicks - can inform the decision whether to deploy a treatment or not, heterogeneous treatment effect estimation allows us to decide who should get treated. These algorithms aim to maximize realized outcomes, and thus focus on assigning treatment to individuals with positive (estimated) treatment effects. Yet in practice, the deployment of assignment policies often only happens after passing a test that the assignment produces a positive net effect relative to some status quo. For example, a drug manufacturer may have to demonstrate that the drug is effective on the target population by submitting a hypothesis test to the FDA for approval.

Phase retrieval is the inverse problem of recovering a signal from magnitude-only Fourier measurements, and underlies numerous imaging modalities, such as Coherent Diffraction Imaging (CDI). A variant of this setup, known as holography, includes a reference object that is placed adjacent to the specimen of interest before measurements are collected. The resulting inverse problem, known as holographic phase retrieval, is well-known to have improved problem conditioning relative to the original. This innovation, i.e. Holographic CDI, becomes crucial at the nanoscale, where imaging specimens such as viruses, proteins, and crystals require low-photon measurements. This data is highly corrupted by Poisson shot noise, and often lacks low-frequency content as well. In this work, we introduce a dataset-free deep learning framework for holographic phase retrieval adapted to these challenges. The key ingredients of our approach are the explicit and flexible incorporation of the physical forward model into an automatic differentiation procedure, the Poisson log-likelihood objective function, and an optional untrained deep image prior. We perform extensive evaluation under realistic conditions. Compared to competing classical methods, our method recovers signal from higher noise levels and is more resilient to suboptimal reference design, as well as to large missing regions of low frequencies in the observations. To the best of our knowledge, this is the first work to consider a dataset-free machine learning approach for holographic phase retrieval.