This paper considers an empirical risk minimization problem under heavy-tailed settings, where data does not have finite variance, but only has $p$-th moment with $p \in (1,2)$. Instead of using estimation procedure based on truncated observed data, we choose the optimizer by minimizing the risk value. Those risk values can be robustly estimated via using the remarkable Catoni's method (Catoni, 2012). Thanks to the structure of Catoni-type influence functions, we are able to establish excess risk upper bounds via using generalized generic chaining methods. Moreover, we take computational issues into consideration. We especially theoretically investigate two types of optimization methods, robust gradient descent algorithm and empirical risk-based methods. With an extensive numerical study, we find that the optimizer based on empirical risks via Catoni-style estimation indeed shows better performance than other baselines. It indicates that estimation directly based on truncated data may lead to unsatisfactory results.

翻译：本文研究了重尾设置下的经验风险最小化问题，其中数据不具有有限方差，仅具有$p$阶矩（$p \in (1,2)$）。我们未采用基于截断观测数据的估计方法，而是通过最小化风险值来选择优化器。这些风险值可通过著名的Catoni方法（Catoni, 2012）进行稳健估计。得益于Catoni型影响函数的结构，我们能够利用广义泛化链式方法建立超额风险的上界。此外，我们考虑了计算问题，并特别从理论上研究了两种优化方法：鲁棒梯度下降算法和基于经验风险的方法。通过广泛的数值研究，我们发现基于Catoni型估计的经验风险优化器确实表现出优于其他基线的性能，这表明直接基于截断数据的估计可能导致不理想的结果。