The field of image reconstruction and inverse problems in imaging have been revolutionized by the introduction of methods which learn to solve inverse problems. This thesis investigates a variety of methods for learning to solve inverse problems by leveraging data: first by exploring the online sparse linear bandit setting, and then by investigating modern methods for leveraging training data to learn to solve inverse problems. In addition, this thesis explores a multi-model method of leveraging human descriptions of change in time series of images to regularize a graph-cut-based change-point detection method. Recent research into learning to solve inverse problems has been dominated by "unrolled optimization" approaches, which unroll a fixed number of iterations of an iterative optimization algorithm, replacing one or more elements of that algorithm with a neural network. These methods have several attractive properties: they can leverage even limited training data to learn accurate reconstructions, they tend to have lower runtime and require fewer iterations than more standard methods which leverage non-learned regularizers, and they are simple to implement and understand. However, learned iterative methods, like most learned inverse problem solvers, are sensitive to small changes in the data measurement model; they are uninterpretable, suffering reduced reconstruction quality if run for more or fewer iterations than were used at train time; and they are limited by memory and numerical constraints to small numbers of iterations, potentially lowering the ceiling for best available reconstruction quality using these methods. This thesis proposes an alternative architecture design based on a Neumann series, which is attractive from a practical perspective for its sample complexity performance and ease to train compared to methods based on unrolled iterative optimization. In addition, this thesis proposes and tests two techniques to adapt arbitrary trained inverse problem solvers to different measurement models, enabling deployment of a single learned model on a variety of forward models without sacrificing performance or requiring potentially-costly new data. Finally, this thesis demonstrates how to train iterative solvers that are unrolled for an arbitrary number of iterations. The proposed technique for the first time permits deep iterative solvers that admit practical convergence guarantees, while allowing flexibility in trading off computation for performance.