  We consider the problem of helping agents improve by setting short-term
goals. Given a set of target skill levels, we assume each agent will try to
improve from their initial skill level to the closest target level within reach
or do nothing if no target level is within reach. We consider two models: the
common improvement capacity model, where agents have the same limit on how much
they can improve, and the individualized improvement capacity model, where
agents have individualized limits. Our goal is to optimize the target levels
for social welfare and fairness objectives, where social welfare is defined as
the total amount of improvement, and fairness objectives are considered where
the agents belong to different underlying populations. A key technical
challenge of this problem is the non-monotonicity of social welfare in the set
of target levels, i.e., adding a new target level may decrease the total amount
of improvement as it may get easier for some agents to improve. This is
especially challenging when considering multiple groups because optimizing
target levels in isolation for each group and outputting the union may result
in arbitrarily low improvement for a group, failing the fairness objective.
Considering these properties, we provide algorithms for optimal and
near-optimal improvement for both social welfare and fairness objectives. These
algorithmic results work for both the common and individualized improvement
capacity models. Furthermore, we show a placement of target levels exists that
is approximately optimal for the social welfare of each group. Unlike the
algorithmic results, this structural statement only holds in the common
improvement capacity model, and we show counterexamples in the individualized
improvement capacity model. Finally, we extend our algorithms to learning
settings where we have only sample access to the initial skill levels of
agents.
