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 a superquantile method. 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.

中文翻译：

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机器学习中的倾斜损失：理论与应用

指数倾斜是一种常用于统计、概率、信息论和优化等领域的技术，以创建参数分布偏移。尽管倾斜在相关领域很流行，但在机器学习中并没有广泛使用。在这项工作中，我们旨在通过探索在风险最小化中使用倾斜来弥合这一差距。我们研究了 ERM 的一个简单扩展——倾斜经验风险最小化 (TERM)——它使用指数倾斜来灵活调整个人损失的影响。由此产生的框架有几个有用的特性：我们表明 TERM 可以分别增加或减少异常值的影响，以实现公平性或稳健性；具有有助于泛化的方差减少特性；并且可以看作是超分位数方法的平滑近似。我们的工作在 TERM 和相关目标之间建立了严格的联系，例如风险价值、条件风险价值和分布稳健优化 (DRO)。我们开发了用于求解 TERM 的批量和随机一阶优化方法，为求解器提供收敛保证，并表明相对于常见替代方案，该框架可以有效求解。最后，我们证明了 TERM 可用于机器学习中的多种应用，例如加强子组之间的公平性、减轻异常值的影响以及处理类不平衡。尽管 TERM 对传统 ERM 目标进行了直接修改，但我们发现该框架可以始终优于 ERM，并通过最先进的、针对特定问题的方法提供具有竞争力的性能。