  Exponential tilting is a technique commonly used in fields such as
statistics, probability, information theory, and optimization to create
parametric distribution shifts. Despite its prevalence in related fields,
tilting has not seen widespread use in machine learning. In this work, we aim
to bridge this gap by exploring the use of tilting in risk minimization. We
study a simple extension to ERM -- tilted empirical risk minimization (TERM) --
which uses exponential tilting to flexibly tune the impact of individual
losses. The resulting framework has several useful properties: We show that
TERM can increase or decrease the influence of outliers, respectively, to
enable fairness or robustness; has variance-reduction properties that can
benefit generalization; and can be viewed as a smooth approximation to the tail
probability of losses. Our work makes rigorous connections between TERM and
related objectives, such as Value-at-Risk, Conditional Value-at-Risk, and
distributionally robust optimization (DRO). We develop batch and stochastic
first-order optimization methods for solving TERM, provide convergence
guarantees for the solvers, and show that the framework can be efficiently
solved relative to common alternatives. Finally, we demonstrate that TERM can
be used for a multitude of applications in machine learning, such as enforcing
fairness between subgroups, mitigating the effect of outliers, and handling
class imbalance. Despite the straightforward modification TERM makes to
traditional ERM objectives, we find that the framework can consistently
outperform ERM and deliver competitive performance with state-of-the-art,
problem-specific approaches.
