  Explanations shed light on a machine learning model's rationales and can aid
in identifying deficiencies in its reasoning process. Explanation generation
models are typically trained in a supervised way given human explanations. When
such annotations are not available, explanations are often selected as those
portions of the input that maximise a downstream task's performance, which
corresponds to optimising an explanation's Faithfulness to a given model.
Faithfulness is one of several so-called diagnostic properties, which prior
work has identified as useful for gauging the quality of an explanation without
requiring annotations. Other diagnostic properties are Data Consistency, which
measures how similar explanations are for similar input instances, and
Confidence Indication, which shows whether the explanation reflects the
confidence of the model. In this work, we show how to directly optimise for
these diagnostic properties when training a model to generate sentence-level
explanations, which markedly improves explanation quality, agreement with human
rationales, and downstream task performance on three complex reasoning tasks.
