Title: Benefits of Online Tilted Empirical Risk Minimization: A Case Study of Outlier Detection and Robust Regression Show Chinese title

Title: 在线倾斜经验风险最小化的优点：异常值检测与稳健回归的案例研究

Abstract: Empirical Risk Minimization (ERM) is a foundational framework for supervised learning but primarily optimizes average-case performance, often neglecting fairness and robustness considerations. Tilted Empirical Risk Minimization (TERM) extends ERM by introducing an exponential tilt hyperparameter $t$ to balance average-case accuracy with worst-case fairness and robustness. However, in online or streaming settings where data arrive one sample at a time, the classical TERM objective degenerates to standard ERM, losing tilt sensitivity. We address this limitation by proposing an online TERM formulation that removes the logarithm from the classical objective, preserving tilt effects without additional computational or memory overhead. This formulation enables a continuous trade-off controlled by $t$, smoothly interpolating between ERM ($t \to 0$), fairness emphasis ($t > 0$), and robustness to outliers ($t < 0$). We empirically validate online TERM on two representative streaming tasks: robust linear regression with adversarial outliers and minority-class detection in binary classification. Our results demonstrate that negative tilting effectively suppresses outlier influence, while positive tilting improves recall with minimal impact on precision, all at per-sample computational cost equivalent to ERM. Online TERM thus recovers the full robustness-fairness spectrum of classical TERM in an efficient single-sample learning regime. Show Chinese abstract

Abstract: 经验风险最小化（ERM）是监督学习的基础框架，但主要优化平均情况性能，通常忽视公平性和鲁棒性考虑。倾斜经验风险最小化（TERM）通过引入一个指数倾斜超参数$t$来平衡平均情况准确性与最坏情况下的公平性和鲁棒性。然而，在在线或流式设置中，数据逐个样本到达时，经典的 TERM 目标退化为标准 ERM，失去倾斜敏感性。我们通过提出一种在线 TERM 公式来解决这一限制，该公式从经典目标中移除了对数，无需额外的计算或内存开销即可保留倾斜效果。这种公式使得由$t$控制的连续权衡成为可能，平滑地在 ERM（$t \to 0$）、公平性强调（$t > 0$）和对异常值的鲁棒性（$t < 0$）之间进行插值。我们在两个典型的流任务上对在线 TERM 进行了实证验证：具有对抗性异常值的鲁棒线性回归和二分类中的少数类检测。我们的结果表明，负向倾斜有效抑制了异常值的影响，而正向倾斜在对精度影响最小的情况下提高了召回率，所有这些都具有与 ERM 相同的每个样本计算成本。因此，在线 TERM 在高效的单样本学习制度中恢复了经典 TERM 的完整鲁棒性-公平性谱。