Title: Gradient Importance Learning for Incomplete Observations. Abstract: Though recent works have developed methods that can generate estimates (or imputations) of the missing entries in a dataset to facilitate downstream analysis, most depend on assumptions that may not align with real-world applications and could suffer from poor performance in subsequent tasks such as classification. This is particularly true if the data have large missingness rates or a small sample size. More importantly, the imputation error could be propagated into the prediction step that follows, which may constrain the capabilities of the prediction model. In this work, we introduce the gradient importance learning (GIL) method to train multilayer perceptrons (MLPs) and long short-term memories (LSTMs) to directly perform inference from inputs containing missing values without imputation. Specifically, we employ reinforcement learning (RL) to adjust the gradients used to train these models via back-propagation. This allows the model to exploit the underlying information behind missingness patterns. We test the approach on real-world time-series (i.e., MIMIC-III), tabular data obtained from an eye clinic, and a standard dataset (i.e., MNIST), where our imputation-free predictions outperform the traditional two-step imputation-based predictions using state-of-the-art imputation methods.