We propose data thinning, an approach for splitting an observation into two or more independent parts that sum to the original observation, and that follow the same distribution as the original observation, up to a (known) scaling of a parameter. This very general proposal is applicable to any convolution-closed distribution, a class that includes the Gaussian, Poisson, negative binomial, gamma, and binomial distributions, among others. Data thinning has a number of applications to model selection, evaluation, and inference. For instance, cross-validation via data thinning provides an attractive alternative to the usual approach of cross-validation via sample splitting, especially in settings in which the latter is not applicable. In simulations and in an application to single-cell RNA-sequencing data, we show that data thinning can be used to validate the results of unsupervised learning approaches, such as k-means clustering and principal components analysis, for which traditional sample splitting is unattractive or unavailable.

Existing gradient-based optimization methods for hyperparameter tuning can only guarantee theoretical convergence to stationary solutions when the bilevel program satisfies the condition that for fixed upper-level variables, the lower-level is strongly convex (LLSC) and smooth (LLS). This condition is not satisfied for bilevel programs arising from tuning hyperparameters in many machine learning algorithms. In this work, we develop a sequentially convergent Value Function based Difference-of-Convex Algorithm with inexactness (VF-iDCA). We then ask: can this algorithm achieve stationary solutions without LLSC and LLS assumptions? We provide a positive answer to this question for bilevel programs from a broad class of hyperparameter tuning applications. Extensive experiments justify our theoretical results and demonstrate the superiority of the proposed VF-iDCA when applied to tune hyperparameters.

We consider conducting inference on the output of the Classification and Regression Tree (CART) (Breiman et al., 1984) algorithm. A naive approach to inference that does not account for the fact that the tree was estimated from the data will not achieve standard guarantees, such as Type 1 error rate control and nominal coverage. Thus, we propose a selective inference framework for conducting inference on a fitted CART tree. In a nutshell, we condition on the fact that the tree was estimated from the data. We propose a test for the difference in the mean response between a pair of terminal nodes that controls the selective Type 1 error rate, and a confidence interval for the mean response within a single terminal node that attains the nominal selective coverage. Efficient algorithms for computing the necessary conditioning sets are provided. We apply these methods in simulation and to a dataset involving the association between portion control interventions and caloric intake.